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 [;] (the_unity_wrt F),T = 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 [;] (the_unity_wrt F),T = 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 [;] (the_unity_wrt F),T = T
let F be BinOp of D; :: thesis: ( F is having_a_unity implies F [;] (the_unity_wrt F),T = T )
assume A1: F is having_a_unity ; :: thesis: F [;] (the_unity_wrt F),T = T

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

suppose i = 0 ; :: thesis: F [;] (the_unity_wrt F),T = T

then ( T = <*> D & F [;] (the_unity_wrt F),T = <*> D ) by Lm3, FINSEQ_2:113;
hence F [;] (the_unity_wrt F),T = T ; :: thesis: verum

end;

suppose i <> 0 ; :: thesis: F [;] (the_unity_wrt F),T = T

then reconsider C = Seg i as non empty set ;
T is Function of C,D by Lm5;
hence F [;] (the_unity_wrt F),T = T by A1, Th45; :: thesis: verum

end;

end;