reserve r, s, t, g for Real,

          r3, r1, r2, q3, p3 for Real;
reserve T for TopStruct,
  f for RealMap of T;
reserve p for Point of TOP-REAL 2,
  P for Subset of TOP-REAL 2,
  Z for non empty Subset of TOP-REAL 2,
  X for non empty compact Subset of TOP-REAL 2;

theorem Th21:
  for r3, q3 st P = {|[ r1, r2 ]| : r3 < r2 & r2 < q3} holds P is open
proof
  deffunc F(Real,Real) = |[ $1,$2 ]|;
  let r3, q3;
  defpred P1[Real,Real] means r3 < $2;
  defpred P2[Real,Real] means $2 < q3;
  reconsider Q1 = {|[ r1,r2 ]|: r3 < r2}, Q2 = {|[ r1,r2 ]|: r2 < q3} as
  open Subset of TOP-REAL 2 by JORDAN1:22,23;
  assume
A1: P = {|[ r1, r2 ]| : r3 < r2 & r2 < q3};
  now
    let x be object;
    hereby
      assume x in P;
      then ex r1, r2 being Real
      st x = |[ r1, r2 ]| & r3 < r2 & r2 < q3 by A1;
      then x in Q1 & x in Q2;
      hence x in Q1/\Q2 by XBOOLE_0:def 4;
    end;
    assume
A2: x in Q1/\Q2;
    then x in Q1 by XBOOLE_0:def 4;
    then consider r1, r2 being Real such that
A3: x = |[ r1, r2 ]| & r3 < r2;
    x in Q2 by A2,XBOOLE_0:def 4;
    then consider r19, r29 being Real such that
A4: x = |[ r19, r29 ]| & r29 < q3;
    |[ r1, r2 ]|`2 = r2 & |[ r19, r29 ]|`2 = r29 by EUCLID:52;
    hence x in P by A1,A3,A4;
  end;
  then
A5: P = Q1/\Q2 by TARSKI:2;
  thus thesis by A5,TOPS_1:11;
end;
