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 Th6:
  {|[ s,t ]|: s1<s & s<s2 & t1<t & t<t2} =
  {|[ s3,t3 ]|:s1<s3} /\ {|[s4,t4]|:s4<s2} /\ {|[s5,t5]|:t1<t5} /\
  {|[s6,t6]|:t6<t2}
proof
  now
    let x be object;
    assume x in { |[ s,t ]|: s1<s & s<s2 & t1<t & t<t2};
    then
A1: ex s,t st |[s,t]|=x & s1<s & s<s2 & t1<t & t<t2;
    then
A2: x in { |[ s3,t3 ]|:s1<s3};
    x in { |[ s4,t4 ]|:s4<s2} by A1;
    then
A3: x in { |[ s3,t3 ]|:s1<s3} /\ { |[ s4,t4 ]|:s4<s2} by A2,XBOOLE_0:def 4;
A4: x in { |[ s5,t5 ]|:t1<t5} by A1;
A5: x in { |[ s6,t6 ]|:t6<t2} by A1;
    x in { |[ s3,t3 ]|:s1<s3} /\ {|[s4,t4]|:s4<s2}/\ {|[s5,t5]|:t1<t5} by A3,A4
,XBOOLE_0:def 4;
    hence
    x in { |[ s3,t3 ]|:s1<s3} /\ {|[s4,t4]|:s4<s2}/\ {|[s5,t5]|:t1<t5}
    /\ {|[s6,t6]|:t6<t2} by A5,XBOOLE_0:def 4;
  end;
  then
A6: { |[ s,t ]|: s1<s & s<s2 & t1<t & t<t2} c=
  { |[ s3,t3 ]|:s1<s3} /\ {|[s4,t4]|:s4<s2}/\ {|[s5,t5]|:t1<t5}
  /\ {|[s6,t6]|:t6<t2};
  now
    let x be object;
    assume
    A7: x in { |[ s3,t3 ]|:s1<s3} /\ {|[s4,t4]|:s4<s2}/\ {|[s5,t5]|:t1<t5 }
    /\ {|[s6,t6]|:t6<t2};
    then
A8: x in { |[ s3,t3 ]|:s1<s3} /\ {|[s4,t4]|:s4<s2}/\ {|[s5,t5]|:t1<t5} by
XBOOLE_0:def 4;
    then
A9: x in { |[ s3,t3 ]|:s1<s3} /\ { |[ s4,t4]|:s4<s2} by XBOOLE_0:def 4;
A10: x in { |[ s6,t6 ]|:t6<t2} by A7,XBOOLE_0:def 4;
A11: x in { |[ s3,t3 ]|:s1<s3} by A9,XBOOLE_0:def 4;
A12: x in { |[ s4,t4 ]|:s4<s2} by A9,XBOOLE_0:def 4;
A13: x in { |[ s5,t5 ]|:t1<t5} by A8,XBOOLE_0:def 4;
A14: ex sa,ta st |[sa,ta]|=x & s1<sa by A11;
A15: ex sb,tb st |[sb,tb]|=x & sb< s2 by A12;
A16: ex sc,tc st |[sc,tc]|=x & t1<tc by A13;
A17: ex sd,td st |[sd,td]|=x & td<t2 by A10;
    consider sa,ta such that
A18: |[sa,ta]|=x and
A19: s1<sa by A11;
    reconsider p=x as Point of TOP-REAL 2 by A14;
A20: p`1=sa by A18,EUCLID:52;
A21: p`2=ta by A18,EUCLID:52;
A22: sa<s2 by A15,A20,EUCLID:52;
A23: t1<ta by A16,A21,EUCLID:52;
    ta<t2 by A17,A21,EUCLID:52;
    hence x in { |[ s,t ]|:s1<s & s<s2 & t1<t & t<t2} by A18,A19,A22,A23;
  end;
  then { |[ s3,t3 ]|:s1<s3} /\ {|[s4,t4]|:s4<s2}/\ {|[s5,t5]|:t1<t5}
  /\ {|[s6,t6]|:t6<t2}c={ |[ s,t ]|:s1<s & s<s2 & t1<t & t<t2};
  hence thesis by A6;
end;
