
theorem T1:
for L being non empty 1-sorted
for f being Function of L,L holds f`^0 = id L & f`^1 = f & f`^2 = f * f
proof
let R be non empty 1-sorted, f be Function of R,R;
consider F being Funcs(R,R)-valued sequence such that
A: f`^0 = F.0 & F.0 = id R &
   for i being Nat holds F.(i+1) = (F.i) * f by dd;
thus f`^0 = id R by A;
thus f`^1 = f;
consider F being Funcs(R,R)-valued sequence such that
A: f`^2 = F.2 & F.0 = id R &
   for i being Nat holds F.(i+1) = (F.i) * f by dd;
F.(0+1) = (id R) * f by A .= f; then
F.(1+1) = f * f by A;
hence thesis by A;
end;
