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)))

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 <> {} )
;

end;

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

end;

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

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