let f, g be PartFunc of REAL ,REAL ; :: thesis: for C being non empty Subset of REAL holds (f || C) (#) (g || C) = (f (#) g) || C
let C be non empty Subset of REAL ; :: thesis: (f || C) (#) (g || C) = (f (#) g) || C
A1: dom ((f || C) (#) (g || C)) = (dom (f | C)) /\ (dom (g || C)) by VALUED_1:def 4
.= ((dom f) /\ C) /\ (dom (g | C)) by RELAT_1:90
.= ((dom f) /\ C) /\ ((dom g) /\ C) by RELAT_1:90
.= (((dom f) /\ C) /\ C) /\ (dom g) by XBOOLE_1:16
.= ((dom f) /\ (C /\ C)) /\ (dom g) by XBOOLE_1:16
.= ((dom f) /\ C) /\ (dom g) ;
dom ((f (#) g) || C) = (dom (f (#) g)) /\ C by RELAT_1:90
.= ((dom f) /\ (dom g)) /\ C by VALUED_1:def 4 ;
then A2: dom ((f || C) (#) (g || C)) = dom ((f (#) g) || C) by A1, XBOOLE_1:16;
for c being Element of C st c in dom ((f || C) (#) (g || C)) holds
((f || C) (#) (g || C)) . c = ((f (#) g) || C) . c

proof

let c be Element of C; :: thesis: ( c in dom ((f || C) (#) (g || C)) implies ((f || C) (#) (g || C)) . c = ((f (#) g) || C) . c )
assume A3: c in dom ((f || C) (#) (g || C)) ; :: thesis: ((f || C) (#) (g || C)) . c = ((f (#) g) || C) . c
then c in (dom (f || C)) /\ (dom (g || C)) by VALUED_1:def 4;
then A4: ( c in dom (f || C) & c in dom (g || C) ) by XBOOLE_0:def 4;
A5: ((f || C) (#) (g || C)) . c = ((f | C) . c) * ((g | C) . c) by VALUED_1:5;
((f (#) g) || C) . c = (f (#) g) . c by A2, A3, FUNCT_1:70
.= (f . c) * (g . c) by VALUED_1:5
.= ((f | C) . c) * (g . c) by A4, FUNCT_1:70 ;
hence ((f || C) (#) (g || C)) . c = ((f (#) g) || C) . c by A4, A5, FUNCT_1:70; :: thesis: verum

end;

hence (f || C) (#) (g || C) = (f (#) g) || C by A2, PARTFUN1:34; :: thesis: verum