# Theory TopologicalGroup_ZF

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theory TopologicalGroup_ZF
imports Topology_ZF_3 Group_ZF_1 Semigroup_ZF
(*     This file is a part of IsarMathLib -     a library of formalized mathematics written for Isabelle/Isar.    Copyright (C) 2009-2013  Slawomir Kolodynski    This program is free software; Redistribution and use in source and binary forms,     with or without modification, are permitted provided that the following conditions are met:   1. Redistributions of source code must retain the above copyright notice,    this list of conditions and the following disclaimer.   2. Redistributions in binary form must reproduce the above copyright notice,    this list of conditions and the following disclaimer in the documentation and/or    other materials provided with the distribution.   3. The name of the author may not be used to endorse or promote products    derived from this software without specific prior written permission.THIS SOFTWARE IS PROVIDED BY THE AUTHOR AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES,INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR APARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT,INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOTLIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES LOSS OF USE, DATA, OR PROFITS ORBUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THEUSE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.*)header{*\isaheader{TopologicalGroup\_ZF.thy}*}theory TopologicalGroup_ZF imports Topology_ZF_3 Group_ZF_1 Semigroup_ZFbegintext{*This theory is about the first subject of algebraic topology:  topological groups.*}section{*Topological group: definition and notation*}text{*Topological group is a group that is a topological space   at the same time. This means that a topological group is a triple of sets,   say $(G,f,T)$ such that $T$ is a topology on $G$, $f$ is a group   operation on $G$ and both $f$ and the operation of taking inverse in $G$   are continuous. Since IsarMathLib defines topology without using the carrier,  (see @{text "Topology_ZF"}), in our setup we just use $\bigcup T$ instead  of $G$ and say that the  pair of sets $(\bigcup T , f)$ is a group.  This way our definition of being a topological group is a statement about two  sets: the topology $T$ and the group operation $f$ on $G=\bigcup T$.  Since the domain of the group operation is $G\times G$, the pair of   topologies in which $f$ is supposed to be continuous is $T$ and  the product topology on $G\times G$ (which we will call $\tau$ below).*}text{*This way we arrive at the following definition of a predicate that  states that pair of sets is a topological group.*}definition  "IsAtopologicalGroup(T,f) ≡ (T {is a topology}) ∧ IsAgroup(\<Union>T,f) ∧  IsContinuous(ProductTopology(T,T),T,f) ∧   IsContinuous(T,T,GroupInv(\<Union>T,f))"text{*We will inherit notation from the @{text "topology0"} locale. That locale assumes that  $T$ is a topology. For convenience we will denote $G=\bigcup T$ and $\tau$ to be   the product topology on $G\times G$. To that we add some  notation specific to groups. We will use additive notation  for the group operation, even though we don't assume that the group is abelian.  The notation $g+A$ will mean the left translation of the set $A$ by element $g$, i.e.  $g+A=\{g+a|a\in A\}$.  The group operation $G$ induces a natural operation on the subsets of $G$  defined as $\langle A,B\rangle \mapsto \{x+y | x\in A, y\in B \}$.  Such operation has been considered in @{text "func_ZF"} and called   $f$ ''lifted to subsets of'' $G$. We will denote the value of such operation   on sets $A,B$ as $A+B$.  The set of neigboorhoods of zero (denoted @{text "\<N>⇣0"}) is the   collection of (not necessarily open) sets whose interior contains the neutral  element of the group.*}locale topgroup = topology0 +  fixes G  defines G_def [simp]: "G ≡ \<Union>T"  fixes prodtop ("τ")  defines prodtop_def [simp]: "τ ≡ ProductTopology(T,T)"  fixes f  assumes Ggroup: "IsAgroup(G,f)"  assumes fcon: "IsContinuous(τ,T,f)"  assumes inv_cont: "IsContinuous(T,T,GroupInv(G,f))"  fixes grop (infixl "\<ra>" 90)  defines grop_def [simp]: "x\<ra>y ≡ f⟨x,y⟩"  fixes grinv ("\<rm> _" 89)  defines grinv_def [simp]: "(\<rm>x) ≡ GroupInv(G,f)(x)"  fixes grsub (infixl "\<rs>" 90)  defines grsub_def [simp]: "x\<rs>y ≡ x\<ra>(\<rm>y)"  fixes setinv ("\<sm> _" 72)  defines setninv_def [simp]: "\<sm>A ≡ GroupInv(G,f)(A)"  fixes ltrans (infix "\<ltr>" 73)  defines ltrans_def [simp]: "x \<ltr> A ≡ LeftTranslation(G,f,x)(A)"  fixes rtrans (infix "\<rtr>" 73)  defines rtrans_def [simp]: "A \<rtr> x ≡ RightTranslation(G,f,x)(A)"  fixes setadd (infixl "\<sad>" 71)  defines setadd_def [simp]: "A\<sad>B ≡ (f {lifted to subsets of} G)⟨A,B⟩"  fixes gzero ("\<zero>")  defines gzero_def [simp]: "\<zero> ≡ TheNeutralElement(G,f)"  fixes zerohoods ("\<N>⇣0")  defines zerohoods_def [simp]: "\<N>⇣0 ≡ {A ∈ Pow(G). \<zero> ∈ int(A)}"  fixes listsum ("∑ _" 70)  defines listsum_def[simp]: "∑k ≡ Fold1(f,k)"text{*The first lemma states that we indeeed talk about topological group  in the context of @{text "topgroup"} locale.*}lemma (in topgroup) topGroup: shows "IsAtopologicalGroup(T,f)"  using topSpaceAssum Ggroup fcon inv_cont IsAtopologicalGroup_def   by simp;text{* If a pair of sets $(T,f)$ forms a topological group, then all theorems proven in the @{text "topgroup"} context are valid as applied to  $(T,f)$.*}lemma topGroupLocale: assumes "IsAtopologicalGroup(T,f)"  shows "topgroup(T,f)"  using assms IsAtopologicalGroup_def topgroup_def     topgroup_axioms.intro topology0_def by simp;text{*We can use the @{text "group0"} locale in the context of @{text "topgroup"}.*}lemma (in topgroup) group0_valid_in_tgroup: shows "group0(G,f)"  using Ggroup group0_def by simptext{*We can use @{text "semigr0"} locale in the context of @{text "topgroup"}. *}lemma (in topgroup) semigr0_valid_in_tgroup: shows "semigr0(G,f)"  using Ggroup IsAgroup_def IsAmonoid_def semigr0_def by simp text{*We can use the @{text "prod_top_spaces0"} locale in the context of @{text "topgroup"}.*}lemma (in topgroup) prod_top_spaces0_valid: shows "prod_top_spaces0(T,T,T)"  using topSpaceAssum prod_top_spaces0_def by simptext{*Negative of a group element is in group.*}lemma (in topgroup) neg_in_tgroup: assumes "g∈G" shows "(\<rm>g) ∈ G"proof -  from assms have "GroupInv(G,f)(g) ∈ G"     using group0_valid_in_tgroup group0.inverse_in_group by blast  thus ?thesis by simpqedtext{*Zero is in the group.*}lemma (in topgroup) zero_in_tgroup: shows "\<zero>∈G"proof -  have "TheNeutralElement(G,f) ∈ G"     using group0_valid_in_tgroup group0.group0_2_L2 by blast  then show "\<zero>∈G" by simpqedtext{*Of course the product topology is a topology (on $G\times G$).*}lemma (in topgroup) prod_top_on_G:  shows "τ {is a topology}" and "\<Union>τ = G×G"  using topSpaceAssum Top_1_4_T1 by auto;text{*Let's recall that $f$ is a binary operation on $G$ in this context.*}lemma (in topgroup) topgroup_f_binop: shows "f : G×G -> G"  using Ggroup group0_def group0.group_oper_assocA by simp;text{*A subgroup of a topological group is a topological group   with relative topology  and restricted operation. Relative topology is the same  as @{text "T {restricted to} H"}  which is defined to be $\{V \cap H: V\in T\}$ in @{text "ZF1"} theory.*}lemma (in topgroup) top_subgroup: assumes A1: "IsAsubgroup(H,f)"  shows "IsAtopologicalGroup(T {restricted to} H,restrict(f,H×H))"proof -  let ?τ⇣0 = "T {restricted to} H"  let ?f⇣H = "restrict(f,H×H)"  have "\<Union>?τ⇣0 = G ∩ H" using union_restrict by simp;  also from A1 have "… = H"     using group0_valid_in_tgroup group0.group0_3_L2 by blast;  finally have "\<Union>?τ⇣0 = H" by simp;  have "?τ⇣0 {is a topology}" using Top_1_L4 by simp;  moreover from A1 \<Union>?τ⇣0 = H have "IsAgroup(\<Union>?τ⇣0,?f⇣H)"    using IsAsubgroup_def by simp;  moreover have "IsContinuous(ProductTopology(?τ⇣0,?τ⇣0),?τ⇣0,?f⇣H)"  proof -    have "two_top_spaces0(τ, T,f)"      using topSpaceAssum prod_top_on_G topgroup_f_binop prod_top_on_G	two_top_spaces0_def by simp;    moreover     from A1 have "H ⊆ G" using group0_valid_in_tgroup group0.group0_3_L2      by simp;    then have "H×H ⊆ \<Union>τ" using prod_top_on_G by auto;    moreover have "IsContinuous(τ,T,f)" using fcon by simp;    ultimately have       "IsContinuous(τ {restricted to} H×H, T {restricted to} ?f⇣H(H×H),?f⇣H)"      using two_top_spaces0.restr_restr_image_cont by simp;    moreover have      "ProductTopology(?τ⇣0,?τ⇣0) = τ {restricted to} H×H"      using topSpaceAssum prod_top_restr_comm by simp;    moreover from A1 have "?f⇣H(H×H) = H" using image_subgr_op      by simp;    ultimately show ?thesis by simp;  qed   moreover have "IsContinuous(?τ⇣0,?τ⇣0,GroupInv(\<Union>?τ⇣0,?f⇣H))"  proof -    let ?g = "restrict(GroupInv(G,f),H)"    have "GroupInv(G,f) : G -> G"      using Ggroup group0_2_T2 by simp;    then have "two_top_spaces0(T,T,GroupInv(G,f))"      using topSpaceAssum two_top_spaces0_def by simp;    moreover from A1 have "H ⊆ \<Union>T"       using group0_valid_in_tgroup group0.group0_3_L2      by simp;    ultimately have       "IsContinuous(?τ⇣0,T {restricted to} ?g(H),?g)"      using inv_cont two_top_spaces0.restr_restr_image_cont      by simp;    moreover from A1 have "?g(H) = H"      using group0_valid_in_tgroup group0.restr_inv_onto      by simp;      moreover    from A1 have "GroupInv(H,?f⇣H) = ?g"      using group0_valid_in_tgroup group0.group0_3_T1      by simp;    with \<Union>?τ⇣0 = H have "?g = GroupInv(\<Union>?τ⇣0,?f⇣H)" by simp;    ultimately show ?thesis by simp;  qed  ultimately show ?thesis unfolding IsAtopologicalGroup_def by simp;qedsection{*Interval arithmetic, translations and inverse of set*}text{*In this section we list some properties of operations of translating a  set and reflecting it around the neutral element of the group. Many of the results  are proven in other theories, here we just collect them and rewrite in notation  specific to the @{text "topgroup"} context.*}text{*Different ways of looking at adding sets.*}lemma (in topgroup) interval_add: assumes "A⊆G" "B⊆G" shows  "A\<sad>B ⊆ G" and "A\<sad>B = f(A×B)"  "A\<sad>B = (\<Union>x∈A. x\<ltr>B)"proof -  from assms show "A\<sad>B ⊆ G" and "A\<sad>B = f(A×B)"     using topgroup_f_binop lift_subsets_explained by auto  from assms show "A\<sad>B = (\<Union>x∈A. x\<ltr>B)"    using group0_valid_in_tgroup group0.image_ltrans_union by simpqedtext{*Right and left translations are continuous.*}lemma (in topgroup) trans_cont: assumes "g∈G" shows  "IsContinuous(T,T,RightTranslation(G,f,g))" and  "IsContinuous(T,T,LeftTranslation(G,f,g))"using assms group0_valid_in_tgroup group0.trans_eq_section  topgroup_f_binop fcon prod_top_spaces0_valid   prod_top_spaces0.fix_1st_var_cont prod_top_spaces0.fix_2nd_var_cont  by autotext{*Left and right translations of an open set are open.*}lemma (in topgroup) open_tr_open: assumes "g∈G" and "V∈T"  shows "g\<ltr>V ∈ T" and  "V\<rtr>g ∈ T"  using assms neg_in_tgroup trans_cont IsContinuous_def     group0_valid_in_tgroup group0.trans_image_vimage by autotext{*Right and left translations are homeomorphisms.*}lemma (in topgroup) tr_homeo: assumes "g∈G" shows  "IsAhomeomorphism(T,T,RightTranslation(G,f,g))" and  "IsAhomeomorphism(T,T,LeftTranslation(G,f,g))"  using assms group0_valid_in_tgroup group0.trans_bij trans_cont open_tr_open    bij_cont_open_homeo by autotext{*Translations preserve interior.*}lemma (in topgroup) trans_interior: assumes A1: "g∈G" and A2: "A⊆G"   shows "g \<ltr> int(A) = int(g\<ltr>A)"proof -  from assms have "A ⊆ \<Union>T" and "IsAhomeomorphism(T,T,LeftTranslation(G,f,g))"    using tr_homeo by auto  then show ?thesis using int_top_invariant by simpqed(*text{*Translating by an inverse and then by an element cancels out.*}lemma (in topgroup) trans_inverse_elem: assumes "g∈G" and "A⊆G"  shows "g\<ltr>((\<rm>g)\<ltr>A) = A"using assms neg_in_tgroup group0_valid_in_tgroup group0.trans_comp_image  group0.group0_2_L6 group0.trans_neutral image_id_same by simp*)text{*Inverse of an open set is open.*}lemma (in topgroup) open_inv_open: assumes "V∈T" shows "(\<sm>V) ∈ T"  using assms group0_valid_in_tgroup group0.inv_image_vimage    inv_cont IsContinuous_def by simptext{*Inverse is a homeomorphism.*}lemma (in topgroup) inv_homeo: shows "IsAhomeomorphism(T,T,GroupInv(G,f))"  using group0_valid_in_tgroup group0.group_inv_bij inv_cont open_inv_open  bij_cont_open_homeo by simptext{*Taking negative preserves interior.*}lemma (in topgroup) int_inv_inv_int: assumes "A ⊆ G"   shows "int(\<sm>A) = \<sm>(int(A))"  using assms inv_homeo int_top_invariant by simpsection{*Neighborhoods of zero*}text{*Zero neighborhoods are (not necessarily open) sets whose interior  contains the neutral element of the group. In the @{text "topgroup"} locale  the collection of neighboorhoods of zero is denoted @{text "\<N>⇣0"}. *}text{*The whole space is a neighborhood of zero.*}lemma (in topgroup) zneigh_not_empty: shows "G ∈ \<N>⇣0"  using topSpaceAssum IsATopology_def Top_2_L3 zero_in_tgroup  by simptext{*Any element belongs to the interior of any neighboorhood of zero  translated by that element.*}lemma (in topgroup) elem_in_int_trans:  assumes A1: "g∈G" and A2: "H ∈ \<N>⇣0"  shows "g ∈ int(g\<ltr>H)"proof -  from A2 have "\<zero> ∈ int(H)" and "int(H) ⊆ G" using Top_2_L2 by auto  with A1 have "g ∈ g \<ltr> int(H)"    using group0_valid_in_tgroup group0.neut_trans_elem by simp  with assms show ?thesis using trans_interior by simpqedtext{*Negative of a neighborhood of zero is a neighborhood of zero.*}lemma (in topgroup) neg_neigh_neigh: assumes "H ∈ \<N>⇣0"  shows "(\<sm>H) ∈ \<N>⇣0"proof -  from assms have "int(H) ⊆ G" and "\<zero> ∈ int(H)" using Top_2_L1 by auto  with assms have "\<zero> ∈ int(\<sm>H)" using group0_valid_in_tgroup group0.neut_inv_neut    int_inv_inv_int by simp  moreover  have "GroupInv(G,f):G->G" using Ggroup group0_2_T2 by simp  then have "(\<sm>H) ⊆ G" using func1_1_L6 by simp  ultimately show ?thesis by simpqedtext{*Translating an open set by a negative of a point that belongs to it  makes it a neighboorhood of zero.*}lemma (in topgroup) open_trans_neigh: assumes A1: "U∈T" and "g∈U"  shows "(\<rm>g)\<ltr>U ∈ \<N>⇣0"proof -  let ?H = "(\<rm>g)\<ltr>U"  from assms have "g∈G" by auto  then have "(\<rm>g) ∈ G" using neg_in_tgroup by simp  with A1 have "?H∈T" using open_tr_open by simp  hence "?H ⊆ G" by auto  moreover have "\<zero> ∈ int(?H)"  proof -    from assms have "U⊆G" and "g∈U" by auto    with ?H∈T show "\<zero> ∈ int(?H)"       using group0_valid_in_tgroup group0.elem_trans_neut Top_2_L3        by auto  qed  ultimately show ?thesis by simpqedsection{*Closure in topological groups*}text{*This section is devoted to a characterization of closure  in topological groups.*}text{*Closure of a set is contained in the sum of the set and any  neighboorhood of zero.*}lemma (in topgroup) cl_contains_zneigh:  assumes A1: "A⊆G" and A2: "H ∈ \<N>⇣0"  shows "cl(A) ⊆ A\<sad>H"proof  fix x assume "x ∈ cl(A)"  from A1 have "cl(A) ⊆ G" using Top_3_L11 by simp  with x ∈ cl(A) have "x∈G" by auto  have "int(H) ⊆ G" using Top_2_L2 by auto  let ?V = "int(x \<ltr> (\<sm>H))"  have "?V = x \<ltr> (\<sm>int(H))"  proof -    from A2 x∈G have "?V = x \<ltr> int(\<sm>H)"       using neg_neigh_neigh trans_interior by simp    with A2 show ?thesis  using int_inv_inv_int by simp  qed  have "A∩?V ≠ 0"  proof -    from A2 x∈G x ∈ cl(A) have "?V∈T" and "x ∈ cl(A) ∩ ?V"       using neg_neigh_neigh elem_in_int_trans Top_2_L2 by auto    with A1 show "A∩?V ≠ 0" using cl_inter_neigh by simp  qed  then obtain y where "y∈A" and "y∈?V" by auto  with ?V = x \<ltr> (\<sm>int(H)) int(H) ⊆ G x∈G have "x ∈ y\<ltr>int(H)"    using group0_valid_in_tgroup group0.ltrans_inv_in by simp  with y∈A have "x ∈ (\<Union>y∈A. y\<ltr>H)" using Top_2_L1 func1_1_L8 by auto  with assms show "x ∈ A\<sad>H" using interval_add by simpqedtext{*The next theorem provides a characterization of closure in topological  groups in terms of neighborhoods of zero.*}theorem (in topgroup) cl_topgroup:  assumes "A⊆G" shows "cl(A) = (\<Inter>H∈\<N>⇣0. A\<sad>H)"proof  from assms show "cl(A) ⊆ (\<Inter>H∈\<N>⇣0. A\<sad>H)"     using zneigh_not_empty cl_contains_zneigh by autonext  { fix x assume "x ∈ (\<Inter>H∈\<N>⇣0. A\<sad>H)"    then have "x ∈ A\<sad>G" using zneigh_not_empty by auto    with assms have "x∈G" using interval_add by blast    have "∀U∈T. x∈U --> U∩A ≠ 0"    proof -      { fix U assume "U∈T" and "x∈U"        let ?H = "\<sm>((\<rm>x)\<ltr>U)"        from U∈T and x∈U have "(\<rm>x)\<ltr>U ⊆ G" and "?H ∈ \<N>⇣0"           using open_trans_neigh neg_neigh_neigh by auto        with x ∈ (\<Inter>H∈\<N>⇣0. A\<sad>H) have "x ∈ A\<sad>?H" by auto        with assms ?H ∈ \<N>⇣0 obtain y where "y∈A" and "x ∈ y\<ltr>?H"          using interval_add by auto        have "y∈U"        proof -          from assms y∈A have "y∈G" by auto          with (\<rm>x)\<ltr>U ⊆ G and x ∈ y\<ltr>?H have "y ∈ x\<ltr>((\<rm>x)\<ltr>U)"            using group0_valid_in_tgroup group0.ltrans_inv_in by simp          with U∈T x∈G show "y∈U"             using neg_in_tgroup group0_valid_in_tgroup group0.trans_comp_image              group0.group0_2_L6 group0.trans_neutral image_id_same              by auto        qed        with y∈A have "U∩A ≠ 0" by auto      } thus ?thesis by simp    qed    with assms x∈G have "x ∈ cl(A)" using inter_neigh_cl by simp  } thus "(\<Inter>H∈\<N>⇣0. A\<sad>H) ⊆ cl(A)" by autoqedsection{*Sums of sequences of elements and subsets*}text{* In this section we consider properties of the function $G^n\rightarrow G$,   $x=(x_0,x_1,...,x_{n-1})\mapsto \sum_{i=0}^{n-1}x_i$. We will model the cartesian product  $G^n$ by the space of sequences $n\rightarrow G$, where $n=\{0,1,...,n-1]\}$ is a natural number.   This space is equipped with a natural product topology defined in @{text "Topology_ZF_3"}. *}text{*Let's recall first that the sum of elements of a group is an element of the group.*}lemma (in topgroup) sum_list_in_group:  assumes "n ∈ nat" and "x: succ(n)->G"  shows "(∑x) ∈ G"proof -  from assms have "semigr0(G,f)" and "n ∈ nat" "x: succ(n)->G"    using semigr0_valid_in_tgroup by auto  then have "Fold1(f,x) ∈ G" by (rule semigr0.prod_type)  thus "(∑x) ∈ G" by simpqedtext{*In this context @{text"x\<ra>y"} is the same as the value of the group operation  on the elements $x$ and $y$. Normally we shouldn't need to state this a s separate lemma.  *}lemma (in topgroup) grop_def1: shows "f⟨x,y⟩ = x\<ra>y" by simp text{*Another theorem from @{text "Semigroup_ZF"} theory that is useful to have in the  additive notation. *}lemma (in topgroup) shorter_set_add:  assumes "n ∈ nat" and "x: succ(succ(n))->G"  shows "(∑x) = (∑Init(x)) \<ra> (x(succ(n)))"proof -  from assms have "semigr0(G,f)" and "n ∈ nat" "x: succ(succ(n))->G"    using semigr0_valid_in_tgroup by auto  then have "Fold1(f,x) = f⟨Fold1(f,Init(x)),x(succ(n))⟩"    by (rule semigr0.shorter_seq)  thus ?thesis by simp   qedtext{*Sum is a continuous function in the product topology.*}theorem (in topgroup) sum_continuous: assumes "n ∈ nat"  shows "IsContinuous(SeqProductTopology(succ(n),T),T,{⟨x,∑x⟩.x∈succ(n)->G})"  proof -    note n ∈ nat    moreover have "IsContinuous(SeqProductTopology(succ(0),T),T,{⟨x,∑x⟩.x∈succ(0)->G})"    proof -      have "{⟨x,∑x⟩.x∈succ(0)->G} = {⟨x,x(0)⟩. x∈1->G}"        using semigr0_valid_in_tgroup semigr0.prod_of_1elem by simp      moreover have        "IsAhomeomorphism(SeqProductTopology(1,T),T,{⟨x,x(0)⟩. x∈1->\<Union>T})"        using topSpaceAssum singleton_prod_top1 by simp      ultimately show ?thesis using IsAhomeomorphism_def by simp    qed    moreover have "∀k∈nat.      IsContinuous(SeqProductTopology(succ(k),T),T,{⟨x,∑x⟩.x∈succ(k)->G})      -->      IsContinuous(SeqProductTopology(succ(succ(k)),T),T,{⟨x,∑x⟩.x∈succ(succ(k))->G})"      proof -        { fix k assume "k ∈ nat"          let ?s = "{⟨x,∑x⟩.x∈succ(k)->G}"          let ?g = "{⟨p,⟨?s(fst(p)),snd(p)⟩⟩. p ∈ (succ(k)->G)×G}"          let ?h = "{⟨x,⟨Init(x),x(succ(k))⟩⟩. x ∈ succ(succ(k))->G}"          let ?φ = "SeqProductTopology(succ(k),T)"          let ?ψ = "SeqProductTopology(succ(succ(k)),T)"          assume "IsContinuous(?φ,T,?s)"          from k ∈ nat have "?s: (succ(k)->G) -> G"            using sum_list_in_group ZF_fun_from_total by simp           have "?h: (succ(succ(k))->G)->(succ(k)->G)×G"          proof -            { fix x assume "x ∈ succ(succ(k))->G"              with k ∈ nat have "Init(x) ∈ (succ(k)->G)"                using init_props by simp              with k ∈ nat x : succ(succ(k))->G                 have "⟨Init(x),x(succ(k))⟩ ∈ (succ(k)->G)×G"                using apply_funtype by blast            } then show ?thesis using ZF_fun_from_total by simp          qed          moreover have "?g:((succ(k)->G)×G)->(G×G)"          proof -            { fix p assume "p ∈ (succ(k)->G)×G"              hence "fst(p): succ(k)->G" and "snd(p) ∈ G" by auto              with ?s: (succ(k)->G) -> G have "⟨?s(fst(p)),snd(p)⟩ ∈ G×G"                using apply_funtype by blast             } then show "?g:((succ(k)->G)×G)->(G×G)" using ZF_fun_from_total              by simp          qed          moreover have "f : G×G -> G" using topgroup_f_binop by simp          ultimately have "f O ?g O ?h :(succ(succ(k))->G)->G" using comp_fun            by blast           from k ∈ nat have "IsContinuous(?ψ,ProductTopology(?φ,T),?h)"            using topSpaceAssum finite_top_prod_homeo IsAhomeomorphism_def            by simp          moreover have "IsContinuous(ProductTopology(?φ,T),τ,?g)"          proof -            from topSpaceAssum have                "T {is a topology}" "?φ {is a topology}" "\<Union>?φ = succ(k)->G"               using seq_prod_top_is_top by auto            moreover from \<Union>?φ = succ(k)->G ?s: (succ(k)->G) -> G               have "?s:\<Union>?φ->\<Union>T" by simp             moreover note IsContinuous(?φ,T,?s)            moreover from \<Union>?φ = succ(k)->G               have "?g = {⟨p,⟨?s(fst(p)),snd(p)⟩⟩. p ∈ \<Union>?φ×\<Union>T}"              by simp            ultimately have "IsContinuous(ProductTopology(?φ,T),ProductTopology(T,T),?g)"              using cart_prod_cont1 by blast             thus ?thesis by simp          qed                   moreover have "IsContinuous(τ,T,f)" using fcon by simp          moreover have "{⟨x,∑x⟩.x∈succ(succ(k))->G} = f O ?g O ?h"          proof -            let ?d = "{⟨x,∑x⟩.x∈succ(succ(k))->G}"            from k∈nat have "∀x∈succ(succ(k))->G. (∑x) ∈ G"              using sum_list_in_group by blast             then have "?d:(succ(succ(k))->G)->G"               using sum_list_in_group ZF_fun_from_total by simp            moreover note f O ?g O ?h :(succ(succ(k))->G)->G            moreover have "∀x∈succ(succ(k))->G. ?d(x) = (f O ?g O ?h)(x)"            proof              fix x assume "x∈succ(succ(k))->G"              then have I: "?h(x) = ⟨Init(x),x(succ(k))⟩"                using ZF_fun_from_tot_val1 by simp              moreover from k∈nat x∈succ(succ(k))->G                 have "Init(x): succ(k)->G"                 using init_props by simp              moreover from k∈nat x:succ(succ(k))->G                 have II: "x(succ(k)) ∈ G"                using apply_funtype by blast              ultimately have "?h(x) ∈ (succ(k)->G)×G" by simp              then have "?g(?h(x)) = ⟨?s(fst(?h(x))),snd(?h(x))⟩"                using ZF_fun_from_tot_val1 by simp              with I have "?g(?h(x)) = ⟨?s(Init(x)),x(succ(k))⟩"                by simp              with Init(x): succ(k)->G have "?g(?h(x)) = ⟨∑Init(x),x(succ(k))⟩"                using ZF_fun_from_tot_val1 by simp              with k ∈ nat x: succ(succ(k))->G                 have "f(?g(?h(x))) = (∑x)"                using shorter_set_add by simp              with x ∈ succ(succ(k))->G have "f(?g(?h(x))) = ?d(x)"                using ZF_fun_from_tot_val1 by simp              moreover from                 ?h: (succ(succ(k))->G)->(succ(k)->G)×G                ?g:((succ(k)->G)×G)->(G×G)                f:(G×G)->G x∈succ(succ(k))->G                have "(f O ?g O ?h)(x) = f(?g(?h(x)))" by (rule func1_1_L18)              ultimately show "?d(x) = (f O ?g O ?h)(x)" by simp             qed            ultimately show "{⟨x,∑x⟩.x∈succ(succ(k))->G} = f O ?g O ?h"               using func_eq by simp          qed          moreover note IsContinuous(τ,T,f)          ultimately have "IsContinuous(?ψ,T,{⟨x,∑x⟩.x∈succ(succ(k))->G})"            using comp_cont3 by simp        } thus ?thesis by simp      qed    ultimately show ?thesis by (rule ind_on_nat)  qedend