set ad = addcomplex || REAL;
[:REAL,REAL:] c= [:COMPLEX,COMPLEX:] by NUMBERS:11, ZFMISC_1:96;
then A1: [:REAL,REAL:] c= dom addcomplex by FUNCT_2:def 1;
then A2: dom (addcomplex || REAL) = [:REAL,REAL:] by RELAT_1:62;
A3: dom addreal = [:REAL,REAL:] by FUNCT_2:def 1;
for z being object st z in dom (addcomplex || REAL) holds
(addcomplex || REAL) . z = addreal . z

proof

let z be object ; :: thesis: ( z in dom (addcomplex || REAL) implies (addcomplex || REAL) . z = addreal . z )
assume A4: z in dom (addcomplex || REAL) ; :: thesis: (addcomplex || REAL) . z = addreal . z
then consider x, y being object such that
A5: ( x in REAL & y in REAL & z = [x,y] ) by A2, ZFMISC_1:def 2;
reconsider x1 = x, y1 = y as Real by A5;
thus (addcomplex || REAL) . z = addcomplex . (x1,y1) by A4, A5, A2, FUNCT_1:49
.= x1 + y1 by BINOP_2:def 3
.= addreal . (x1,y1) by BINOP_2:def 9
.= addreal . z by A5 ; :: thesis: verum

end;

hence addcomplex || REAL = addreal by A1, A3, RELAT_1:62; :: thesis: verum