 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, Y being complex-membered set
  holds multRel(X /\ Y,z) = multRel(X,z) /\ multRel(Y,z)
proof
  let X, Y be complex-membered set;
  now
    let z0 be object;
    hereby
      assume A1: z0 in multRel(X /\ Y,z);
      then consider x,y being object such that
        A2: z0 = [x,y] by RELAT_1:def 1;
      reconsider a=x,b=y as set by TARSKI:1;
      [a,b] in multRel(X /\ Y,z) by A1, A2;
      then a in X /\ Y & b in X /\ Y by MMLQUER2:4;
      then reconsider a,b as Complex;
      [a,b] in multRel(X /\ Y,z) by A1, A2;
      then a in X /\ Y & b in X /\ Y & b = z * a by Th42;
      then a in X & b in X & a in Y & b in Y & b = z * a by XBOOLE_0:def 4;
      hence z0 in multRel(X,z) & z0 in multRel(Y,z) by A2, Th42;
    end;
    assume A3: z0 in multRel(X,z) & z0 in multRel(Y,z);
    then consider x,y being object such that
      A4: z0 = [x,y] by RELAT_1:def 1;
    reconsider a=x,b=y as set by TARSKI:1;
    [a,b] in multRel(X,z) by A3, A4;
    then a in X & b in X by MMLQUER2:4;
    then reconsider a,b as Complex;
    [a,b] in multRel(X,z) & [a,b] in multRel(Y,z) by A3, A4;
    then a in X & b in X & a in Y & b in Y & b = z * a by Th42;
    then a in X /\ Y & b in X /\ Y & b = z * a by XBOOLE_0:def 4;
    hence z0 in multRel(X /\ Y,z) by A4, Th42;
  end;
  hence thesis by XBOOLE_0:def 4;
end;
