theorem Th30: :: JORDAN1:36

for s1, t1, s2, t2 being Real
for P, P1, P2 being Subset of (TOP-REAL 2) st s1 < s2 & t1 < t2 & P = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) } & P1 = { pa where pa is Point of (TOP-REAL 2) : ( s1 < pa `1 & pa `1 < s2 & t1 < pa `2 & pa `2 < t2 ) } & P2 = { pb where pb is Point of (TOP-REAL 2) : ( not s1 <= pb `1 or not pb `1 <= s2 or not t1 <= pb `2 or not pb `2 <= t2 ) } holds
( P ` = P1 \/ P2 & P ` <> {} & P1 misses P2 & ( for P1A, P2B being Subset of ((TOP-REAL 2) | (P `)) st P1A = P1 & P2B = P2 holds
( P1A is a_component & P2B is a_component ) ) )