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 (#) g) || C) = (dom (f (#) g)) /\ C by RELAT_1:61
.= ((dom f) /\ (dom g)) /\ C by VALUED_1:def 4 ;
A2: dom ((f || C) (#) (g || C)) = (dom (f | C)) /\ (dom (g || C)) by VALUED_1:def 4
.= ((dom f) /\ C) /\ (dom (g | C)) by RELAT_1:61
.= ((dom f) /\ C) /\ ((dom g) /\ C) by RELAT_1:61
.= (((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) ;
then A3: 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 )
A4: ((f || C) (#) (g || C)) . c = ((f | C) . c) * ((g | C) . c) by VALUED_1:5;
assume A5: c in dom ((f || C) (#) (g || C)) ; :: thesis: ((f || C) (#) (g || C)) . c = ((f (#) g) || C) . c
then A6: c in (dom (f || C)) /\ (dom (g || C)) by VALUED_1:def 4;
then A7: c in dom (f || C) by XBOOLE_0:def 4;
A8: c in dom (g || C) by A6, XBOOLE_0:def 4;
((f (#) g) || C) . c = (f (#) g) . c by A3, A5, FUNCT_1:47
.= (f . c) * (g . c) by VALUED_1:5
.= ((f | C) . c) * (g . c) by A7, FUNCT_1:47 ;
hence ((f || C) (#) (g || C)) . c = ((f (#) g) || C) . c by A8, A4, FUNCT_1:47; :: thesis: verum

end;

hence (f || C) (#) (g || C) = (f (#) g) || C by A2, A1, PARTFUN1:5, XBOOLE_1:16; :: thesis: verum