let i1, i2 be Integer; :: thesis: (addRel (INT,i1)) * (addRel (INT,i2)) = addRel (INT,(i1 + i2))
A1: (addRel (INT,i1)) * (addRel (INT,i2)) c= addRel (INT,(i1 + i2)) by Th17;

now :: thesis: for x, y being object st [x,y] in addRel (INT,(i1 + i2)) holds
[x,y] in (addRel (INT,i1)) * (addRel (INT,i2))

let x, y be object ; :: thesis: ( [x,y] in addRel (INT,(i1 + i2)) implies [x,y] in (addRel (INT,i1)) * (addRel (INT,i2)) )
reconsider a = x, b = y as set by TARSKI:1;
assume A2: [x,y] in addRel (INT,(i1 + i2)) ; :: thesis: [x,y] in (addRel (INT,i1)) * (addRel (INT,i2))
then [a,b] in addRel (INT,(i1 + i2)) ;
then A3: ( a in INT & b in INT ) by MMLQUER2:4;
then reconsider a = a, b = b as Integer ;
A4: b = (i1 + i2) + a by A2, Th11;
set c = i1 + a;
( i1 + a in INT & b = i2 + (i1 + a) ) by A4, INT_1:def 2;
then ( [a,(i1 + a)] in addRel (INT,i1) & [(i1 + a),b] in addRel (INT,i2) ) by A3, Th11;
hence [x,y] in (addRel (INT,i1)) * (addRel (INT,i2)) by RELAT_1:def 8; :: thesis: verum

end;

then addRel (INT,(i1 + i2)) c= (addRel (INT,i1)) * (addRel (INT,i2)) by RELAT_1:def 3;
hence (addRel (INT,i1)) * (addRel (INT,i2)) = addRel (INT,(i1 + i2)) by A1, XBOOLE_0:def 10; :: thesis: verum