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 Th7:
  the carrier of N = the carrier of M
  implies the multMagma of N = the multMagma of M
proof
  assume A1: the carrier of N = the carrier of M;
  per cases;
  suppose the carrier of M = {};
    hence thesis by A1;
  end;
  suppose the carrier of M <> {};
    A2: the multF of M
    = (the multF of M)|[:the carrier of M,the carrier of M:]
    .= (the multF of M)||(the carrier of M) by REALSET1:def 2; then
    reconsider M9=M as multSubmagma of M by Def9;
    the multF of M9 = (the multF of N)||the carrier of M9 by A1,A2,Def9; then
    M9 is multSubmagma of N by A1,Def9;
    hence thesis by Th6;
  end;
end;
