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;

theorem
  f is multiplicative & g is multiplicative
  implies {v where v is Element of M: f.v=g.v} is stable Subset of M
proof
  assume A1: f is multiplicative;
  assume A2: g is multiplicative;
  set X = {v where v is Element of M: f.v=g.v};
  for x being object st x in X holds x in the carrier of M
  proof
    let x be object;
    assume x in X; then
    consider v be Element of M such that
    A3: x=v & f.v=g.v;
    thus x in the carrier of M by A3;
  end; then
  reconsider X as Subset of M by TARSKI:def 3;
  for v,w being Element of M st v in X & w in X holds v*w in X
  proof
    let v,w be Element of M;
    assume v in X; then
    consider v9 be Element of M such that
    A4: v=v9 & f.v9=g.v9;
    assume w in X; then
    consider w9 be Element of M such that
    A5: w=w9 & f.w9=g.w9;
    f.(v*w) = g.v*g.w by A4,A5,A1,GROUP_6:def 6
    .= g.(v*w) by A2,GROUP_6:def 6;
    hence v*w in X;
  end;
  hence thesis by Def10;
end;
