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 Th20:
  for s1,t1,s2,t2,P,Q st P = { |[ sa,ta ]|:s1<sa & sa<s2 & t1<ta & ta<t2} &
  Q = { |[ sb,tb ]|:not (s1<=sb & sb<=s2 & t1<=tb & tb<=t2)} holds P misses Q
proof
  let s1,t1,s2,t2,P,Q;
  assume that
A1: P = { |[ sa,ta ]|:s1<sa & sa<s2 & t1<ta & ta<t2} and
A2: Q = { |[ sb,tb ]|:not (s1<=sb & sb<=s2 & t1<=tb & tb<=t2)};
  assume not thesis;
  then consider x be object such that
A3: x in P and
A4: x in Q by XBOOLE_0:3;
  consider sa,ta such that
A5: |[sa,ta]|=x and
A6: s1<sa and
A7: sa<s2 and
A8: t1<ta and
A9: ta<t2 by A1,A3;
A10: ex sb,tb st ( |[sb,tb]|=x)&( not (s1<=sb & sb<=s2 & t1<=tb
  & tb<=t2)) by A2,A4;
  set p= |[sa,ta]|;
A11: p`1=sa by EUCLID:52;
  p`2=ta by EUCLID:52;
  hence contradiction by A5,A6,A7,A8,A9,A10,A11,EUCLID:52;
end;
