reserve M,N for non empty multMagma,
  f for Function of M, N;

theorem Th4:
  f is multiplicative implies equ_kernel f is compatible
proof
  assume A1: f is multiplicative;
  set R = equ_kernel f;
  for v,v9,w,w9 being Element of M st v in Class(R,v9) & w in Class(R,w9) holds
  v*w in Class(R,v9*w9)
  proof
    let v,v9,w,w9 be Element of M;
    assume v in Class(R,v9); then
    A2: [v9,v] in R by EQREL_1:18;
    assume w in Class(R,w9); then
    [w9,w] in R by EQREL_1:18; then
    A3: f.w9 = f.w by Def8;
    f.(v9*w9) = f.v9 * f.w9 by A1,GROUP_6:def 6
    .= f.v * f.w by A2,A3,Def8
    .= f.(v*w) by A1,GROUP_6:def 6; then
    [v9*w9,v*w] in R by Def8;
    hence v*w in Class(R,v9*w9) by EQREL_1:18;
  end;
  hence equ_kernel f is compatible;
end;
