let n1, n2 be Nat; :: thesis: (multRel (NAT,n1)) * (multRel (NAT,n2)) = multRel (NAT,(n1 * n2))
A1: (multRel (NAT,n1)) * (multRel (NAT,n2)) c= multRel (NAT,(n1 * n2)) by Th51;

now :: thesis: for x, y being object st [x,y] in multRel (NAT,(n1 * n2)) holds
[x,y] in (multRel (NAT,n1)) * (multRel (NAT,n2))

let x, y be object ; :: thesis: ( [x,y] in multRel (NAT,(n1 * n2)) implies [x,y] in (multRel (NAT,n1)) * (multRel (NAT,n2)) )
reconsider a = x, b = y as set by TARSKI:1;
assume A2: [x,y] in multRel (NAT,(n1 * n2)) ; :: thesis: [x,y] in (multRel (NAT,n1)) * (multRel (NAT,n2))
then [a,b] in multRel (NAT,(n1 * n2)) ;
then A3: ( a in NAT & b in NAT ) by MMLQUER2:4;
then reconsider a = a, b = b as Nat ;
A4: b = (n1 * n2) * a by A2, Th42;
set c = n1 * a;
( n1 * a in NAT & b = n2 * (n1 * a) ) by A4, ORDINAL1:def 12;
then ( [a,(n1 * a)] in multRel (NAT,n1) & [(n1 * a),b] in multRel (NAT,n2) ) by A3, Th42;
hence [x,y] in (multRel (NAT,n1)) * (multRel (NAT,n2)) by RELAT_1:def 8; :: thesis: verum

end;

then multRel (NAT,(n1 * n2)) c= (multRel (NAT,n1)) * (multRel (NAT,n2)) by RELAT_1:def 3;
hence (multRel (NAT,n1)) * (multRel (NAT,n2)) = multRel (NAT,(n1 * n2)) by A1, XBOOLE_0:def 10; :: thesis: verum