reserve k,n,i,j for Nat;

theorem Th22:
  for G being Group,a being Element of G holds (<*a*>)" =<* a" *>
proof
  let G be Group,a be Element of G;
A1: len ((<*a*>)")=len (<* a *>) by Def3
    .=1 by FINSEQ_1:39
    .=len (<* a" *>) by FINSEQ_1:39;
  for i being Nat st 1<=i & i<= len <*a" *> holds ((<*a*>)").i=(<* a" *>). i
  proof
    let i be Nat;
    assume
A2: 1<=i & i<= len <*a" *>;
A3: len <*a" *>=1 by FINSEQ_1:39;
    then
A4: i=1 by A2,XXREAL_0:1;
    len (<* a *>) =1 by FINSEQ_1:39;
    then i in Seg len (<*a*>) by A2,A3;
    then
A5: i in dom (<*a*>) by FINSEQ_1:def 3;
    i in Seg len ((<*a*>)") by A1,A2;
    then i in dom ((<*a*>)") by FINSEQ_1:def 3;
    then
A6: ((<*a*>)").i=((<*a*>)")/.i by PARTFUN1:def 6
      .= ((<* a *>)/.i)" by A5,Def3;
    (<* a *>)/.i= ((<* a *>).i) by A5,PARTFUN1:def 6
      .= a by A4;
    hence thesis by A4,A6;
  end;
  hence thesis by A1;
end;
