reserve x,y for set;
reserve s,r for Real;
reserve r1,r2 for Real;
reserve n for Nat;
reserve p,q,q1,q2 for Point of TOP-REAL 2;

theorem
  for P being Subset of R^1, r,s being Real st P=].r,s.[ holds P is open
proof
  let P be Subset of R^1, r,s be Real;
  assume
A1: P=].r,s.[;
  {r1:r<r1} c= REAL
  proof
    let x be object;
    assume x in {r1:r<r1};
    then consider r1 such that
A2:   ( r1=x)&( r<r1);
     r1 in REAL by XREAL_0:def 1;
    hence thesis by A2;
  end;
  then reconsider P1={r1 where r1 is Real:r<r1} as Subset of R^1
  by TOPMETR:17;
  {r2:s>r2} c= REAL
  proof
    let x be object;
    assume x in {r2:s>r2};
    then consider r2 such that
A3:   ( r2=x)&( s>r2);
     r2 in REAL by XREAL_0:def 1;
    hence thesis by A3;
  end;
  then reconsider P2={r2 where r2 is Real:s>r2} as Subset of R^1
  by TOPMETR:17;
A4: P1 is open by JORDAN2B:25;
A5: P2 is open by JORDAN2B:24;
A6: P c= P1 /\ P2
  proof
    let x be object;
    assume
A7: x in P;
    then reconsider r1=x as Real;
A8: r<r1 by A1,A7,XXREAL_1:4;
A9: r1<s by A1,A7,XXREAL_1:4;
A10: x in P1 by A8;
    x in P2 by A9;
    hence thesis by A10,XBOOLE_0:def 4;
  end;
  P1 /\ P2 c= P
  proof
    let x be object;
    assume
A11: x in P1 /\ P2;
    then
A12: x in P1 by XBOOLE_0:def 4;
A13: x in P2 by A11,XBOOLE_0:def 4;
A14: ex r1 st ( r1=x)&( r<r1) by A12;
    ex r2 st ( r2=x)&( s>r2) by A13;
    hence thesis by A1,A14,XXREAL_1:4;
  end;
  then P=P1 /\ P2 by A6;
  hence thesis by A4,A5,TOPS_1:11;
end;
