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 Th18:
  for s1,t1,s2,t2 holds
  { |[ s,t ]|: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 ]|:s1<s & s<s2 & t1<t & t<t2};
  reconsider P1={|[s,t]|:s1<s} as Subset of TOP-REAL 2 by Lm2,Lm5;
  reconsider P2={|[s,t]|:s<s2} as Subset of TOP-REAL 2 by Lm2,Lm3;
  reconsider P3={|[s,t]|:t1<t} as Subset of TOP-REAL 2 by Lm2,Lm6;
  reconsider P4={|[s,t]|:t<t2} as Subset of TOP-REAL 2 by Lm2,Lm4;
A1: P=P1 /\ P2 /\ P3 /\ P4 by Th6;
A2: P1 is open by Th14;
  P2 is open by Th15;
  then
A3: P1 /\ P2 is open by A2,TOPS_1:11;
A4: P3 is open by Th16;
A5: P4 is open by Th17;
  P1 /\ P2 /\ P3 is open by A3,A4,TOPS_1:11;
  hence thesis by A1,A5,TOPS_1:11;
end;
