let D be non empty set ; :: thesis:
let i be natural Number ; :: thesis:
let T1, T2 be Tuple of i,D; :: thesis:
let F be BinOp of D; :: thesis:
assume A1: ( F is associative & F is having_an_inverseOp & F is having_a_unity & F .: (T1,T2) = i |-> (the_unity_wrt F) ) ; :: thesis:

suppose i = 0 ; :: thesis:

then ( T1 = <*> D & T2 = <*> D ) ;
hence ( T1 = (the_inverseOp_wrt F) * T2 & (the_inverseOp_wrt F) * T1 = T2 ) ; :: thesis:

end;

suppose i <> 0 ; :: thesis:

then reconsider C = Seg i as non empty set ;
( T1 is Function of C,D & T2 is Function of C,D ) by Lm4;
hence ( T1 = (the_inverseOp_wrt F) * T2 & (the_inverseOp_wrt F) * T1 = T2 ) by A1, Th72; :: thesis:

end;

end;