let M be non empty set ; :: thesis: for V being ComplexNormSpace
for f being PartFunc of M, the carrier of V holds - f = (- 1r) (#) f
let V be ComplexNormSpace; :: thesis: for f being PartFunc of M, the carrier of V holds - f = (- 1r) (#) f
let f be PartFunc of M, the carrier of V; :: thesis: - f = (- 1r) (#) f
A1: dom (- f) = dom f by Def6
.= dom ((- 1r) (#) f) by Def4 ;

now

let c be Element of M; :: thesis: ( c in dom ((- 1r) (#) f) implies (- f) /. c = ((- 1r) (#) f) /. c )
assume A2: c in dom ((- 1r) (#) f) ; :: thesis: (- f) /. c = ((- 1r) (#) f) /. c
hence (- f) /. c = - (f /. c) by A1, Def6
.= (- 1r) * (f /. c) by CLVECT_1:4
.= ((- 1r) (#) f) /. c by A2, Def4 ;
:: thesis: verum

end;

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