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 Th19:
  for s1,t1,s2,t2 holds
{ |[ s,t ]|:not (s1<=s & s<=s2 & t1<=t & t<=t2)} is open Subset of TOP-REAL 2
proof
  let s1,t1,s2,t2;
  set P={ |[ s,t ]|:not (s1<=s & s<=s2 & t1<=t & t<=t2)};
A1: P={ |[ s3,t3 ]|:s3<s1} \/ {|[s4,t4]|:t4<t1}\/ {|[s5,t5]|:s2<s5}
  \/ {|[s6,t6]|:t2<t6} by Th7;
  reconsider A0={ |[ s,t ]|:s<s1} as Subset of TOP-REAL 2 by Lm2,Lm3;
  reconsider A1={ |[ s,t ]|:t<t1} as Subset of TOP-REAL 2 by Lm2,Lm4;
  reconsider A2={ |[ s,t ]|:s2<s} as Subset of TOP-REAL 2 by Lm2,Lm5;
  reconsider A3={ |[ s,t ]|:t2<t} as Subset of TOP-REAL 2 by Lm2,Lm6;
A2: A0 is open by Th15;
  A1 is open by Th17;
  then
A3: A0 \/ A1 is open by A2,TOPS_1:10;
  A2 is open by Th14;
  then
A4: A0 \/ A1 \/ A2 is open by A3,TOPS_1:10;
  A3 is open by Th16;
  hence thesis by A1,A4,TOPS_1:10;
end;
