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 Th21:
  for s1,s2,t1,t2 be Real holds
  {p where p is Point of TOP-REAL 2:s1<p`1 & p`1<s2 & t1<p`2 & p`2<t2} =
  {|[sa,ta]|:s1<sa & sa<s2 & t1<ta & ta<t2}
proof
  let s1,s2,t1,t2;
  now
    let x be object;
A1: now
      assume x in
      {p where p is Point of TOP-REAL 2:s1<p`1 & p`1<s2 & t1<p`2 & p`2<t2};
      then consider pp being Point of TOP-REAL 2 such that
A2:   pp=x and
A3:   s1<pp`1 and
A4:   pp`1<s2 and
A5:   t1<pp`2 and
A6:   pp`2<t2;
      |[pp`1,pp`2]|=x by A2,EUCLID:53;
      hence
      x in {|[s1a,t1a]|:s1<s1a & s1a<s2 & t1<t1a & t1a<t2} by A3,A4,A5,A6;
    end;
    now
      assume x in {|[sa,ta]|:s1<sa & sa<s2 & t1<ta & ta<t2};
      then consider sa,ta being Real such that
A7:   |[sa,ta]|=x and
A8:   s1<sa and
A9:   sa<s2 and
A10:  t1<ta and
A11:  ta<t2;
      set pa=|[sa,ta]|;
A12:  s1<pa`1 by A8,EUCLID:52;
A13:  pa`1<s2 by A9,EUCLID:52;
A14:  t1<pa`2 by A10,EUCLID:52;
      pa`2<t2 by A11,EUCLID:52;
      hence x in
      {p where p is Point of TOP-REAL 2:s1<p`1 & p`1<s2 & t1<p`2 & p`2<t2}
      by A7,A12,A13,A14;
    end;
    hence x in
    {p where p is Point of TOP-REAL 2:s1<p`1 & p`1<s2 & t1<p`2 & p`2<t2}
    iff x in {|[sa,ta]|:s1<sa & sa<s2 & t1<ta & ta<t2} by A1;
  end;
  hence thesis by TARSKI:2;
end;
