let X, Y be non empty set ; :: thesis: for Z being non empty RelStr
for S being non empty SubRelStr of Z |^ [:X,Y:]
for T being non empty SubRelStr of (Z |^ Y) |^ X
for f being Function of T,S st f is uncurrying & f is one-to-one & f is onto holds
f " is currying
let Z be non empty RelStr ; :: thesis: for S being non empty SubRelStr of Z |^ [:X,Y:]
for T being non empty SubRelStr of (Z |^ Y) |^ X
for f being Function of T,S st f is uncurrying & f is one-to-one & f is onto holds
f " is currying
let S be non empty SubRelStr of Z |^ [:X,Y:]; :: thesis: for T being non empty SubRelStr of (Z |^ Y) |^ X
for f being Function of T,S st f is uncurrying & f is one-to-one & f is onto holds
f " is currying
let T be non empty SubRelStr of (Z |^ Y) |^ X; :: thesis: for f being Function of T,S st f is uncurrying & f is one-to-one & f is onto holds
f " is currying
let f be Function of T,S; :: thesis: ( f is uncurrying & f is one-to-one & f is onto implies f " is currying )
A1: ( Funcs (X, the carrier of (Z |^ Y)) = the carrier of ((Z |^ Y) |^ X) & Funcs (Y, the carrier of Z) = the carrier of (Z |^ Y) ) by YELLOW_1:28;
assume A2: ( f is uncurrying & f is one-to-one & f is onto ) ; :: thesis: f " is currying
then A3: rng f = the carrier of S by FUNCT_2:def 3;
A4: f " = f " by A2, TOPS_2:def 4;
A5: Funcs ([:X,Y:], the carrier of Z) = the carrier of (Z |^ [:X,Y:]) by YELLOW_1:28;

hereby :: according to WAYBEL27:def 2 :: thesis: for b₁ being set holds
( not b₁ in proj1 (f ") or (f ") . b₁ = curry b₁ )

let x be set ; :: thesis: ( x in dom (f ") implies ( x is Function & proj1 x is Relation ) )
assume x in dom (f ") ; :: thesis: ( x is Function & proj1 x is Relation )
then x is Element of (Z |^ [:X,Y:]) by YELLOW_0:58;
then reconsider g = x as Function of [:X,Y:], the carrier of Z by A5, FUNCT_2:66;
dom g = proj1 g ;
hence ( x is Function & proj1 x is Relation ) ; :: thesis: verum

end;

let g be Function; :: thesis: ( not g in proj1 (f ") or (f ") . g = curry g )
assume g in dom (f ") ; :: thesis: (f ") . g = curry g
then consider h being object such that
A6: h in dom f and
A7: g = f . h by A3, FUNCT_1:def 3;
reconsider h = h as Function by A6;
h is Element of ((Z |^ Y) |^ X) by A6, YELLOW_0:58;
then h is Function of X,(Funcs (Y, the carrier of Z)) by A1, FUNCT_2:66;
then A8: rng h c= Funcs (Y, the carrier of Z) by RELAT_1:def 19;
g = uncurry h by A2, A6, A7;
then curry g = h by A8, FUNCT_5:48;
hence (f ") . g = curry g by A2, A4, A6, A7, FUNCT_1:32; :: thesis: verum