set ad = multreal || RAT;
[:RAT,RAT:] c= [:REAL,REAL:] by NUMBERS:12, ZFMISC_1:96;
then A1: [:RAT,RAT:] c= dom multreal by FUNCT_2:def 1;
then A2: dom (multreal || RAT) = [:RAT,RAT:] by RELAT_1:62;
A3: dom multrat = [:RAT,RAT:] by FUNCT_2:def 1;
for z being object st z in dom (multreal || RAT) holds
(multreal || RAT) . z = multrat . z

proof

let z be object ; :: thesis: ( z in dom (multreal || RAT) implies (multreal || RAT) . z = multrat . z )
assume A4: z in dom (multreal || RAT) ; :: thesis: (multreal || RAT) . z = multrat . z
then consider x, y being object such that
A5: ( x in RAT & y in RAT & z = [x,y] ) by A2, ZFMISC_1:def 2;
reconsider x1 = x, y1 = y as Rational by A5;
thus (multreal || RAT) . z = multreal . (x1,y1) by A4, A5, A2, FUNCT_1:49
.= x1 * y1 by BINOP_2:def 11
.= multrat . (x1,y1) by BINOP_2:def 17
.= multrat . z by A5 ; :: thesis: verum

end;

hence multreal || RAT = multrat by A1, A3, FUNCT_1:2, RELAT_1:62; :: thesis: verum