let q1, q2 be Rational; :: thesis: (addRel (RAT,q1)) * (addRel (RAT,q2)) = addRel (RAT,(q1 + q2))
A1: (addRel (RAT,q1)) * (addRel (RAT,q2)) c= addRel (RAT,(q1 + q2)) by Th17;

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

let x, y be object ; :: thesis: ( [x,y] in addRel (RAT,(q1 + q2)) implies [x,y] in (addRel (RAT,q1)) * (addRel (RAT,q2)) )
reconsider a = x, b = y as set by TARSKI:1;
assume A2: [x,y] in addRel (RAT,(q1 + q2)) ; :: thesis: [x,y] in (addRel (RAT,q1)) * (addRel (RAT,q2))
then [a,b] in addRel (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, Th11;
set c = q1 + a;
( q1 + a in RAT & b = q2 + (q1 + a) ) by A4, RAT_1:def 2;
then ( [a,(q1 + a)] in addRel (RAT,q1) & [(q1 + a),b] in addRel (RAT,q2) ) by A3, Th11;
hence [x,y] in (addRel (RAT,q1)) * (addRel (RAT,q2)) by RELAT_1:def 8; :: thesis: verum

end;

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