let X be set ; :: thesis: for C being non empty set
for V being RealNormSpace
for f being PartFunc of C,V holds
( (- f) | X = - (f | X) & ||.f.|| | X = ||.(f | X).|| )
let C be non empty set ; :: thesis: for V being RealNormSpace
for f being PartFunc of C,V holds
( (- f) | X = - (f | X) & ||.f.|| | X = ||.(f | X).|| )
let V be RealNormSpace; :: thesis: for f being PartFunc of C,V holds
( (- f) | X = - (f | X) & ||.f.|| | X = ||.(f | X).|| )
let f be PartFunc of C,V; :: thesis: ( (- f) | X = - (f | X) & ||.f.|| | X = ||.(f | X).|| )

A1: now :: thesis: for c being Element of C st c in dom ((- f) | X) holds
((- f) | X) /. c = (- (f | X)) /. c

let c be Element of C; :: thesis: ( c in dom ((- f) | X) implies ((- f) | X) /. c = (- (f | X)) /. c )
assume A2: c in dom ((- f) | X) ; :: thesis: ((- f) | X) /. c = (- (f | X)) /. c
then A3: c in (dom (- f)) /\ X by RELAT_1:61;
then A4: c in X by XBOOLE_0:def 4;
A5: c in dom (- f) by A3, XBOOLE_0:def 4;
then c in dom f by Def5;
then c in (dom f) /\ X by A4, XBOOLE_0:def 4;
then A6: c in dom (f | X) by RELAT_1:61;
then A7: c in dom (- (f | X)) by Def5;
thus ((- f) | X) /. c = (- f) /. c by A2, PARTFUN2:15
.= - (f /. c) by A5, Def5
.= - ((f | X) /. c) by A6, PARTFUN2:15
.= (- (f | X)) /. c by A7, Def5 ; :: thesis: verum

end;

dom ((- f) | X) = (dom (- f)) /\ X by RELAT_1:61
.= (dom f) /\ X by Def5
.= dom (f | X) by RELAT_1:61
.= dom (- (f | X)) by Def5 ;
hence (- f) | X = - (f | X) by A1, PARTFUN2:1; :: thesis: ||.f.|| | X = ||.(f | X).||
A8: dom (||.f.|| | X) = (dom ||.f.||) /\ X by RELAT_1:61
.= (dom f) /\ X by NORMSP_0:def 3
.= dom (f | X) by RELAT_1:61
.= dom ||.(f | X).|| by NORMSP_0:def 3 ;

now :: thesis: for c being Element of C st c in dom (||.f.|| | X) holds
(||.f.|| | X) . c = ||.(f | X).|| . c

let c be Element of C; :: thesis: ( c in dom (||.f.|| | X) implies (||.f.|| | X) . c = ||.(f | X).|| . c )
assume A9: c in dom (||.f.|| | X) ; :: thesis: (||.f.|| | X) . c = ||.(f | X).|| . c
then A10: c in dom (f | X) by A8, NORMSP_0:def 3;
c in (dom ||.f.||) /\ X by A9, RELAT_1:61;
then A11: c in dom ||.f.|| by XBOOLE_0:def 4;
thus (||.f.|| | X) . c = ||.f.|| . c by A9, FUNCT_1:47
.= ||.(f /. c).|| by A11, NORMSP_0:def 3
.= ||.((f | X) /. c).|| by A10, PARTFUN2:15
.= ||.(f | X).|| . c by A8, A9, NORMSP_0:def 3 ; :: thesis: verum

end;

hence ||.f.|| | X = ||.(f | X).|| by A8, PARTFUN1:5; :: thesis: verum