let n be Nat; :: thesis: for T being non empty TopSpace
for f being Function of T,(TOP-REAL n)
for g being Function of T,R^1 holds f <#> g = f <##> (incl g,n)
let T be non empty TopSpace; :: thesis: for f being Function of T,(TOP-REAL n)
for g being Function of T,R^1 holds f <#> g = f <##> (incl g,n)
let f be Function of T,(TOP-REAL n); :: thesis: for g being Function of T,R^1 holds f <#> g = f <##> (incl g,n)
let g be Function of T,R^1 ; :: thesis: f <#> g = f <##> (incl g,n)
set I = incl g,n;
reconsider h = f <#> g as Function of T,(TOP-REAL n) by Th45;
reconsider G = f <##> (incl g,n) as Function of T,(TOP-REAL n) by Th41;
h = G

proof

let t be Point of T; :: according to FUNCT_2:def 9 :: thesis: h . t = G . t
A1: dom h = the carrier of T by FUNCT_2:def 1;
A2: (f . t) (#) ((incl g,n) . t) = (f . t) (#) (g . t)

proof

A3: ( dom (f . t) = Seg n & dom ((incl g,n) . t) = Seg n ) by EUCLID:3;
A4: dom ((f . t) (#) ((incl g,n) . t)) = (dom (f . t)) /\ (dom ((incl g,n) . t)) by VALUED_1:def 4
.= Seg n by A3 ;
hence dom ((f . t) (#) ((incl g,n) . t)) = dom ((f . t) (#) (g . t)) by EUCLID:3; :: according to FUNCT_1:def 17 :: thesis: for b₁ being set holds
( not b₁ in proj1 ((f . t) (#) ((incl g,n) . t)) or ((f . t) (#) ((incl g,n) . t)) . b₁ = ((f . t) (#) (g . t)) . b₁ )
let x be set ; :: thesis: ( not x in proj1 ((f . t) (#) ((incl g,n) . t)) or ((f . t) (#) ((incl g,n) . t)) . x = ((f . t) (#) (g . t)) . x )
assume A5: x in dom ((f . t) (#) ((incl g,n) . t)) ; :: thesis: ((f . t) (#) ((incl g,n) . t)) . x = ((f . t) (#) (g . t)) . x
hence ((f . t) (#) ((incl g,n) . t)) . x = ((f . t) . x) * (((incl g,n) . t) . x) by VALUED_1:def 4
.= ((f . t) . x) * (g . t) by A4, A5, Th47
.= ((f . t) (#) (g . t)) . x by VALUED_1:6 ;
:: thesis: verum

end;

dom G = the carrier of T by FUNCT_2:def 1;
hence G . t = (f . t) (#) ((incl g,n) . t) by VALUED_2:def 47
.= h . t by A1, A2, VALUED_2:def 43 ;
:: thesis: verum

end;

hence f <#> g = f <##> (incl g,n) ; :: thesis: verum