let X be non empty set ; for F being BinOp of X
for x being Element of X
for g being Function of X,X holds (F .: ((id X),g)) . x = F . (x,(g . x))
let F be BinOp of X; for x being Element of X
for g being Function of X,X holds (F .: ((id X),g)) . x = F . (x,(g . x))
let x be Element of X; for g being Function of X,X holds (F .: ((id X),g)) . x = F . (x,(g . x))
let g be Function of X,X; (F .: ((id X),g)) . x = F . (x,(g . x))
thus (F .: ((id X),g)) . x =
F . (((id X) . x),(g . x))
by Th37
.=
F . (x,(g . x))
; verum