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 Th7:
  {|[ s,t ]|:not (s1<=s & s<=s2 & t1<=t & t<=t2)} =
  {|[ s3,t3 ]|:s3<s1} \/ {|[s4,t4]|:t4<t1} \/ {|[s5,t5]|:s2<s5} \/
  {|[s6,t6]|:t2<t6}
proof
  now
    let x be object;
    assume x in { |[ s,t ]|:not (s1<=s & s<=s2 & t1<=t & t<=t2)};
    then ex s,t st |[s,t]|=x & not (s1<=s & s<=s2 & t1<=t & t<=t2);
    then x in { |[ s3,t3 ]|:s3<s1}or x in { |[ s4,t4 ]|:t4<t1}
    or x in { |[ s5,t5 ]|:s2<s5} or x in { |[ s6,t6 ]|:t2<t6};
    then x in { |[ s3,t3 ]|:s3<s1} \/ { |[ s4,t4 ]|:t4<t1}
or x in { |[ s5,t5 ]|:s2<s5} or x in { |[ s6,t6 ]|:t2<t6} by XBOOLE_0:def 3;
    then x in { |[ s3,t3 ]|:s3<s1} \/ {|[s4,t4]|:t4<t1}\/ {|[s5,t5]|:s2<s5}
    or x in { |[ s6,t6 ]|:t2<t6} by XBOOLE_0:def 3;
    hence
    x in { |[ s3,t3 ]|:s3<s1} \/ {|[s4,t4]|:t4<t1}\/ {|[s5,t5]|:s2<s5}
    \/ {|[s6,t6]|:t2<t6} by XBOOLE_0:def 3;
  end;
  then
A1: { |[ s,t ]|:not (s1<=s & s<=s2 & t1<=t & t<=t2)} c=
  { |[ s3,t3 ]|:s3<s1} \/ {|[s4,t4]|:t4<t1}\/ {|[s5,t5]|:s2<s5}
  \/ {|[s6,t6]|:t2<t6};
  now
    let x be object;
    assume x in { |[ s3,t3 ]|:s3<s1} \/ {|[s4,t4]|:t4<t1}\/ {|[s5,t5]|:s2< s5
    } \/ {|[s6,t6]|:t2<t6};
    then x in { |[ s3,t3 ]|:s3<s1} \/ {|[s4,t4]|:t4<t1}\/ {|[s5,t5]|:s2<s5}
    or x in { |[ s6,t6 ]|:t2<t6} by XBOOLE_0:def 3;
    then x in { |[ s3,t3 ]|:s3<s1} \/ { |[ s4,t4]|:t4<t1}
    or x in { |[ s5,t5 ]|:s2<s5} or x in { |[ s6,t6 ]|:t2<t6} by XBOOLE_0:def 3
    ;
    then x in { |[ s3,t3 ]|:s3<s1} or x in { |[ s4,t4 ]|:t4<t1}
    or x in { |[ s5,t5 ]|:s2<s5} or x in { |[ s6,t6 ]|:t2<t6} by XBOOLE_0:def 3
    ;
    then (ex sa,ta st |[sa,ta]|=x & sa<s1)
    or (ex sc,tc st |[sc,tc]|=x & tc<t1) or (ex sb,tb st |[sb,tb]|=x & s2<sb)
    or ex sd,td st |[sd,td]|=x & t2<td;
    hence x in { |[ s,t ]|:not (s1<=s & s<=s2 & t1<=t & t<=t2)};
  end;
  then { |[ s3,t3 ]|:s3<s1} \/ {|[s4,t4]|:t4<t1}\/ {|[s5,t5]|:s2<s5}
  \/ {|[s6,t6]|:t2<t6}c={ |[ s,t ]|:not (s1<=s & s<=s2 & t1<=t & t<=t2)};
  hence thesis by A1;
end;
