reserve M,N for non empty multMagma,
  f for Function of M, N;
reserve M for multMagma;
reserve N,K for multSubmagma of M;
reserve M,N for non empty multMagma,
  A for Subset of M,
  f,g for Function of M,N,
  X for stable Subset of M,
  Y for stable Subset of N;
reserve X for set;
reserve x,y,Y for set;
reserve n,m,p for Nat;

theorem Th27:
  X c= Y implies free_magma_carrier X c= free_magma_carrier Y
proof
  assume A1: X c= Y;
  per cases;
  suppose X = {}; then
    free_magma_carrier X = {};
    hence free_magma_carrier X c= free_magma_carrier Y;
  end;
  suppose A2: X <> {};
      let x be object;
      assume A3: x in free_magma_carrier X;
      reconsider X9=X as non empty set by A2;
      reconsider w=x as Element of free_magma_carrier X9 by A3;
      A4: w in [:free_magma(X9,w`2),{w`2}:] by Th25; then
      A5: w`1 in free_magma(X9,w`2) & w`2 in {w`2} by MCART_1:10;
      reconsider Y9=Y as non empty set by A2,A1;
      A6: free_magma(X9,w`2) c= free_magma(Y9,w`2) by A1,Th23;
      w = [w`1,w`2] by A4,MCART_1:21; then
      A7: w in [:free_magma(Y9,w`2),{w`2}:] by A6,A5,ZFMISC_1:def 2;
      [:free_magma(Y9,w`2),{w`2}:] c= free_magma_carrier Y9 by Lm1;
      hence x in free_magma_carrier Y by A7;
  end;
end;
