reserve x for set,
  i,j,k,n for Nat,
  K for Field;
reserve a,b,c,d for Element of K;

theorem
  Per <*<*a*>*> = a
proof
  set M = <*<*a*>*>;
A1: (PPath_product M).(idseq 1)=a
  proof
    reconsider p = idseq 1 as Element of Permutations 1 by MATRIX_1:def 12;
A2: len Path_matrix(p,M) = 1 by MATRIX_3:def 7;
    then
A3: dom Path_matrix(p,M) = Seg 1 by FINSEQ_1:def 3;
    then
A4: 1 in dom Path_matrix(p,M);
    then 1 = p.1 by A3,FUNCT_1:18;
    then Path_matrix(p,M).1 = M*(1,1) by A4,MATRIX_3:def 7;
    then Path_matrix(p,M).1 = a by MATRIX_0:49;
    then
A5: Path_matrix(p,M) = <*a*> by A2,FINSEQ_1:40;
    (PPath_product M).p = (the multF of K) $$ Path_matrix(p,M) & <*a*> = 1
    |->a by Def1,FINSEQ_2:59;
    hence thesis by A5,FINSOP_1:16;
  end;
  Permutations 1 in Fin Permutations 1 by FINSUB_1:def 5; then
  In (Permutations 1, Fin Permutations 1) = Permutations 1 by SUBSET_1:def 8;
  then
  In (Permutations 1, Fin Permutations 1) = {idseq 1} &
    idseq 1 in Permutations 1 by MATRIX_1:10,TARSKI:def 1;
  hence thesis by A1,SETWISEO:17;
end;
