theorem Th41:
  for p,Perm,q st q=p*Perm" holds (the multF of K) $$ (Path_matrix
  (p,A*Perm)) = (the multF of K) $$ (Path_matrix(q,A))
proof
  let p,Perm,q such that
A1: q=p*Perm";
  set mm=the multF of K;
  set P2= Path_matrix(q,A);
  set P1=Path_matrix(p,A*Perm);
  now
    per cases;
    case
A2:   n=0;
      then len P1=0 by MATRIX_3:def 7;
      then
A3:   mm $$ P1=the_unity_wrt mm by FINSOP_1:def 1;
      len P2=0 by A2,MATRIX_3:def 7;
      hence thesis by A3,FINSOP_1:def 1;
    end;
    case
      n+0>0;
      then
A4:   n>=1 by NAT_1:19;
A5:   len P2=n by MATRIX_3:def 7;
A6:   Perm is Element of Permutations(n) by MATRIX_1:def 12;
      P1 = P2*Perm by A1,Th40;
      hence thesis by A4,A5,A6,MATRIX_7:33;
    end;
  end;
  hence thesis;
end;
