let X be set ; :: thesis: for Y being complex-functions-membered set
for f being PartFunc of X,Y
for g, h being complex-valued Function holds (f <+> g) <+> h = f <+> (g + h)
let Y be complex-functions-membered set ; :: thesis: for f being PartFunc of X,Y
for g, h being complex-valued Function holds (f <+> g) <+> h = f <+> (g + h)
let f be PartFunc of X,Y; :: thesis: for g, h being complex-valued Function holds (f <+> g) <+> h = f <+> (g + h)
let g, h be complex-valued Function; :: thesis: (f <+> g) <+> h = f <+> (g + h)
set f1 = f <+> g;
A1: dom (g + h) = (dom g) /\ (dom h) by VALUED_1:def 1;
A2: dom ((f <+> g) <+> h) = (dom (f <+> g)) /\ (dom h) by Def41;
( dom (f <+> g) = (dom f) /\ (dom g) & dom (f <+> (g + h)) = (dom f) /\ (dom (g + h)) ) by Def41;
hence A3: dom ((f <+> g) <+> h) = dom (f <+> (g + h)) by A2, A1, XBOOLE_1:16; :: according to FUNCT_1:def 11 :: thesis: for b₁ being set holds
( not b₁ in proj1 ((f <+> g) <+> h) or ((f <+> g) <+> h) . b₁ = (f <+> (g + h)) . b₁ )
let x be set ; :: thesis: ( not x in proj1 ((f <+> g) <+> h) or ((f <+> g) <+> h) . x = (f <+> (g + h)) . x )
assume A4: x in dom ((f <+> g) <+> h) ; :: thesis: ((f <+> g) <+> h) . x = (f <+> (g + h)) . x
then A5: x in dom (f <+> g) by A2, XBOOLE_0:def 4;
A6: x in dom (g + h) by A3, A4, XBOOLE_0:def 4;
thus ((f <+> g) <+> h) . x = ((f <+> g) . x) + (h . x) by A4, Def41
.= ((f . x) + (g . x)) + (h . x) by A5, Def41
.= (f . x) + ((g . x) + (h . x)) by Th12
.= (f . x) + ((g + h) . x) by A6, VALUED_1:def 1
.= (f <+> (g + h)) . x by A3, A4, Def41 ; :: thesis: verum