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;
reserve n for non zero Nat;
reserve n     for non zero Nat,
        x,y,z for Element of REAL n;
reserve p for Element of EMINFTY n;

theorem Th70:
  for p being Point of TOP-REAL n, q being Element of EMINFTY n st
  q = p holds cl_Ball(q,r) = ClosedHypercube(p,n |-> r)
  proof
    let p be Point of TOP-REAL n,
        q be Element of EMINFTY n;
    assume
A1: q = p;
A8: cl_Ball(q,r) c= ClosedHypercube(p,n|->r)
    proof
      let x be object;
      assume x in cl_Ball(q,r);
      then consider f being Function such that
A2:   x = f and
A3:   dom f = Seg n and
A4:   for i being Nat st i in Seg n holds f.i in [.(@q).i-r,(@q).i+r.]
        by Th69;
      reconsider rq = @q as Element of REAL n;
A5:   now
        let i be Nat;
        assume i in Seg n;
        then f.i in [.rq.i-r,rq.i+r.] & (n|->r).i = r by A4,FINSEQ_2:57;
        hence f.i in [.p.i-(n|->r).i,p.i+(n|->r).i.] by A1;
      end;
      rng f c= REAL
      proof
        let u be object;
        assume u in rng f;
        then consider v be object such that
A6:     v in dom f and
A7:     u = f.v by FUNCT_1:def 3;
        f.v in [.rq.v-r,rq.v+r.] by A6,A3,A4;
        hence u in REAL by A7;
      end;
      then f in Funcs(Seg n,REAL) by A3,FUNCT_2:def 2;
      then f in REAL n by FINSEQ_2:93;
      then reconsider fx = f as Element of TOP-REAL n by EUCLID:22;
      fx in ClosedHypercube(p, n|->r) by A5,TIETZE_2:def 2;
      hence thesis by A2;
    end;
    ClosedHypercube(p,n|->r) c= cl_Ball(q,r)
    proof
      let x be object;
      assume
A9:   x in ClosedHypercube(p,n|->r);
      then reconsider y = x as Element of TOP-REAL n;
      reconsider f = y as Function;
      now
        take f;
        thus y = f;
        y in TOP-REAL n;
        then y in REAL n by EUCLID:22;
        then y is Tuple of n,REAL by FINSEQ_2:131;
        hence dom f = Seg n by FINSEQ_2:124;
        hereby
          let i be Nat;
          assume i in Seg n;
          then y.i in [.p.i-(n|->r).i,p.i+(n|->r).i.] & (n|->r).i = r
            by FINSEQ_2:57,A9,TIETZE_2:def 2;
          hence f.i in [.(@q).i-r,(@q).i+r.] by A1;
        end;
      end;
      hence thesis by Th69;
    end;
    hence thesis by A8;
  end;
