set ad = multreal | [:INT,INT:];
[:INT,INT:] c= [:REAL,REAL:] by NUMBERS:15, ZFMISC_1:96;
then A1: [:INT,INT:] c= dom multreal by FUNCT_2:def 1;
then A2: dom (multreal | [:INT,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 (multreal | [:INT,INT:]) holds
(multreal | [:INT,INT:]) . z = multint . z

proof

let z be object ; :: thesis: ( z in dom (multreal | [:INT,INT:]) implies (multreal | [:INT,INT:]) . z = multint . z )
assume A4: z in dom (multreal | [:INT,INT:]) ; :: thesis: (multreal | [:INT,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 (multreal | [:INT,INT:]) . z = multreal . (x1,y1) by A4, A5, A2, FUNCT_1:49
.= x1 * y1 by BINOP_2:def 11
.= multint . (x1,y1) by BINOP_2:def 22
.= multint . z by A5 ; :: thesis: verum

end;

hence multint = multreal | [:INT,INT:] by A1, A3, FUNCT_1:2, RELAT_1:62; :: thesis: verum