reserve
  n, m for Nat,
  T for non empty TopSpace,
  M, M1, M2 for non empty MetrSpace;

theorem
  for f being Function of R^1,TOP-REAL m holds f is open iff
  for p being Point of R^1, r being positive Real
  ex s being positive Real st Ball(f.p,s) c= f.:].p-r,p+r.[
  proof
    let f be Function of R^1,TOP-REAL m;
A1: the TopStruct of TOP-REAL m = TopSpaceMetr Euclid m by EUCLID:def 8;
    then reconsider f1 = f as Function of R^1,TopSpaceMetr Euclid m;
    thus f is open implies
    for p being Point of R^1, r being positive Real
    ex s being positive Real st Ball(f.p,s) c= f.:].p-r,p+r.[
    proof
      assume
A2:   f is open;
      let p be Point of R^1, r be positive Real;
      reconsider p1 = p as Point of RealSpace;
      reconsider q1 = f.p as Point of Euclid m by EUCLID:67;
      f1 is open by A1,A2,Th1;
      then consider s being positive Real such that
A3:   Ball(q1,s) c= f1.:Ball(p1,r) by Th6;
      ].p-r,p+r.[ = Ball(p1,r) & Ball(f.p,s) = Ball(q1,s)
      by FRECHET:7,TOPREAL9:13;
      hence thesis by A3;
    end;
    assume
A4: for p being Point of R^1, r being positive Real
    ex s being positive Real st Ball(f.p,s) c= f.:].p-r,p+r.[;
    for p being Point of RealSpace, q being Point of Euclid m,
        r being positive Real st q = f1.p
    ex s being positive Real st Ball(q,s) c= f1.:Ball(p,r)
    proof
      let p be Point of RealSpace, q be Point of Euclid m,
          r be positive Real such that
A5:   q = f1.p;
      reconsider p1 = p as Point of R^1;
      consider s being positive Real such that
A6:   Ball(f.p1,s) c= f.:].p1-r,p1+r.[ by A4;
      ].p1-r,p1+r.[ = Ball(p,r) & Ball(f.p1,s) = Ball(q,s)
      by A5,FRECHET:7,TOPREAL9:13;
      hence thesis by A6;
    end;
    then f1 is open by Th6;
    hence thesis by A1,Th1;
  end;
