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

for x being Element of X

for g being Function of X,X holds (F .: (g,(id X))) . x = F . ((g . x),x)

let F be BinOp of X; :: thesis: for x being Element of X

for g being Function of X,X holds (F .: (g,(id X))) . x = F . ((g . x),x)

let x be Element of X; :: thesis: for g being Function of X,X holds (F .: (g,(id X))) . x = F . ((g . x),x)

let g be Function of X,X; :: thesis: (F .: (g,(id X))) . x = F . ((g . x),x)

thus (F .: (g,(id X))) . x = F . ((g . x),((id X) . x)) by Th37

.= F . ((g . x),x) ; :: thesis: verum

for x being Element of X

for g being Function of X,X holds (F .: (g,(id X))) . x = F . ((g . x),x)

let F be BinOp of X; :: thesis: for x being Element of X

for g being Function of X,X holds (F .: (g,(id X))) . x = F . ((g . x),x)

let x be Element of X; :: thesis: for g being Function of X,X holds (F .: (g,(id X))) . x = F . ((g . x),x)

let g be Function of X,X; :: thesis: (F .: (g,(id X))) . x = F . ((g . x),x)

thus (F .: (g,(id X))) . x = F . ((g . x),((id X) . x)) by Th37

.= F . ((g . x),x) ; :: thesis: verum