let X be non empty set ; :: thesis: for Y being set
for F being BinOp of X
for f, g, h being Function of Y,X st F is associative holds
F .: ((F .: (f,g)),h) = F .: (f,(F .: (g,h)))
let Y be set ; :: thesis: for F being BinOp of X
for f, g, h being Function of Y,X st F is associative holds
F .: ((F .: (f,g)),h) = F .: (f,(F .: (g,h)))
let F be BinOp of X; :: thesis: for f, g, h being Function of Y,X st F is associative holds
F .: ((F .: (f,g)),h) = F .: (f,(F .: (g,h)))
let f, g, h be Function of Y,X; :: thesis: ( F is associative implies F .: ((F .: (f,g)),h) = F .: (f,(F .: (g,h))) )
assume A1: F is associative ; :: thesis: F .: ((F .: (f,g)),h) = F .: (f,(F .: (g,h)))

per cases ( Y = {} or Y <> {} ) ;

suppose Y = {} ; :: thesis: F .: ((F .: (f,g)),h) = F .: (f,(F .: (g,h)))

hence F .: ((F .: (f,g)),h) = F .: (f,(F .: (g,h))) ; :: thesis: verum

end;

suppose A2: Y <> {} ; :: thesis: F .: ((F .: (f,g)),h) = F .: (f,(F .: (g,h)))

now :: thesis: for y being Element of Y holds (F .: ((F .: (f,g)),h)) . y = F . ((f . y),((F .: (g,h)) . y))

let y be Element of Y; :: thesis: (F .: ((F .: (f,g)),h)) . y = F . ((f . y),((F .: (g,h)) . y))
reconsider x1 = f . y, x2 = g . y, x3 = h . y as Element of X by A2, FUNCT_2:5;
thus (F .: ((F .: (f,g)),h)) . y = F . (((F .: (f,g)) . y),(h . y)) by A2, Th37
.= F . ((F . ((f . y),(g . y))),(h . y)) by A2, Th37
.= F . (x1,(F . (x2,x3))) by A1
.= F . ((f . y),((F .: (g,h)) . y)) by A2, Th37 ; :: thesis: verum

end;

hence F .: ((F .: (f,g)),h) = F .: (f,(F .: (g,h))) by A2, Th38; :: thesis: verum

end;

end;