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

theorem Th5:
  f is multiplicative implies
  ex r being Relators of M st equ_kernel f = equ_rel r
proof
  assume A1: f is multiplicative;
  set I = {[v,w] where v,w is Element of M: f.v = f.w};
  set r = id I;
  for y being object st y in rng r
  holds y in [: the carrier of M, the carrier of M:]
  proof
    let y be object;
    assume y in rng r; then
    consider x be object such that
    A2: x in dom r & y = r.x by FUNCT_1:def 3;
    y = x by A2,FUNCT_1:17; then
    y in I by A2; then
    consider v9,w9 be Element of M such that
    A3: y = [v9,w9] & f.v9 = f.w9;
    thus thesis by A3,ZFMISC_1:def 2;
  end; then
  rng r c= [: the carrier of M, the carrier of M:]; then
  reconsider r as Relators of M by FUNCT_2:2;
  take r;
  reconsider R=equ_kernel f as compatible Equivalence_Relation of M by A1,Th4;
  A4: for i being set, v,w being Element of M
  st i in dom r & r.i = [v,w] holds v in Class(R,w)
  proof
    let i be set;
    let v,w be Element of M;
    assume A5: i in dom r & r.i = [v,w]; then
    A6: r.i = i by FUNCT_1:17;
    consider v9,w9 be Element of M such that
    A7: i=[v9,w9] & f.v9 = f.w9 by A5;
    [v,w] in equ_kernel f by A7,Def8,A5,A6;
    hence v in Class(R,w) by EQREL_1:19;
  end; then
  A8: equ_rel r c= R by Th3;
  for z being object st z in R holds z in equ_rel r
  proof
    let z be object;
    assume A9: z in R;
    then consider x,y be object such that
    A10: x in the carrier of M & y in the carrier of M &
    z = [x,y] by ZFMISC_1:def 2;
    A11: f.x = f.y by A9,A10,Def8;
    reconsider x,y as Element of M by A10;
    set X9 = {R9 where R9 is compatible Equivalence_Relation of M:
    for i being set, v,w being Element of M st i in dom r & r.i = [v,w]
    holds v in Class(R9,w)};
    A12: R in X9 by A4;
    for Y being set st Y in X9 holds z in Y
    proof
      let Y be set;
      assume Y in X9; then
      consider R9 be compatible Equivalence_Relation of M such that
      A13: R9=Y & for i being set, v,w being Element of M
      st i in dom r & r.i = [v,w] holds v in Class(R9,w);
      set i = [x,y];
      A14: i in I by A11; then
      r.i = [x,y] by FUNCT_1:17; then
      x in Class(R9,y) by A14,A13;
      hence z in Y by A10,A13,EQREL_1:19;
    end;
    hence thesis by A12,SETFAM_1:def 1;
  end; then
  R c= equ_rel r;
  hence thesis by A8,XBOOLE_0:def 10;
end;
