let D be non empty set ; :: thesis: for i being Nat
for T being Element of i -tuples_on 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 Nat; :: thesis: for T being Element of i -tuples_on 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 Element of i -tuples_on 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 )
set e = the_unity_wrt F;

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

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

then ( F .: (i |-> (the_unity_wrt F)),T = <*> D & T = <*> D & F .: T,(i |-> (the_unity_wrt F)) = <*> D ) by Lm2, FINSEQ_2:113;
hence ( F .: (i |-> (the_unity_wrt F)),T = T & F .: T,(i |-> (the_unity_wrt F)) = T ) ; :: 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 & i |-> (the_unity_wrt F) = (Seg i) --> (the_unity_wrt F) ) by Lm5;
hence ( F .: (i |-> (the_unity_wrt F)),T = T & F .: T,(i |-> (the_unity_wrt F)) = T ) by A1, Th44; :: thesis: verum

end;

end;