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

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

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

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

end;

suppose A2: Y <> {} ; :: thesis: F [:] f,(F . x1,x2) = F [:] (F [:] f,x1),x2

now

let y be Element of Y; :: thesis: (F [:] f,(F . x1,x2)) . y = F . ((F [:] f,x1) . y),x2
reconsider x3 = f . y as Element of X by A2, FUNCT_2:7;
thus (F [:] f,(F . x1,x2)) . y = F . (f . y),(F . x1,x2) by A2, Th60
.= F . (F . x3,x1),x2 by A1, BINOP_1:def 3
.= F . ((F [:] f,x1) . y),x2 by A2, Th60 ; :: thesis: verum

end;

hence F [:] f,(F . x1,x2) = F [:] (F [:] f,x1),x2 by A2, Th61; :: thesis: verum

end;

end;