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 Th49:
  a <= c & d <= b implies ClosedHyperInterval(c,d) c= ClosedHyperInterval(a,b)
  proof
    assume that
A1: a <= c and
A3: d <= b;
    now
      let t be object;
      assume t in ClosedHyperInterval(c,d);
      then consider t1 be Element of REAL n such that
A4:   t = t1 and
A5:   for i be Nat st i in Seg n holds t1.i in [.c.i,d.i.] by Def3;
      for i be Nat st i in Seg n holds t1.i in [.a.i,b.i.]
      proof
        let i be Nat;
        assume
A6:     i in Seg n;
        then a.i <= c.i & d.i <= b.i by A1,A3;
        then [.c.i,d.i.] c= [.a.i,b.i.] by XXREAL_1:34;
        hence thesis by A6,A5;
      end;
      hence t in ClosedHyperInterval(a,b) by A4,Def3;
    end;
    hence thesis;
  end;
