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 )

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

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;

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;

end;