let E, F, G be non empty set ; :: thesis: for f being Function of [:E,F:],G

for x, y being object st x in E & y in F holds

((curry' f) . y) . x = f . (x,y)

let f be Function of [:E,F:],G; :: thesis: for x, y being object st x in E & y in F holds

((curry' f) . y) . x = f . (x,y)

let x, y be object ; :: thesis: ( x in E & y in F implies ((curry' f) . y) . x = f . (x,y) )

assume that

A1: x in E and

A2: y in F ; :: thesis: ((curry' f) . y) . x = f . (x,y)

dom f = [:E,F:] by FUNCT_2:def 1;

then ex g being Function st

( (curry' f) . y = g & dom g = E & rng g c= rng f & ( for x being object st x in E holds

g . x = f . (x,y) ) ) by A2, FUNCT_5:32, ZFMISC_1:90;

hence ((curry' f) . y) . x = f . (x,y) by A1; :: thesis: verum

for x, y being object st x in E & y in F holds

((curry' f) . y) . x = f . (x,y)

let f be Function of [:E,F:],G; :: thesis: for x, y being object st x in E & y in F holds

((curry' f) . y) . x = f . (x,y)

let x, y be object ; :: thesis: ( x in E & y in F implies ((curry' f) . y) . x = f . (x,y) )

assume that

A1: x in E and

A2: y in F ; :: thesis: ((curry' f) . y) . x = f . (x,y)

dom f = [:E,F:] by FUNCT_2:def 1;

then ex g being Function st

( (curry' f) . y = g & dom g = E & rng g c= rng f & ( for x being object st x in E holds

g . x = f . (x,y) ) ) by A2, FUNCT_5:32, ZFMISC_1:90;

hence ((curry' f) . y) . x = f . (x,y) by A1; :: thesis: verum