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 Th13:
  for s1,t1,s2,t2,P st P = { |[ s,t ]|:not (s1<=s & s<=s2 & t1<=t & t<=t2)}
  holds P is connected
proof
  let s1,t1,s2,t2,P;
  assume P={ |[ s,t ]|:not (s1<=s & s<=s2 & t1<=t & t<=t2)};
  then
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}, A1={ |[ s,t ]|:t<t1},
  A2={ |[ s,t ]|:s2<s}, A3={ |[ s,t ]|:t2<t}
  as Subset of TOP-REAL 2 by Lm2,Lm3,Lm4,Lm5,Lm6;
A2: s1-1<s1 by XREAL_1:44;
A3: t1-1<t1 by XREAL_1:44;
A4: |[s1-1,t1-1]| in A0 by A2;
  |[s1-1,t1-1]| in A1 by A3;
  then A0 /\ A1 <> {} by A4,XBOOLE_0:def 4;
  then
A5: A0 meets A1;
A6: s2< s2+1 by XREAL_1:29;
A7: |[s2+1,t1-1]| in A1 by A3;
  |[s2+1,t1-1]| in A2 by A6;
  then A1 /\ A2 <> {} by A7,XBOOLE_0:def 4;
  then
A8: A1 meets A2;
A9: t2< t2+1 by XREAL_1:29;
A10: |[s2+1,t2+1]| in A2 by A6;
  |[s2+1,t2+1]| in A3 by A9;
  then A2 /\ A3 <> {} by A10,XBOOLE_0:def 4;
  then
A11: A2 meets A3;
A12: A0 is convex by Th10;
A13: A1 is convex by Th12;
A14: A2 is convex by Th9;
  A3 is convex by Th11;
  hence thesis by A1,A5,A8,A11,A12,A13,A14,Th5;
end;
