let X, Y be non empty set ; :: thesis: for F being BinOp of X
for f being Function of Y,X
for y being Element of Y st F is idempotent holds
(F [:] (f,(f . y))) . y = f . y
let F be BinOp of X; :: thesis: for f being Function of Y,X
for y being Element of Y st F is idempotent holds
(F [:] (f,(f . y))) . y = f . y
let f be Function of Y,X; :: thesis: for y being Element of Y st F is idempotent holds
(F [:] (f,(f . y))) . y = f . y
let y be Element of Y; :: thesis: ( F is idempotent implies (F [:] (f,(f . y))) . y = f . y )
assume A1: F is idempotent ; :: thesis: (F [:] (f,(f . y))) . y = f . y
thus (F [:] (f,(f . y))) . y = F . ((f . y),(f . y)) by Th48
.= f . y by A1 ; :: thesis: verum