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 Th65:
  for x,y being Element of REAL 1 holds (Infty_dist 1).(x,y) = |.x.1-y.1.|
  proof
    let x,y be Element of REAL 1;
    consider S being ext-real-membered set such that
A1: S = the set of all |. x.i - y.i .| where i is Element of Seg 1 and
A2: (Infty_dist 1).(x,y) = sup S by Th57;
    S = {|. x.1 - y.1 .|}
    proof
      for t be object st t in S holds t in {|.x.1-y.1.|}
      proof
        let t be object;
        assume t in S;
        then consider i be Element of Seg 1 such that
A3:     t = |.x.i-y.i.| by A1;
        i = 1 by TARSKI:def 1,FINSEQ_1:2;
        hence thesis by A3,TARSKI:def 1;
      end; then
A4:   S c= {|.x.1-y.1.|};
      for t be object st t in {|.x.1-y.1.|} holds t in S
      proof
        let t be object;
        assume t in {|.x.1-y.1.|}; then
A5:     t = |.x.1-y.1.| by TARSKI:def 1;
        1 is Element of Seg 1 by TARSKI:def 1,FINSEQ_1:2;
        hence thesis by A5,A1;
      end;
      then {|.x.1-y.1.|} c= S;
      hence thesis by A4;
    end;
    hence thesis by A2,XXREAL_2:11;
  end;
