let C be non empty set ; :: thesis: for f1, f2 being PartFunc of C,COMPLEX holds (f1 (#) f2) ^ = (f1 ^ ) (#) (f2 ^ )
let f1, f2 be PartFunc of C,COMPLEX ; :: thesis: (f1 (#) f2) ^ = (f1 ^ ) (#) (f2 ^ )
A1: dom ((f1 (#) f2) ^ ) = (dom (f1 (#) f2)) \ ((f1 (#) f2) " {0c }) by Def2
.= ((dom f1) \ (f1 " {0c })) /\ ((dom f2) \ (f2 " {0c })) by Th16
.= (dom (f1 ^ )) /\ ((dom f2) \ (f2 " {0c })) by Def2
.= (dom (f1 ^ )) /\ (dom (f2 ^ )) by Def2
.= dom ((f1 ^ ) (#) (f2 ^ )) by Th5 ;

now

let c be Element of C; :: thesis: ( c in dom ((f1 (#) f2) ^ ) implies ((f1 (#) f2) ^ ) /. c = ((f1 ^ ) (#) (f2 ^ )) /. c )
assume A2: c in dom ((f1 (#) f2) ^ ) ; :: thesis: ((f1 (#) f2) ^ ) /. c = ((f1 ^ ) (#) (f2 ^ )) /. c
then c in (dom (f1 ^ )) /\ (dom (f2 ^ )) by A1, Th5;
then A3: ( c in dom (f1 ^ ) & c in dom (f2 ^ ) ) by XBOOLE_0:def 4;
c in (dom (f1 (#) f2)) \ ((f1 (#) f2) " {0c }) by A2, Def2;
then A4: c in dom (f1 (#) f2) by XBOOLE_0:def 5;
thus ((f1 (#) f2) ^ ) /. c = ((f1 (#) f2) /. c) " by A2, Def2
.= ((f1 /. c) * (f2 /. c)) " by A4, Th5
.= ((f1 /. c) " ) * ((f2 /. c) " ) by XCMPLX_1:205
.= ((f1 ^ ) /. c) * ((f2 /. c) " ) by A3, Def2
.= ((f1 ^ ) /. c) * ((f2 ^ ) /. c) by A3, Def2
.= ((f1 ^ ) (#) (f2 ^ )) /. c by A1, A2, Th5 ; :: thesis: verum

end;

hence (f1 (#) f2) ^ = (f1 ^ ) (#) (f2 ^ ) by A1, PARTFUN2:3; :: thesis: verum