 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 Th51:
  for X being complex-membered set
  holds multRel(X,z1) * multRel(X,z2) c= multRel(X,z1*z2)
proof
  let X be complex-membered set;
  now
    let x,y be object;
    assume [x,y] in multRel(X,z1) * multRel(X,z2);
    then consider z being object such that
      A1: [x,z] in multRel(X,z1) & [z,y] in multRel(X,z2) by RELAT_1:def 8;
    reconsider a = x, b = y, c = z as set by TARSKI:1;
    [a,c] in multRel(X,z1) & [c,b] in multRel(X,z2) by A1;
    then A2: a in X & c in X & b in X by MMLQUER2:4;
    then reconsider a,b,c as Complex;
    c = z1 * a & b = z2 * c by A1, Th42;
    then b = z1 * z2 * a;
    hence [x,y] in multRel(X,z1*z2) by A2, Th42;
  end;
  hence thesis by RELAT_1:def 3;
end;
