let q1, q2 be Rational; :: thesis: (multRel (RAT,q1)) * (multRel (RAT,q2)) = multRel (RAT,(q1 * q2))
A1: (multRel (RAT,q1)) * (multRel (RAT,q2)) c= multRel (RAT,(q1 * q2)) by Th51;

now :: thesis: for x, y being object st [x,y] in multRel (RAT,(q1 * q2)) holds
[x,y] in (multRel (RAT,q1)) * (multRel (RAT,q2))

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

end;

then multRel (RAT,(q1 * q2)) c= (multRel (RAT,q1)) * (multRel (RAT,q2)) by RELAT_1:def 3;
hence (multRel (RAT,q1)) * (multRel (RAT,q2)) = multRel (RAT,(q1 * q2)) by A1, XBOOLE_0:def 10; :: thesis: verum