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
  b < a & n is non zero implies ClosedHyperInterval(a,b) is empty
  proof
    assume that
A1: b < a and
A2: n is non zero;
    assume not ClosedHyperInterval(a,b) is empty;
    then consider x be object such that
A3: x in ClosedHyperInterval(a,b);
    consider y be Element of REAL n such that
    x = y and
A4: for i be Nat st i in Seg n holds y.i in [.a.i,b.i.] by A3,Def3;
    1 <= n by A2,NAT_1:14;
    then b.1 < a.1 & y.1 in [.a.1,b.1.] by A1,A4,FINSEQ_1:1;
    hence contradiction by XXREAL_1:29;
  end;
