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 Th8:
  the carrier of N is stable Subset of M
proof
  for v,w being Element of M st v in the carrier of N & w in the carrier of N
  holds v*w in the carrier of N
  proof
    let v,w be Element of M;
    assume A1: v in the carrier of N & w in the carrier of N; then
    reconsider v9=v, w9=w as Element of N;
    A2: [v,w] in [:the carrier of N,the carrier of N:] by A1,ZFMISC_1:def 2;
    v9*w9 = (the multF of N).[v9,w9] by BINOP_1:def 1
    .= ((the multF of M)||the carrier of N).[v9,w9] by Def9
    .= ((the multF of M)|[:the carrier of N,the carrier of N:]).[v,w]
    by REALSET1:def 2
    .= (the multF of M).[v,w] by A2,FUNCT_1:49
    .= v*w by BINOP_1:def 1;
    hence v*w in the carrier of N by A1;
  end;
  hence the carrier of N is stable Subset of M by Def9,Def10;
end;
