A0: INT = the carrier of INT.Ring by INT_3:def 3;
set scmult = multcomplex | [:INT,G_INT_SET:];
[:INT,G_INT_SET:] c= [:COMPLEX,COMPLEX:] by NUMBERS:16, ZFMISC_1:96;
then [:INT,G_INT_SET:] c= dom multcomplex by FUNCT_2:def 1;
then A1: dom (multcomplex | [:INT,G_INT_SET:]) = [:INT,G_INT_SET:] by RELAT_1:62;
for z being object st z in [:INT,G_INT_SET:] holds
(multcomplex | [:INT,G_INT_SET:]) . z in G_INT_SET

proof

let z be object ; :: thesis: ( z in [:INT,G_INT_SET:] implies (multcomplex | [:INT,G_INT_SET:]) . z in G_INT_SET )
assume A2: z in [:INT,G_INT_SET:] ; :: thesis: (multcomplex | [:INT,G_INT_SET:]) . z in G_INT_SET
consider x, y being object such that
A3: ( x in INT & y in G_INT_SET & z = [x,y] ) by A2, ZFMISC_1:def 2;
reconsider x1 = x as Element of INT by A3;
reconsider y1 = y as G_INTEG by Th2, A3;
(multcomplex | [:INT,G_INT_SET:]) . z = multcomplex . (x1,y1) by A2, A3, FUNCT_1:49
.= x1 * y1 by BINOP_2:def 5 ;
hence (multcomplex | [:INT,G_INT_SET:]) . z in G_INT_SET ; :: thesis: verum

end;

hence multcomplex | [: the carrier of INT.Ring,G_INT_SET:] is Function of [: the carrier of INT.Ring,G_INT_SET:],G_INT_SET by A1, FUNCT_2:3, A0; :: thesis: verum