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,P2 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} &
  P2 = {pb where pb is Point of TOP-REAL 2:
  not( s1<=pb`1 & pb`1<=s2 & t1<=pb`2 & pb`2<=t2)} holds
  P2 is Subset of (TOP-REAL 2)|P`
proof
  let s1,s2,t1,t2,P,P2;
  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} and
A4: P2 = {pb where pb is Point of TOP-REAL 2: not( s1<=pb`1 & pb`1<=s2 &
  t1<=pb`2 & pb`2<=t2)};
  P2 c= [#]((TOP-REAL 2)|P`) by A1,A2,A3,A4,Th34;
  hence thesis;
end;
