let C be non empty set ; :: thesis: for f1, f2 being PartFunc of C,COMPLEX holds f1 - f2 = (- 1r) (#) (f2 - f1)
let f1, f2 be PartFunc of C,COMPLEX; :: thesis: f1 - f2 = (- 1r) (#) (f2 - f1)
A1: dom (f1 - f2) = (dom f2) /\ (dom f1) by Th4
.= dom (f2 - f1) by Th4
.= dom ((- 1r) (#) (f2 - f1)) by Th7 ;

now

A2: dom (f1 - f2) = (dom f2) /\ (dom f1) by Th4
.= dom (f2 - f1) by Th4 ;
let c be Element of C; :: thesis: ( c in dom (f1 - f2) implies (f1 - f2) /. c = ((- 1r) (#) (f2 - f1)) /. c )
assume A3: c in dom (f1 - f2) ; :: thesis: (f1 - f2) /. c = ((- 1r) (#) (f2 - f1)) /. c
thus (f1 - f2) /. c = (f1 /. c) - (f2 /. c) by A3, Th4
.= (- 1) * ((f2 /. c) - (f1 /. c))
.= (- 1r) * ((f2 - f1) /. c) by A3, A2, Th4, COMPLEX1:def 4
.= ((- 1r) (#) (f2 - f1)) /. c by A1, A3, Th7 ; :: thesis: verum

end;

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