let X, Y be non empty set ; :: thesis: for F being BinOp of X

for f being Function of Y,X

for x being Element of X

for y being Element of Y holds (F [:] (f,x)) . y = F . ((f . y),x)

let F be BinOp of X; :: thesis: for f being Function of Y,X

for x being Element of X

for y being Element of Y holds (F [:] (f,x)) . y = F . ((f . y),x)

let f be Function of Y,X; :: thesis: for x being Element of X

for y being Element of Y holds (F [:] (f,x)) . y = F . ((f . y),x)

let x be Element of X; :: thesis: for y being Element of Y holds (F [:] (f,x)) . y = F . ((f . y),x)

let y be Element of Y; :: thesis: (F [:] (f,x)) . y = F . ((f . y),x)

dom (F [:] (f,x)) = Y by FUNCT_2:def 1;

hence (F [:] (f,x)) . y = F . ((f . y),x) by Th27; :: thesis: verum

for f being Function of Y,X

for x being Element of X

for y being Element of Y holds (F [:] (f,x)) . y = F . ((f . y),x)

let F be BinOp of X; :: thesis: for f being Function of Y,X

for x being Element of X

for y being Element of Y holds (F [:] (f,x)) . y = F . ((f . y),x)

let f be Function of Y,X; :: thesis: for x being Element of X

for y being Element of Y holds (F [:] (f,x)) . y = F . ((f . y),x)

let x be Element of X; :: thesis: for y being Element of Y holds (F [:] (f,x)) . y = F . ((f . y),x)

let y be Element of Y; :: thesis: (F [:] (f,x)) . y = F . ((f . y),x)

dom (F [:] (f,x)) = Y by FUNCT_2:def 1;

hence (F [:] (f,x)) . y = F . ((f . y),x) by Th27; :: thesis: verum