let C be non empty set ; :: thesis: for V being non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-unital RLSStruct
for f1, f2 being PartFunc of C,V holds f1 - f2 = (- 1) (#) (f2 - f1)
let V be non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-unital RLSStruct ; :: thesis: for f1, f2 being PartFunc of C,V holds f1 - f2 = (- 1) (#) (f2 - f1)
let f1, f2 be PartFunc of C,V; :: thesis: f1 - f2 = (- 1) (#) (f2 - f1)
A1: dom (f1 - f2) = (dom f2) /\ (dom f1) by Def2
.= dom (f2 - f1) by Def2
.= dom ((- 1) (#) (f2 - f1)) by Def4 ;

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

A2: dom (f1 - f2) = (dom f2) /\ (dom f1) by Def2
.= dom (f2 - f1) by Def2 ;
let c be Element of C; :: thesis: ( c in dom (f1 - f2) implies (f1 - f2) /. c = ((- 1) (#) (f2 - f1)) /. c )
assume A3: c in dom (f1 - f2) ; :: thesis: (f1 - f2) /. c = ((- 1) (#) (f2 - f1)) /. c
thus (f1 - f2) /. c = (f1 /. c) - (f2 /. c) by A3, Def2
.= - ((f2 /. c) - (f1 /. c)) by RLVECT_1:33
.= (- 1) * ((f2 /. c) - (f1 /. c)) by RLVECT_1:16
.= (- 1) * ((f2 - f1) /. c) by A3, A2, Def2
.= ((- 1) (#) (f2 - f1)) /. c by A1, A3, Def4 ; :: thesis: verum

end;

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