 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
  multRel(RAT,q1) * multRel(RAT,q2) = multRel(RAT,q1*q2)
proof
  A1: multRel(RAT,q1) * multRel(RAT,q2) c= multRel(RAT,q1*q2) by Th51;
  now
    let x,y be object;
    reconsider a=x,b=y as set by TARSKI:1;
    assume A2: [x,y] in multRel(RAT,q1*q2);
    then [a,b] in multRel(RAT,q1*q2);
    then A3: a in RAT & b in RAT by MMLQUER2:4;
    then reconsider a,b as Rational;
    A4: b = q1 * q2 * a by A2, Th42;
    set c = q1 * a;
    c in RAT & b = q2 * c by A4, RAT_1:def 2;
    then [a,c] in multRel(RAT,q1) & [c,b] in multRel(RAT,q2) by A3, Th42;
    hence [x,y] in multRel(RAT,q1) * multRel(RAT,q2) by RELAT_1:def 8;
  end;
  then multRel(RAT,q1*q2) c= multRel(RAT,q1) * multRel(RAT,q2)
    by RELAT_1:def 3;
  hence thesis by A1, XBOOLE_0:def 10;
end;
