set mu = multrat || INT;
[:INT,INT:] c= [:RAT,RAT:] by NUMBERS:14, ZFMISC_1:96;
then A1: [:INT,INT:] c= dom multrat by FUNCT_2:def 1;
then A2: dom (multrat || INT) = [:INT,INT:] by RELAT_1:62;
A3: dom multint = [:INT,INT:] by FUNCT_2:def 1;
for z being object st z in dom (multrat || INT) holds
(multrat || INT) . z = multint . z

proof

let z be object ; :: thesis: ( z in dom (multrat || INT) implies (multrat || INT) . z = multint . z )
assume A4: z in dom (multrat || INT) ; :: thesis: (multrat || INT) . z = multint . 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 (multrat || INT) . z = multrat . (x1,y1) by A4, A5, A2, FUNCT_1:49
.= x1 * y1 by BINOP_2:def 17
.= multint . (x1,y1) by BINOP_2:def 22
.= multint . z by A5 ; :: thesis: verum

end;

hence multrat || INT = multint by A1, A3, RELAT_1:62; :: thesis: verum