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 Th48:
  ClosedHyperInterval(a,a) = {a}
  proof
A1: ClosedHyperInterval(a,a) c= {a}
    proof
      let x be object;
      assume x in ClosedHyperInterval(a,a);
      then consider y be Element of REAL n such that
A2:   x = y and
A3:   for i be Nat st i in Seg n holds y.i in [.a.i,a.i.] by Def3;
      reconsider y1 = y,a1 = a as Function;
A4:   dom y = Seg n & dom a1 = Seg n by FINSEQ_2:124;
      for z be object st z in dom y1 holds y1.z = a1.z
      proof
        let z be object;
        assume z in dom y1;
        then z in Seg n by FINSEQ_2:124;
        then y1.z in [.a.z,a.z.] by A3;
        then y1.z in {a.z} by XXREAL_1:17;
        hence thesis by TARSKI:def 1;
      end;
      then y1 = a1 by A4,FUNCT_1:def 11;
      hence thesis by A2,TARSKI:def 1;
    end;
    {a} c= ClosedHyperInterval(a,a)
    proof
      let x be object;
      assume
A5:   x in {a};
      then reconsider x1 = x as Element of REAL n by TARSKI:def 1;
      for i be Nat st i in Seg n holds x1.i in [.a.i,a.i.]
      proof
        let i be Nat;
        assume i in Seg n;
        x1.i = a.i by A5,TARSKI:def 1;
        then x1.i in {a.i} by TARSKI:def 1;
        hence thesis by XXREAL_1:17;
      end;
      hence thesis by Def3;
    end;
    hence thesis by A1;
  end;
