let b be BinOp of INT; :: thesis: ( b = addint iff for i1, i2 being Element of INT holds b . (i1,i2) = addreal . (i1,i2) )

hereby :: thesis: ( ( for i1, i2 being Element of INT holds b . (i1,i2) = addreal . (i1,i2) ) implies b = addint )

assume A1: b = addint ; :: thesis: for i1, i2 being Element of INT holds b . (i1,i2) = addreal . (i1,i2)
let i1, i2 be Element of INT ; :: thesis: b . (i1,i2) = addreal . (i1,i2)
thus b . (i1,i2) = i1 + i2 by A1, BINOP_2:def 20
.= addreal . (i1,i2) by BINOP_2:def 9 ; :: thesis: verum

end;

assume A3: for i1, i2 being Element of INT holds b . (i1,i2) = addreal . (i1,i2) ; :: thesis: b = addint

now

let i1, i2 be Element of INT ; :: thesis: b . (i1,i2) = addint . (i1,i2)
thus b . (i1,i2) = addreal . (i1,i2) by A3
.= i1 + i2 by BINOP_2:def 9
.= addint . (i1,i2) by BINOP_2:def 20 ; :: thesis: verum

end;

hence b = addint by BINOP_1:2; :: thesis: verum