let D be non empty set ; :: thesis: for i being natural Number

for T being Tuple of i,D

for F being BinOp of D st F is having_a_unity holds

( F .: ((i |-> (the_unity_wrt F)),T) = T & F .: (T,(i |-> (the_unity_wrt F))) = T )

let i be natural Number ; :: thesis: for T being Tuple of i,D

for F being BinOp of D st F is having_a_unity holds

( F .: ((i |-> (the_unity_wrt F)),T) = T & F .: (T,(i |-> (the_unity_wrt F))) = T )

let T be Tuple of i,D; :: thesis: for F being BinOp of D st F is having_a_unity holds

( F .: ((i |-> (the_unity_wrt F)),T) = T & F .: (T,(i |-> (the_unity_wrt F))) = T )

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

assume A1: F is having_a_unity ; :: thesis: ( F .: ((i |-> (the_unity_wrt F)),T) = T & F .: (T,(i |-> (the_unity_wrt F))) = T )

for T being Tuple of i,D

for F being BinOp of D st F is having_a_unity holds

( F .: ((i |-> (the_unity_wrt F)),T) = T & F .: (T,(i |-> (the_unity_wrt F))) = T )

let i be natural Number ; :: thesis: for T being Tuple of i,D

for F being BinOp of D st F is having_a_unity holds

( F .: ((i |-> (the_unity_wrt F)),T) = T & F .: (T,(i |-> (the_unity_wrt F))) = T )

let T be Tuple of i,D; :: thesis: for F being BinOp of D st F is having_a_unity holds

( F .: ((i |-> (the_unity_wrt F)),T) = T & F .: (T,(i |-> (the_unity_wrt F))) = T )

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

assume A1: F is having_a_unity ; :: thesis: ( F .: ((i |-> (the_unity_wrt F)),T) = T & F .: (T,(i |-> (the_unity_wrt F))) = T )

per cases
( i = 0 or i <> 0 )
;

end;

suppose A2:
i = 0
; :: thesis: ( F .: ((i |-> (the_unity_wrt F)),T) = T & F .: (T,(i |-> (the_unity_wrt F))) = T )

then
T = <*> D
;

hence ( F .: ((i |-> (the_unity_wrt F)),T) = T & F .: (T,(i |-> (the_unity_wrt F))) = T ) by A2, Lm1; :: thesis: verum

end;hence ( F .: ((i |-> (the_unity_wrt F)),T) = T & F .: (T,(i |-> (the_unity_wrt F))) = T ) by A2, Lm1; :: thesis: verum

suppose
i <> 0
; :: thesis: ( F .: ((i |-> (the_unity_wrt F)),T) = T & F .: (T,(i |-> (the_unity_wrt F))) = T )

then reconsider C = Seg i as non empty set ;

T is Function of C,D by Lm4;

hence ( F .: ((i |-> (the_unity_wrt F)),T) = T & F .: (T,(i |-> (the_unity_wrt F))) = T ) by A1, Th43; :: thesis: verum

end;T is Function of C,D by Lm4;

hence ( F .: ((i |-> (the_unity_wrt F)),T) = T & F .: (T,(i |-> (the_unity_wrt F))) = T ) by A1, Th43; :: thesis: verum