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 Th22:
  for s1,s2,t1,t2 holds {qc where qc is Point of TOP-REAL 2:
  not (s1<=qc`1 & qc`1<=s2 & t1<=qc`2 & qc`2<=t2)} =
  {|[sb,tb]| : not (s1<=sb & sb<=s2 & t1<=tb & tb<=t2)}
proof
  let s1,s2,t1,t2;
  now
    let x be object;
A1: now
      assume x in {qc where qc is Point of TOP-REAL 2:
      not (s1<=qc`1 & qc`1<=s2 & t1<=qc`2 & qc`2<=t2)};
      then consider q being Point of TOP-REAL 2 such that
A2:   q=x and
A3:   not (s1<=q`1 & q`1<=s2 & t1<=q`2 & q`2<=t2);
      |[q`1,q`2]|=x by A2,EUCLID:53;
      hence
      x in {|[s2a,t2a]| : not (s1<=s2a & s2a<=s2 & t1<=t2a & t2a<=t2)} by A3;
    end;
    now
      assume x in {|[sb,tb]| : not (s1<=sb & sb<=s2 & t1<=tb & tb<=t2)};
      then consider sb,tb being Real such that
A4:   |[sb,tb]|=x and
A5:   not (s1<=sb & sb<=s2 & t1<=tb & tb<=t2);
      set qa=|[sb,tb]|;
      not (s1<=qa`1 & qa`1<=s2 & t1<=qa`2 & qa`2<=t2) by A5,EUCLID:52;
      hence x in {qc where qc is Point of TOP-REAL 2:
      not (s1<=qc`1 & qc`1<=s2 & t1<=qc`2 & qc`2<=t2)} by A4;
    end;
    hence x in {qc where qc is Point of TOP-REAL 2:
    not (s1<=qc`1 & qc`1<=s2 & t1<=qc`2 & qc`2<=t2)} iff x in {|[sb,tb]| :
    not (s1<=sb & sb<=s2 & t1<=tb & tb<=t2)} by A1;
  end;
  hence thesis by TARSKI:2;
end;
