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

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

suppose Y = {} ; :: thesis: F [;] x,f = F [:] f,x

hence F [;] x,f = F [:] f,x ; :: thesis: verum

end;

suppose A2: Y <> {} ; :: thesis: F [;] x,f = F [:] f,x

now

let y be Element of Y; :: thesis: (F [;] x,f) . y = F . (f . y),x
reconsider x1 = f . y as Element of X by A2, FUNCT_2:7;
thus (F [;] x,f) . y = F . x,x1 by A2, Th66
.= F . (f . y),x by A1, BINOP_1:def 2 ; :: thesis: verum

end;

hence F [;] x,f = F [:] f,x by A2, Th61; :: thesis: verum

end;

end;