intros.
induction H0.
{
apply ltp_to_lep.
apply (pythagoras_obtuse O E E' A B C AC BC AB AC2 BC2 AB2 S2); auto.
}
{
assert(HH:= pythagoras O E E' A B C AC BC AB AC2 BC2 AB2 H H0 H1 H2 H3 H4 H5 H6).
assert(S2 = AB2).
{
apply (sum_uniqueness O E E' AC2 BC2); auto.
}
subst S2.
apply leP_refl.
}
Qed.

Lemma pythagoras_acute : forall O E E' A B C AC BC AB AC2 BC2 AB2 S2,
O<>E -> Acute A C B ->
Length O E E' A B AB -> Length O E E' A C AC -> Length O E E' B C BC ->
Prod O E E' AC AC AC2 -> Prod O E E' BC BC BC2 -> Prod O E E' AB AB AB2 ->
Sum  O E E' AC2 BC2 S2 -> LtP O E E' AB2 S2.