let M be non empty set ; :: thesis: for V being ComplexNormSpace
for f1, f2 being PartFunc of M,the carrier of V holds f1 - f2 = (- 1r ) (#) (f2 - f1)
let V be ComplexNormSpace; :: thesis: for f1, f2 being PartFunc of M,the carrier of V holds f1 - f2 = (- 1r ) (#) (f2 - f1)
let f1, f2 be PartFunc of M,the carrier of V; :: thesis: f1 - f2 = (- 1r ) (#) (f2 - f1)
A1: dom (f1 - f2) = (dom f2) /\ (dom f1) by Def2
.= dom (f2 - f1) by Def2
.= dom ((- 1r ) (#) (f2 - f1)) by Def4 ;

now

let x be Element of M; :: thesis: ( x in dom (f1 - f2) implies (f1 - f2) /. x = ((- 1r ) (#) (f2 - f1)) /. x )
assume A2: x in dom (f1 - f2) ; :: thesis: (f1 - f2) /. x = ((- 1r ) (#) (f2 - f1)) /. x
A3: dom (f1 - f2) = (dom f2) /\ (dom f1) by Def2
.= dom (f2 - f1) by Def2 ;
thus (f1 - f2) /. x = (f1 /. x) - (f2 /. x) by A2, Def2
.= - ((f2 /. x) - (f1 /. x)) by RLVECT_1:47
.= (- 1r ) * ((f2 /. x) - (f1 /. x)) by CLVECT_1:4
.= (- 1r ) * ((f2 - f1) /. x) by A2, A3, Def2
.= ((- 1r ) (#) (f2 - f1)) /. x by A1, A2, Def4 ; :: thesis: verum

end;

hence f1 - f2 = (- 1r ) (#) (f2 - f1) by A1, PARTFUN2:3; :: thesis: verum