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 x1, x2 being Element of X st F is associative holds
F [:] (F [;] x1,f),x2 = F [;] x1,(F [:] f,x2)
let Y be set ; :: thesis: for F being BinOp of X
for f being Function of Y,X
for x1, x2 being Element of X st F is associative holds
F [:] (F [;] x1,f),x2 = F [;] x1,(F [:] f,x2)
let F be BinOp of X; :: thesis: for f being Function of Y,X
for x1, x2 being Element of X st F is associative holds
F [:] (F [;] x1,f),x2 = F [;] x1,(F [:] f,x2)
let f be Function of Y,X; :: thesis: for x1, x2 being Element of X st F is associative holds
F [:] (F [;] x1,f),x2 = F [;] x1,(F [:] f,x2)
let x1, x2 be Element of X; :: thesis: ( F is associative implies F [:] (F [;] x1,f),x2 = F [;] x1,(F [:] f,x2) )
assume A1: F is associative ; :: thesis: F [:] (F [;] x1,f),x2 = F [;] x1,(F [:] f,x2)

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

suppose Y = {} ; :: thesis: F [:] (F [;] x1,f),x2 = F [;] x1,(F [:] f,x2)

hence F [:] (F [;] x1,f),x2 = F [;] x1,(F [:] f,x2) ; :: thesis: verum

end;

suppose S: Y <> {} ; :: thesis: F [:] (F [;] x1,f),x2 = F [;] x1,(F [:] f,x2)

now

let y be Element of Y; :: thesis: (F [:] (F [;] x1,f),x2) . y = F . x1,((F [:] f,x2) . y)
reconsider x3 = f . y as Element of X by S, FUNCT_2:7;
thus (F [:] (F [;] x1,f),x2) . y = F . ((F [;] x1,f) . y),x2 by S, Th60
.= F . (F . x1,x3),x2 by S, Th66
.= F . x1,(F . x3,x2) by A1, BINOP_1:def 3
.= F . x1,((F [:] f,x2) . y) by S, Th60 ; :: thesis: verum

end;

hence F [:] (F [;] x1,f),x2 = F [;] x1,(F [:] f,x2) by S, Th67; :: thesis: verum

end;

end;