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;

theorem Th67:
  r <= 0 implies product Intervals(x,r) is empty
  proof
    assume
A1: r <= 0;
    assume product Intervals(x,r) is non empty;
    then consider t be object such that
A2: t in product Intervals(x,r);
    consider g be Function such that
    g = t and
    dom g = dom Intervals(x,r) and
A3: for y being object st y in dom Intervals(x,r) holds g.y in Intervals(x,r).y
    by A2,CARD_3:def 5;
A4: dom x = Seg n by FINSEQ_2:124; then
A5: dom Intervals(x,r) = Seg n by EUCLID_9:def 3;
A6: n = 1 or n > 1 by NAT_1:53;
    then 1 in dom Intervals(x,r) by A5;
    then g.1 in Intervals(x,r).1 & 1 in dom x by A3,A4,A6;
    then ].x.1-r,x.1+r.[ is non empty by EUCLID_9:def 3;
    hence contradiction by A1;
  end;
