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 Th64:
  (Infty_dist 2).(|[0,0]|,|[1,1]|) = 1
  proof
    2 is non zero Nat & |[0,0]| is Element of REAL 2 &
    |[1,1]| is Element of REAL 2 by EUCLID:22;
    then consider S being ext-real-membered set such that
A1: S = the set of all |. |[0,0]|.i - |[1,1]|.i .| where
      i is Element of Seg 2 and
A2: (Infty_dist 2).(|[0,0]|,|[1,1]|) = sup S by Th57;
    S = {|. 0 - 1 .|}
    proof
      for t be object st t in S holds t in {|.0-1.|}
      proof
        let t be object;
        assume t in S;
        then consider i be Element of Seg 2 such that
A3:     t = |.|[0,0]|.i-|[1,1]|.i.| by A1;
        per cases by FINSEQ_1:2,TARSKI:def 2;
        suppose
A4:       i = 1;
          thus thesis by TARSKI:def 1,A4,A3;
        end;
        suppose
A5:       i = 2;
          thus thesis by TARSKI:def 1,A5,A3;
        end;
      end; then
A6:   S c= {|.0-1.|};
      {|.0 - 1.|} c= S
      proof
        1 in Seg 2;
        then |.|[0,0]|.1 - |[1,1]|.1.| in S by A1;
        hence thesis by TARSKI:def 1;
      end;
      hence thesis by A6;
    end;
    then S = {|. -1 .|} & |.-1.| = - (-1) by ABSVALUE:def 1;
    hence thesis by A2,XXREAL_2:11;
  end;
