let X, Y be non empty set ; :: thesis: for F being BinOp of X
for g, f being Function of Y,X
for x being Element of X st ( for y being Element of Y holds g . y = F . (f . y),x ) holds
g = F [:] f,x
let F be BinOp of X; :: thesis: for g, f being Function of Y,X
for x being Element of X st ( for y being Element of Y holds g . y = F . (f . y),x ) holds
g = F [:] f,x
let g, f be Function of Y,X; :: thesis: for x being Element of X st ( for y being Element of Y holds g . y = F . (f . y),x ) holds
g = F [:] f,x
let x be Element of X; :: thesis: ( ( for y being Element of Y holds g . y = F . (f . y),x ) implies g = F [:] f,x )
assume A1: for y being Element of Y holds g . y = F . (f . y),x ; :: thesis: g = F [:] f,x

now

let y be Element of Y; :: thesis: g . y = (F [:] f,x) . y
thus g . y = F . (f . y),x by A1
.= (F [:] f,x) . y by Th60 ; :: thesis: verum

end;

hence g = F [:] f,x by FUNCT_2:113; :: thesis: verum