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 Th30:
  X = {w`1 where w is Element of free_magma X: length w = 1}
proof
  for x being object holds x in X iff
  x in {w`1 where w is Element of free_magma X: length w = 1}
  proof
    let x be object;
    hereby
      assume A1: x in X; then
      A2: x in free_magma(X,1) by Def13;
      1 in {1} by TARSKI:def 1; then
      A3: [x,1] in [:free_magma(X,1),{1}:] by A2,ZFMISC_1:def 2;
      [:free_magma(X,1),{1}:] c= free_magma_carrier X by Lm1; then
      reconsider w9 = [x,1] as Element of free_magma X by A3;
      1 = [x,1]`2; then
      A4: length w9 = 1 by A1,Def18;
      x = [x,1]`1;
      hence x in {w`1 where w is Element of free_magma X: length w = 1} by A4;
    end;
    assume x in {w`1 where w is Element of free_magma X: length w = 1}; then
    consider w be Element of free_magma X such that
    A5: x = w`1 & length w = 1;
    A6: w`2 = 1 by A5,Def18;
    per cases;
    suppose X is non empty; then
      w in [:free_magma(X,1),{1}:] by A6,Th25; then
      w in [:X,{1}:] by Def13;
      hence x in X by A5,MCART_1:10;
    end;
    suppose X is empty; hence thesis by A5,Def18; end;
  end;
  hence thesis by TARSKI:2;
end;
