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 Th12
.=
(f [+] (c1 + c2)) . x
by A2, A3, Def37
;
verum