let X be set ; for Y being complex-functions-membered set
for c1, c2 being Complex
for f being PartFunc of X,Y holds (f [-] c1) [-] c2 = f [-] (c1 + c2)
let Y be complex-functions-membered set ; for c1, c2 being Complex
for f being PartFunc of X,Y holds (f [-] c1) [-] c2 = f [-] (c1 + c2)
let c1, c2 be Complex; for f being PartFunc of X,Y holds (f [-] c1) [-] c2 = f [-] (c1 + c2)
let f be PartFunc of X,Y; (f [-] c1) [-] c2 = f [-] (c1 + c2)
set f1 = f [-] c1;
A1:
dom ((f [-] c1) [-] c2) = dom (f [-] c1)
by Def37;
dom (f [-] c1) = dom f
by Def37;
hence A2:
dom ((f [-] c1) [-] c2) = dom (f [-] (c1 + c2))
by A1, Def37; FUNCT_1:def 11 for b1 being object holds
( not b1 in dom ((f [-] c1) [-] c2) or ((f [-] c1) [-] c2) . b1 = (f [-] (c1 + c2)) . b1 )
let x be object ; ( not x in dom ((f [-] c1) [-] c2) or ((f [-] c1) [-] c2) . x = (f [-] (c1 + c2)) . x )
assume A3:
x in dom ((f [-] c1) [-] c2)
; ((f [-] c1) [-] c2) . x = (f [-] (c1 + c2)) . x
hence ((f [-] c1) [-] c2) . x =
((f [-] c1) . x) - c2
by Def37
.=
((f . x) - c1) - c2
by A1, A3, Def37
.=
(f . x) - (c1 + c2)
by Th15
.=
(f [-] (c1 + c2)) . x
by A2, A3, Def37
;
verum