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
  for x being Real holds x in [.r,s.] iff
    1 |-> x in ClosedHyperInterval(<*r*>,<*s*>)
  proof
    set  a1 = <*r*>, b1 = <*s*>;
A1: for x be Real st x in [.r,s.] holds  1|-> x in ClosedHyperInterval(a1,b1)
    proof
      let t be Real;
      assume
A2:   t in [.r,s.];
      reconsider t1 =  1|-> t as Element of REAL 1;
      ex y be Element of REAL 1 st
      t1 = y & (for i be Nat st i in Seg 1 holds y.i in [.a1.i,b1.i.])
      proof
        take t1;
        thus t1 = t1;
        thus for i be Nat st i in Seg 1 holds t1.i in [.a1.i,b1.i.]
        proof
          let i be Nat;
          assume
A3:       i in Seg 1; then
A4:       i = 1 by FINSEQ_1:2,TARSKI:def 1;
          now
            t1.1 = (Seg 1 --> t).1 by FINSEQ_2:def 2;
            hence t1.i = t by A3,A4,FUNCOP_1:7;
            thus a1.i = r & b1.i = s by A4;
          end;
          hence thesis by A2;
        end;
      end;
      hence thesis by Def3;
    end;
    for x be Real st 1 |-> x in ClosedHyperInterval(a1,b1) holds x in [.r,s.]
    proof
      let x be Real;
      assume 1 |-> x in ClosedHyperInterval(a1,b1);
      then consider y be Element of REAL 1 such that
A5:   1 |-> x = y and
A6:   for i be Nat st i in Seg 1 holds y.i in [.a1.i,b1.i.] by Def3;
A7:   1 in Seg 1;
      y = Seg 1 --> x & 1 in Seg 1 by A5,FINSEQ_2:def 2; then
A8:   y.1 = x by FUNCOP_1:7;
      a1.1 = r & b1.1 = s;
      hence thesis by A6,A7,A8;
    end;
    hence thesis by A1;
  end;
