set ad = addreal || INT;
[:INT,INT:] c= [:REAL,REAL:] by NUMBERS:15, ZFMISC_1:96;
then A1: [:INT,INT:] c= dom addreal by FUNCT_2:def 1;
then A2: dom (addreal || INT) = [:INT,INT:] by RELAT_1:62;
A3: dom addint = [:INT,INT:] by FUNCT_2:def 1;
for z being object st z in dom (addreal || INT) holds
(addreal || INT) . z = addint . z

proof

let z be object ; :: thesis: ( z in dom (addreal || INT) implies (addreal || INT) . z = addint . z )
assume A4: z in dom (addreal || INT) ; :: thesis: (addreal || INT) . z = addint . z
then consider x, y being object such that
A5: ( x in INT & y in INT & z = [x,y] ) by A2, ZFMISC_1:def 2;
reconsider x1 = x, y1 = y as Integer by A5;
thus (addreal || INT) . z = addreal . (x1,y1) by A4, A5, A2, FUNCT_1:49
.= x1 + y1 by BINOP_2:def 9
.= addint . (x1,y1) by BINOP_2:def 20
.= addint . z by A5 ; :: thesis: verum

end;

hence addreal || INT = addint by A1, A3, FUNCT_1:2, RELAT_1:62; :: thesis: verum