let X be non empty set ; :: thesis: for Y being set

for F being BinOp of X

for f, g being Function of Y,X st F is commutative holds

F .: (f,g) = F .: (g,f)

let Y be set ; :: thesis: for F being BinOp of X

for f, g being Function of Y,X st F is commutative holds

F .: (f,g) = F .: (g,f)

let F be BinOp of X; :: thesis: for f, g being Function of Y,X st F is commutative holds

F .: (f,g) = F .: (g,f)

let f, g be Function of Y,X; :: thesis: ( F is commutative implies F .: (f,g) = F .: (g,f) )

assume A1: F is commutative ; :: thesis: F .: (f,g) = F .: (g,f)

for F being BinOp of X

for f, g being Function of Y,X st F is commutative holds

F .: (f,g) = F .: (g,f)

let Y be set ; :: thesis: for F being BinOp of X

for f, g being Function of Y,X st F is commutative holds

F .: (f,g) = F .: (g,f)

let F be BinOp of X; :: thesis: for f, g being Function of Y,X st F is commutative holds

F .: (f,g) = F .: (g,f)

let f, g be Function of Y,X; :: thesis: ( F is commutative implies F .: (f,g) = F .: (g,f) )

assume A1: F is commutative ; :: thesis: F .: (f,g) = F .: (g,f)

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

end;

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

end;

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

hence
F .: (f,g) = F .: (g,f)
by A2, Th38; :: thesis: verumlet y be Element of Y; :: thesis: (F .: (f,g)) . y = F . ((g . y),(f . y))

reconsider x1 = f . y, x2 = g . y as Element of X by A2, FUNCT_2:5;

thus (F .: (f,g)) . y = F . (x1,x2) by A2, Th37

.= F . ((g . y),(f . y)) by A1 ; :: thesis: verum

end;reconsider x1 = f . y, x2 = g . y as Element of X by A2, FUNCT_2:5;

thus (F .: (f,g)) . y = F . (x1,x2) by A2, Th37

.= F . ((g . y),(f . y)) by A1 ; :: thesis: verum