reserve M,N for non empty multMagma,
  f for Function of M, N;
reserve M for multMagma;
reserve N,K for multSubmagma of M;

theorem Th6:
  N is multSubmagma of K & K is multSubmagma of N
  implies the multMagma of N = the multMagma of K
proof
  assume that
A1: N is multSubmagma of K and
A2: K is multSubmagma of N;
  set A = the carrier of N;
  set B = the carrier of K;
  set f = the multF of N;
  set g = the multF of K;
A3: A c= B & B c= A by A1,A2,Def9;
  f = g||A by A1,Def9
  .= (f||B)||A by A2,Def9
  .= (f|[:B,B:])||A by REALSET1:def 2
  .= (f|[:B,B:])|[:A,A:] by REALSET1:def 2
  .= f||B by REALSET1:def 2
  .= g by A2,Def9;
  hence thesis by A3,XBOOLE_0:def 10;
end;
