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 - g) || C) = (dom (f - g)) /\ C by RELAT_1:90
.= ((dom f) /\ (dom g)) /\ C by VALUED_1:12 ;
a1: dom ((f || C) - (g || C)) = (dom (f | C)) /\ (dom (g | C)) by VALUED_1:12
.= ((dom f) /\ C) /\ (dom (g | C)) by RELAT_1:90
.= ((dom f) /\ C) /\ ((dom g) /\ C) by RELAT_1:90
.= (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) ;
then A2: 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 A3: 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_1:90;
then A4: c in dom (f - g) by XBOOLE_0:def 4;
A5: ((f - g) || C) . c = (f - g) . c by A3, FUNCT_1:70
.= (f . c) - (g . c) by A4, VALUED_1:13 ;
c in (dom (f || C)) /\ (dom (g || C)) by A2, A3, VALUED_1:12;
then A6: ( c in dom (f | C) & c in dom (g | C) ) by XBOOLE_0:def 4;
((f || C) - (g || C)) . c = ((f || C) . c) - ((g || C) . c) by A2, A3, VALUED_1:13
.= (f . c) - ((g | C) . c) by A6, FUNCT_1:70 ;
hence ((f - g) || C) . c = ((f || C) - (g || C)) . c by A5, A6, FUNCT_1:70; :: thesis: verum

end;

hence (f - g) || C = (f || C) - (g || C) by A1, a1, PARTFUN1:34, XBOOLE_1:16; :: thesis: verum