let C, D be non empty set ; :: thesis: for f being Function of C,D

for F being BinOp of D st F is having_a_unity holds

F [;] ((the_unity_wrt F),f) = f

let f be Function of C,D; :: thesis: for F being BinOp of D st F is having_a_unity holds

F [;] ((the_unity_wrt F),f) = f

let F be BinOp of D; :: thesis: ( F is having_a_unity implies F [;] ((the_unity_wrt F),f) = f )

set e = the_unity_wrt F;

assume A1: F is having_a_unity ; :: thesis: F [;] ((the_unity_wrt F),f) = f

for F being BinOp of D st F is having_a_unity holds

F [;] ((the_unity_wrt F),f) = f

let f be Function of C,D; :: thesis: for F being BinOp of D st F is having_a_unity holds

F [;] ((the_unity_wrt F),f) = f

let F be BinOp of D; :: thesis: ( F is having_a_unity implies F [;] ((the_unity_wrt F),f) = f )

set e = the_unity_wrt F;

assume A1: F is having_a_unity ; :: thesis: F [;] ((the_unity_wrt F),f) = f

now :: thesis: for c being Element of C holds (F [;] ((the_unity_wrt F),f)) . c = f . c

hence
F [;] ((the_unity_wrt F),f) = f
by FUNCT_2:63; :: thesis: verumlet c be Element of C; :: thesis: (F [;] ((the_unity_wrt F),f)) . c = f . c

thus (F [;] ((the_unity_wrt F),f)) . c = F . ((the_unity_wrt F),(f . c)) by FUNCOP_1:53

.= f . c by A1, SETWISEO:15 ; :: thesis: verum

end;thus (F [;] ((the_unity_wrt F),f)) . c = F . ((the_unity_wrt F),(f . c)) by FUNCOP_1:53

.= f . c by A1, SETWISEO:15 ; :: thesis: verum