let r1, r2 be Real; :: thesis: (addRel (REAL,r1)) * (addRel (REAL,r2)) = addRel (REAL,(r1 + r2))
A1: (addRel (REAL,r1)) * (addRel (REAL,r2)) c= addRel (REAL,(r1 + r2)) by Th17;

now :: thesis: for x, y being object st [x,y] in addRel (REAL,(r1 + r2)) holds
[x,y] in (addRel (REAL,r1)) * (addRel (REAL,r2))

let x, y be object ; :: thesis: ( [x,y] in addRel (REAL,(r1 + r2)) implies [x,y] in (addRel (REAL,r1)) * (addRel (REAL,r2)) )
reconsider a = x, b = y as set by TARSKI:1;
assume A2: [x,y] in addRel (REAL,(r1 + r2)) ; :: thesis: [x,y] in (addRel (REAL,r1)) * (addRel (REAL,r2))
then [a,b] in addRel (REAL,(r1 + r2)) ;
then A3: ( a in REAL & b in REAL ) by MMLQUER2:4;
then reconsider a = a, b = b as Real ;
A4: b = (r1 + r2) + a by A2, Th11;
set c = r1 + a;
( r1 + a in REAL & b = r2 + (r1 + a) ) by A4, XREAL_0:def 1;
then ( [a,(r1 + a)] in addRel (REAL,r1) & [(r1 + a),b] in addRel (REAL,r2) ) by A3, Th11;
hence [x,y] in (addRel (REAL,r1)) * (addRel (REAL,r2)) by RELAT_1:def 8; :: thesis: verum

end;

then addRel (REAL,(r1 + r2)) c= (addRel (REAL,r1)) * (addRel (REAL,r2)) by RELAT_1:def 3;
hence (addRel (REAL,r1)) * (addRel (REAL,r2)) = addRel (REAL,(r1 + r2)) by A1, XBOOLE_0:def 10; :: thesis: verum