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 .: ((C --> (the_unity_wrt F)),f) = f & F .: (f,(C --> (the_unity_wrt F))) = f )

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

( F .: ((C --> (the_unity_wrt F)),f) = f & F .: (f,(C --> (the_unity_wrt F))) = f )

let F be BinOp of D; :: thesis: ( F is having_a_unity implies ( F .: ((C --> (the_unity_wrt F)),f) = f & F .: (f,(C --> (the_unity_wrt F))) = f ) )

set e = the_unity_wrt F;

reconsider g = C --> (the_unity_wrt F) as Function of C,D ;

assume A1: F is having_a_unity ; :: thesis: ( F .: ((C --> (the_unity_wrt F)),f) = f & F .: (f,(C --> (the_unity_wrt F))) = f )

for F being BinOp of D st F is having_a_unity holds

( F .: ((C --> (the_unity_wrt F)),f) = f & F .: (f,(C --> (the_unity_wrt F))) = f )

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

( F .: ((C --> (the_unity_wrt F)),f) = f & F .: (f,(C --> (the_unity_wrt F))) = f )

let F be BinOp of D; :: thesis: ( F is having_a_unity implies ( F .: ((C --> (the_unity_wrt F)),f) = f & F .: (f,(C --> (the_unity_wrt F))) = f ) )

set e = the_unity_wrt F;

reconsider g = C --> (the_unity_wrt F) as Function of C,D ;

assume A1: F is having_a_unity ; :: thesis: ( F .: ((C --> (the_unity_wrt F)),f) = f & F .: (f,(C --> (the_unity_wrt F))) = f )

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

hence
F .: ((C --> (the_unity_wrt F)),f) = f
by FUNCT_2:63; :: thesis: F .: (f,(C --> (the_unity_wrt F))) = flet c be Element of C; :: thesis: (F .: (g,f)) . c = f . c

thus (F .: (g,f)) . c = F . ((g . c),(f . c)) by FUNCOP_1:37

.= F . ((the_unity_wrt F),(f . c))

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

end;thus (F .: (g,f)) . c = F . ((g . c),(f . c)) by FUNCOP_1:37

.= F . ((the_unity_wrt F),(f . c))

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

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

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

thus (F .: (f,g)) . c = F . ((f . c),(g . c)) by FUNCOP_1:37

.= F . ((f . c),(the_unity_wrt F))

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

end;thus (F .: (f,g)) . c = F . ((f . c),(g . c)) by FUNCOP_1:37

.= F . ((f . c),(the_unity_wrt F))

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