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
A2: ( i1 in INT & i2 in INT ) ;
thus b . i1,i2 = i1 + i2 by A1, BINOP_2:def 20
.= addreal . i1,i2 by A2, BINOP_2:def 9, NUMBERS:15 ; :: 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
A4: ( i1 in INT & i2 in INT ) ;
thus b . i1,i2 = addreal . i1,i2 by A3
.= i1 + i2 by A4, BINOP_2:def 9, NUMBERS:15
.= addint . i1,i2 by BINOP_2:def 20 ; :: thesis: verum

end;

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