theorem :: JORDAN12:35

for f1, f2, g1, g2 being special FinSequence of (TOP-REAL 2) st f1 ^' f2 is non constant standard special_circular_sequence & g1 ^' g2 is non constant standard special_circular_sequence & L~ f1 misses L~ g2 & L~ f2 misses L~ g1 & f1 ^' f2,g1 ^' g2 are_in_general_position holds
for p1, p2, q1, q2 being Point of (TOP-REAL 2) st f1 . 1 = p1 & f1 . (len f1) = p2 & g1 . 1 = q1 & g1 . (len g1) = q2 & f1 /. (len f1) = f2 /. 1 & g1 /. (len g1) = g2 /. 1 & p1 in (L~ f1) /\ (L~ f2) & q1 in (L~ g1) /\ (L~ g2) & ex C being Subset of (TOP-REAL 2) st
( C is_a_component_of (L~ (f1 ^' f2)) ` & q1 in C & q2 in C ) holds
ex C being Subset of (TOP-REAL 2) st
( C is_a_component_of (L~ (g1 ^' g2)) ` & p1 in C & p2 in C )