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 [;] ((the_unity_wrt F),T) = 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 [;] ((the_unity_wrt F),T) = T
let T be Tuple of i,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 ;
hence F [;] ((the_unity_wrt F),T) = T by Lm2; :: 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 Lm4;
hence F [;] ((the_unity_wrt F),T) = T by A1, Th44; :: thesis: verum

end;

end;