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

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

let y be Element of Y; :: thesis: g . y = (F [;] (x,f)) . y
thus g . y = F . (x,(f . y)) by A1
.= (F [;] (x,f)) . y by Th53 ; :: thesis: verum

end;

hence g = F [;] (x,f) by FUNCT_2:63; :: thesis: verum