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 Th16:
  for s1 holds { |[ s,t ]|:s1<t } is open Subset of TOP-REAL 2
proof
  let s1;
  reconsider P= { |[ s,t ]|:s1<t } as Subset of REAL 2 by Lm6;
  reconsider PP=P as Subset of TopSpaceMetr Euclid 2;
  for pe being Point of Euclid 2 st pe in P ex r being Real
  st r>0 & Ball(pe,r) c= P
  proof
    let pe be Point of Euclid 2;
    assume pe in P;
    then consider s,t such that
A1: |[ s,t ]|=pe and
A2: s1<t;
    set r=(t-s1)/2;
A3: t-s1>0 by A2,XREAL_1:50;
    then
A4: r>0 by XREAL_1:139;
    Ball(pe,r) c= P
    proof
      let x be object;
      assume x in Ball(pe,r);
      then x in {q where q is Element of Euclid 2:dist(pe,q)<r} by METRIC_1:17;
      then consider q being Element of Euclid 2 such that
A5:   q=x and
A6:   dist(pe,q)<r;
      reconsider ppe=pe, pq=q as Point of TOP-REAL 2 by EUCLID:22;
      (Pitag_dist 2).(pe,q)=dist(pe,q) by METRIC_1:def 1;
      then
A7:   sqrt ((ppe`1 - pq`1)^2 + (ppe`2 - pq`2)^2)<r by A6,TOPREAL3:7;
A8:   0 <= (ppe`1 - pq`1)^2 by XREAL_1:63;
      0 <= (ppe`2 - pq`2)^2 by XREAL_1:63;
      then
A9:  0+0 <= (ppe`1 - pq`1)^2 + (ppe`2 - pq`2)^2 by A8,XREAL_1:7;
      then 0 <=sqrt ( (ppe`1 - pq`1)^2 + (ppe`2 - pq`2)^2) by SQUARE_1:def 2;
      then (sqrt ((ppe`1 - pq`1)^2 + (ppe`2 - pq`2)^2))^2<r^2
      by A7,SQUARE_1:16;
      then
A10:  (ppe`1 - pq`1)^2 + (ppe`2 - pq`2)^2 < r^2 by A9,SQUARE_1:def 2;
      (ppe`2 - pq`2)^2 <= (ppe`2 - pq`2)^2+(ppe`1 - pq`1)^2 by XREAL_1:31,63;
      then (ppe`2 - pq`2)^2 <r^2 by A10,XXREAL_0:2;
      then ppe`2 - pq`2 < r by A4,SQUARE_1:15;
      then ppe`2 <pq`2+r by XREAL_1:19;
      then ppe`2 - r < pq`2 by XREAL_1:19;
      then
A11:  t-(t-s1)/2 < pq`2 by A1,EUCLID:52;
      t-(t-s1)/2 =r+s1;
      then
A12:  s1< t-(t-s1)/2 by A3,XREAL_1:29,139;
A13:  |[pq`1,pq`2]|=x by A5,EUCLID:53;
      s1<pq`2 by A11,A12,XXREAL_0:2;
      hence thesis by A13;
    end;
    hence thesis by A3,XREAL_1:139;
  end;
  then PP is open by TOPMETR:15;
  hence thesis by Lm9,PRE_TOPC:30;
end;
