let M be non empty set ; :: thesis: for V being ComplexNormSpace
for f1, f2 being PartFunc of M,V holds f1 - f2 = (- 1r) (#) (f2 - f1)
let V be ComplexNormSpace; :: thesis: for f1, f2 being PartFunc of M,V holds f1 - f2 = (- 1r) (#) (f2 - f1)
let f1, f2 be PartFunc of M,V; :: thesis: f1 - f2 = (- 1r) (#) (f2 - f1)
A1: dom (f1 - f2) = (dom f2) /\ (dom f1) by VFUNCT_1:def 2
.= dom (f2 - f1) by VFUNCT_1:def 2
.= dom ((- 1r) (#) (f2 - f1)) by Def2 ;

now :: thesis: for x being Element of M st x in dom (f1 - f2) holds
(f1 - f2) /. x = ((- 1r) (#) (f2 - f1)) /. x

A2: dom (f1 - f2) = (dom f2) /\ (dom f1) by VFUNCT_1:def 2
.= dom (f2 - f1) by VFUNCT_1:def 2 ;
let x be Element of M; :: thesis: ( x in dom (f1 - f2) implies (f1 - f2) /. x = ((- 1r) (#) (f2 - f1)) /. x )
assume A3: x in dom (f1 - f2) ; :: thesis: (f1 - f2) /. x = ((- 1r) (#) (f2 - f1)) /. x
thus (f1 - f2) /. x = (f1 /. x) - (f2 /. x) by A3, VFUNCT_1:def 2
.= - ((f2 /. x) - (f1 /. x)) by RLVECT_1:33
.= (- 1r) * ((f2 /. x) - (f1 /. x)) by CLVECT_1:3
.= (- 1r) * ((f2 - f1) /. x) by A3, A2, VFUNCT_1:def 2
.= ((- 1r) (#) (f2 - f1)) /. x by A1, A3, Def2 ; :: thesis: verum

end;

hence f1 - f2 = (- 1r) (#) (f2 - f1) by A1, PARTFUN2:1; :: thesis: verum