let f, g be PartFunc of REAL,REAL; for C being non empty Subset of REAL holds (f - g) || C = (f || C) - (g || C)
let C be non empty Subset of REAL; (f - g) || C = (f || C) - (g || C)
A1: dom ((f || C) - (g || C)) =
(dom (f | C)) /\ (dom (g | C))
by VALUED_1:12
.=
((dom f) /\ C) /\ (dom (g | C))
by RELAT_1:61
.=
((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_1:12
;
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;
( c in dom ((f - g) || C) implies ((f - g) || C) . c = ((f || C) - (g || C)) . c )
assume A4:
c in dom ((f - g) || C)
;
((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_1:12;
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_1:13
;
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_1:13
.=
(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;
verum
end;
hence
(f - g) || C = (f || C) - (g || C)
by A2, A1, PARTFUN1:5, XBOOLE_1:16; verum