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 Th20:
  proj2"].r,s.[ = {|[ r1, r2 ]| : r < r2 & r2 < s}
proof
  set Q = proj2"].r,s.[;
  set QQ = {|[ r1,r2 ]|: r < r2 & r2 < s};
  now
    let z be object;
    hereby
      assume
A1:   z in Q;
      then reconsider p = z as Point of TOP-REAL 2;
      proj2.p in ].r,s.[ by A1,FUNCT_2:38;
      then
A2:   ex t being Real st t = proj2.p & r<t & t<s;
      p`2 = proj2.p & p = |[ p`1,p`2 ]| by Def6,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<r2 & r2 <s;
    set p = |[ r1,r2 ]|;
A5: r2 in ].r,s.[ by A4;
    proj2.p = p`2 by Def6
      .= r2 by EUCLID:52;
    hence z in Q by A3,A5,FUNCT_2:38;
  end;
  hence thesis by TARSKI:2;
end;
