 reserve X, Y for set, A for Ordinal;
 reserve z,z1,z2 for Complex;
 reserve r,r1,r2 for Real;
 reserve q,q1,q2 for Rational;
 reserve i,i1,i2 for Integer;
 reserve n,n1,n2 for Nat;

theorem
  for X being complex-membered set st 0 in X
  holds multRel(X,0) = [: X, {0} :]
proof
  let X be complex-membered set;
  assume A0: 0 in X;
  set M = [: X, {0} :];
  now
    let x,y be object;
    reconsider a=x,b=y as set by TARSKI:1;
    hereby
      assume A1: [x,y] in multRel(X,0);
      then [a,b] in multRel(X,0);
      then a in X & b in X by MMLQUER2:4;
      then reconsider a,b as Complex;
      [a,b] in multRel(X,0) by A1;
      then a in X & b in X & b = 0 * a by Th42;
      hence [x,y] in M by ZFMISC_1:106;
    end;
    assume [x,y] in M;
    then A2: x in X & y = 0 by ZFMISC_1:106;
    then reconsider a,b as Complex;
    b = 0 * a by A2;
    hence [x,y] in multRel(X,0) by A0, A2, Th42;
  end;
  hence thesis by RELAT_1:def 2;
end;
