reserve k,n,i,j for Nat;

theorem Th26:
  for ITP being Element of Permutations(n), ITG being Element of
  Group_of_Perm(n) st ITG=ITP & n>=1 holds ITP"=ITG"
proof
  let ITP be Element of Permutations(n), ITG be Element of Group_of_Perm(n);
  assume that
A1: ITG=ITP and
A2: n>=1;
  reconsider qf=ITP as Function of Seg n,Seg n by MATRIX_1:def 12;
  dom qf=Seg n by A2,FUNCT_2:def 1;
  then
A3: ITP"*ITP=id Seg n by FUNCT_1:39;
  ITP is Permutation of Seg n by MATRIX_1:def 12;
  then rng qf = Seg n by FUNCT_2:def 3;
  then
A4: ITP*ITP"=id Seg n by FUNCT_1:39;
  reconsider pf=ITP" as Element of Group_of_Perm n by MATRIX_1:def 13;
A5: idseq n=1_Group_of_Perm n & ITG*pf=(the multF of (Group_of_Perm(n))).(
  ITG,pf ) by MATRIX_1:15;
A6: pf*ITG=(the multF of (Group_of_Perm(n))).(pf,ITG);
  (the multF of (Group_of_Perm n)).(ITG,pf)=(ITP")*ITP & (the multF of (
  Group_of_Perm n)).(pf,ITG)=ITP*(ITP") by A1,MATRIX_1:def 13;
  hence thesis by A3,A4,A5,A6,GROUP_1:def 5;
end;
