let f, g be PartFunc of REAL,REAL; :: thesis: for C being non empty Subset of REAL holds (f + g) || C = (f || C) + (g || C)
let C be non empty Subset of REAL; :: thesis: (f + g) || C = (f || C) + (g || C)
A1: dom ((f || C) + (g || C)) = (dom (f | C)) /\ (dom (g || C)) by VALUED_1:def 1
.= ((dom f) /\ C) /\ (dom (g | C)) by RELAT_1:61
.= ((dom f) /\ C) /\ ((dom g) /\ C) by RELAT_1:61
.= (dom f) /\ (C /\ ((dom g) /\ C)) by XBOOLE_1:16
.= (dom f) /\ ((dom g) /\ (C /\ C)) by XBOOLE_1:16
.= (dom f) /\ ((dom g) /\ C) ;
A2: dom ((f + g) || C) = (dom (f + g)) /\ C by RELAT_1:61
.= ((dom f) /\ (dom g)) /\ C by VALUED_1:def 1 ;
then A3: dom ((f + g) || C) = dom ((f || C) + (g || C)) by A1, XBOOLE_1:16;
for c being Element of C st c in dom ((f + g) || C) holds
((f + g) || C) . c = ((f || C) + (g || C)) . c

proof

let c be Element of C; :: thesis: ( c in dom ((f + g) || C) implies ((f + g) || C) . c = ((f || C) + (g || C)) . c )
assume A4: c in dom ((f + g) || C) ; :: thesis: ((f + g) || C) . c = ((f || C) + (g || C)) . c
then c in (dom (f + g)) /\ C by RELAT_1:61;
then A5: c in dom (f + g) by XBOOLE_0:def 4;
A6: c in (dom (f || C)) /\ (dom (g || C)) by A3, A4, VALUED_1:def 1;
then A7: c in dom (f || C) by XBOOLE_0:def 4;
A8: ((f + g) || C) . c = (f + g) . c by A4, FUNCT_1:47
.= (f . c) + (g . c) by A5, VALUED_1:def 1 ;
A9: c in dom (g || C) by A6, XBOOLE_0:def 4;
((f || C) + (g || C)) . c = ((f | C) . c) + ((g || C) . c) by A3, A4, VALUED_1:def 1
.= (f . c) + ((g | C) . c) by A7, FUNCT_1:47 ;
hence ((f + g) || C) . c = ((f || C) + (g || C)) . c by A8, A9, FUNCT_1:47; :: thesis: verum

end;

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