set M = { (a + b) where a, b is Element of : ( a in I & b in J ) } ;
for x, y being Element of st x in { (a + b) where a, b is Element of : ( a in I & b in J ) } & y in { (a + b) where a, b is Element of : ( a in I & b in J ) } holds
x + y in { (a + b) where a, b is Element of : ( a in I & b in J ) }

proof

let x, y be Element of ; :: thesis: ( x in { (a + b) where a, b is Element of : ( a in I & b in J ) } & y in { (a + b) where a, b is Element of : ( a in I & b in J ) } implies x + y in { (a + b) where a, b is Element of : ( a in I & b in J ) } )
assume that
A1: x in { (a + b) where a, b is Element of : ( a in I & b in J ) } and
A2: y in { (a + b) where a, b is Element of : ( a in I & b in J ) } ; :: thesis: x + y in { (a + b) where a, b is Element of : ( a in I & b in J ) }
consider a', b' being Element of such that
A3: x = a' + b' and
A4: ( a' in I & b' in J ) by A1;
consider c, d being Element of such that
A5: y = c + d and
A6: ( c in I & d in J ) by A2;
A7: (a' + c) + (b' + d) = ((a' + c) + b') + d by RLVECT_1:def 6
.= (c + x) + d by A3, RLVECT_1:def 6
.= x + y by A5, RLVECT_1:def 6 ;
( a' + c in I & b' + d in J ) by A4, A6, Def1;
hence x + y in { (a + b) where a, b is Element of : ( a in I & b in J ) } by A7; :: thesis: verum

end;

hence I + J is add-closed by Def1; :: thesis: verum