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 Th18:
  proj1"].r,s.[ = {|[ r1, r2 ]| : r < r1 & r1 < s}
proof
  set Q = proj1"].r,s.[;
  set QQ = {|[ r1,r2 ]|: r < r1 & r1 < s};
  now
    let z be object;
    hereby
      assume
A1:   z in Q;
      then reconsider p = z as Point of TOP-REAL 2;
      proj1.p in ].r,s.[ by A1,FUNCT_2:38;
      then
A2:   ex t being Real st t = proj1.p & r<t & t<s;
      p`1 = proj1.p & p = |[ p`1,p`2 ]| by Def5,EUCLID:53;
      hence z in QQ by A2;
    end;
    assume z in QQ;
    then consider r1, r2 being Real such that
A3: z = |[ r1,r2 ]| and
A4: r<r1 & r1 <s;
    set p = |[ r1,r2 ]|;
A5: r1 in ].r,s.[ by A4;
    proj1.p = p`1 by Def5
      .= r1 by EUCLID:52;
    hence z in Q by A3,A5,FUNCT_2:38;
  end;
  hence thesis by TARSKI:2;
end;
