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 Th55:
  for p being Point of TOP-REAL m st n in dom p holds
  PROJ(m,n).:Ball(p,r) = ]. p/.n-r , p/.n+r .[
  proof
    let p be Point of TOP-REAL m such that
A1: n in dom p;
    per cases;
    suppose
A2:   r <= 0;
      then ]. p/.n-r , p/.n+r .[ = {};
      hence thesis by A2;
    end;
    suppose
A3:   0 < r;
A4:   dom p = Seg m by FINSEQ_1:89;
      thus PROJ(m,n).:Ball(p,r) c= ]. p/.n-r , p/.n+r .[
      proof
        let y be object;
        assume y in PROJ(m,n).:Ball(p,r);
        then consider x being Element of TOP-REAL m such that
A5:     x in Ball(p,r) and
A6:     y = PROJ(m,n).x by FUNCT_2:65;
A7:     PROJ(m,n).x = x/.n by Def6;
A8:     |.x-p.| < r by A5,TOPREAL9:7;
        0 <= Sum sqr (x-p) by RVSUM_1:86;
        then (sqrt Sum sqr (x-p))^2 = Sum sqr (x-p) by SQUARE_1:def 2;
        then
A9:     Sum sqr (x-p) < r^2 by A8,SQUARE_1:16;
        dom x = Seg m by FINSEQ_1:89;
        then (x/.n - p/.n)^2 <= Sum sqr (x-p) by A4,A1,EUCLID_9:8;
        then (x/.n - p/.n)^2 < r^2 by A9,XXREAL_0:2;
        then -r < x/.n - p/.n & x/.n - p/.n < r by A3,SQUARE_1:48;
        then -r + p/.n < x/.n - p/.n + p/.n &
        x/.n - p/.n + p/.n < r + p/.n by XREAL_1:6;
        hence thesis by A6,A7,XXREAL_1:4;
      end;
      let y be object;
      assume
A10:   y in ]. p/.n-r , p/.n+r .[;
      then reconsider y as Element of REAL;
      set x = p+*(n,y);
      reconsider X = x as FinSequence of REAL by EUCLID:24;
A11:   dom X = dom p by FUNCT_7:30;
A12:   p/.n = p.n by A1,PARTFUN1:def 6;
      p/.n-r < y & y < p/.n+r by A10,XXREAL_1:4;
      then
A13:   y - p/.n < r & -r < y - p/.n by XREAL_1:19,20;
      x-p = (0*m)+*(n,y-p.n) by Th17;
      then |.x-p.| = |.y-p.n.| by A1,A4,Th13;
      then |.x-p.| < r by A12,A13,SEQ_2:1;
      then
A14:   x in Ball(p,r) by TOPREAL9:7;
      PROJ(m,n).x = x/.n by Def6
      .= X.n by A11,A1,PARTFUN1:def 6
      .= y by A1,FUNCT_7:31;
      hence thesis by A14,FUNCT_2:35;
    end;
  end;
