reserve GX,GY for non empty TopSpace,
  x,y for Point of GX,
  r,s for Real;
reserve Q,P1,P2 for Subset of TOP-REAL 2;
reserve P for Subset of TOP-REAL 2;
reserve w1,w2 for Point of TOP-REAL 2;
reserve pa,pb for Point of TOP-REAL 2,
  s1,t1,s2,t2,s,t,s3,t3,s4,t4,s5,t5,s6,t6, l,sa,sd,ta,td for Real;
reserve s1a,t1a,s2a,t2a,s3a,t3a,sb,tb,sc,tc for Real;

theorem
  for s1,s2,t1,t2 for P 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} holds P is Jordan
proof
  let s1,s2,t1,t2;
  let P be Subset of TOP-REAL 2;
  assume that
A1: s1<s2 and
A2: t1<t2 and
A3: 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};
  reconsider P1= {pa where pa is Point of TOP-REAL 2:
  s1<pa`1 & pa`1<s2 & t1 <pa`2 & pa`2 < t2},
  P2= {pc where pc is Point of TOP-REAL 2:
  not( s1<=pc`1 & pc`1<=s2 & t1<=pc`2 & pc`2<=t2)}
  as Subset of TOP-REAL 2 by Th23,Th24;
  reconsider PC = P` as Subset of TOP-REAL 2;
A4: P1= {pa where pa is Point of TOP-REAL 2: s1<pa`1 & pa`1<s2 & t1 <pa`2 &
  pa`2 < t2};
A5: P2= {pc where pc is Point of TOP-REAL 2: not( s1<=pc`1 & pc`1<=s2 & t1<=
  pc`2 & pc`2<=t2)};
A6: PC=P1 \/ P2 by A1,A2,A3,Th30;
A7: PC<>{} by A1,A2,A3,A4,A5,Th30;
A8: P1 misses P2 by A1,A2,A3,Th30;
A9: P=(Cl P1)\P1 by A1,A2,A3,A5,Th31;
A10: P=(Cl P2)\P2 by A1,A2,A3,A4,Th31;
  for P1A,P2B be Subset of (TOP-REAL 2)|P` holds P1A=P1 & P2B=P2 implies
  P1A is a_component & P2B is a_component by A1,A2,A3,Th30;
  hence thesis by A6,A7,A8,A9,A10;
end;
