reserve n   for Nat,
        r,s for Real,
        x,y for Element of REAL n,
        p,q for Point of TOP-REAL n,
        e   for Point of Euclid n;
reserve n for non zero Nat;
reserve n for non zero Nat;
reserve n for Nat,
        X for set,
        S for Subset-Family of X;
reserve n for Nat,
        S for Subset-Family of REAL;
reserve n       for Nat,
        a,b,c,d for Element of REAL n;

theorem Th52:
  for p being Point of TOP-REAL n ex a being Element of REAL n st a = p &
  ClosedHypercube(p,b) = ClosedHyperInterval(a - b,a + b)
  proof
    let p be Point of TOP-REAL n;
    reconsider a = p as Element of REAL n by EUCLID:22;
    take a;
    thus a = p;
A1: ClosedHypercube(p,b) c= ClosedHyperInterval(a - b,a + b)
    proof
      let x be object;
      assume
A2:   x in ClosedHypercube(p,b);
      then reconsider y = x as Element of REAL n by EUCLID:22;
      for i be Nat st i in Seg n holds y.i in [.(a-b).i,(a+b).i.]
      proof
        let i be Nat;
        assume
A3:     i in Seg n;
        a.i - b.i = (a - b).i & (a + b).i = a.i + b.i by RVSUM_1:11,RVSUM_1:27;
        hence thesis by A2,A3,TIETZE_2:def 2;
      end;
      hence thesis by Def3;
    end;
    ClosedHyperInterval(a - b,a + b) c= ClosedHypercube(p,b)
    proof
      let x be object;
      assume
A4:   x in ClosedHyperInterval(a-b,a+b);
      then reconsider q = x as Element of TOP-REAL n by EUCLID:22;
      now
        let i be Nat;
        assume
A5:     i in Seg n;
        consider y being Element of REAL n such that
A6:     x = y and
A7:     (for i being Nat st i in Seg n holds y.i in [.(a-b).i,(a+b).i.])
          by A4,Def3;
        (a-b).i = a.i-b.i & (a+b).i = a.i+b.i by RVSUM_1:11,RVSUM_1:27;
        hence q.i in [.p.i-b.i,p.i+b.i.] by A5,A6,A7;
      end;
      hence thesis by TIETZE_2:def 2;
    end;
    hence thesis by A1;
  end;
