reserve
  x for object, X for set,
  i, n, m for Nat,
  r, s for Real,
  c, c1, c2, d for Complex,
  f, g for complex-valued Function,
  g1 for n-element complex-valued FinSequence,
  f1 for n-element real-valued FinSequence,
  T for non empty TopSpace,
  p for Element of TOP-REAL n;

theorem
  n in Seg m implies PROJ(m,n) is open
  proof
    set f = PROJ(m,n);
    assume
A1: n in Seg m;
    for p being Point of TOP-REAL m, r being positive Real holds
    ex s being positive Real st ].f.p-s,f.p+s.[ c= f.:Ball(p,r)
    proof
      let p be Point of TOP-REAL m, r be positive Real;
      take r;
A2:   dom p = Seg m by FINSEQ_1:89;
      p/.n = f.p by Def6;
      hence thesis by A2,A1,Th55;
    end;
    hence thesis by TOPS_4:13;
  end;
