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,P,P1 st 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} holds
  P1 is Subset of (TOP-REAL 2)|P`
proof
  let s1,s2,t1,t2,P,P1;
  assume that
A1: 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} and
A2: P1 = {pa where pa is Point of TOP-REAL 2: s1<pa`1 & pa`1<s2 & t1 <pa
  `2 & pa`2 < t2};
  P1 c= [#]((TOP-REAL 2)|P`)by A1,A2,Th32;
  hence thesis;
end;
