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 Th58:
  (Infty_dist n).(x,y) = abs (x-y).max_diff_index(x,y)
  proof
    max_diff_index(x,y) in dom x by EUCLID_9:4; then
A1: max_diff_index(x,y) in Seg n by FINSEQ_2:124;
    dom abs(x-y) = Seg n by FINSEQ_2:124; then
A2: abs(x-y).max_diff_index(x,y) in rng abs(x-y) by A1,FUNCT_1:def 3;
    sup rng abs(x-y) is UpperBound of rng abs(x-y) by XXREAL_2:def 3; then
    abs(x-y).max_diff_index(x,y) <= sup rng abs(x-y) by A2,XXREAL_2:def 1; then
A3: abs(x-y).max_diff_index(x,y) <= (Infty_dist n).(x,y) by Def7;
   (Infty_dist n).(x,y) <= abs(x-y).max_diff_index(x,y)
    proof
      now
        let t be ExtReal;
        assume t in rng abs(x-y);
        then ex u be object st u in dom abs(x-y) & t = abs(x-y).u
          by FUNCT_1:def 3;
        hence t <= abs(x-y).max_diff_index(x,y) by EUCLID_9:5;
      end;
      then abs(x-y).max_diff_index(x,y) is UpperBound of rng abs(x-y)
        by XXREAL_2:def 1;
      then sup rng abs(x-y) <= abs(x-y).max_diff_index(x,y) by XXREAL_2:def 3;
      hence thesis by Def7;
    end;
    hence thesis by A3,XXREAL_0:1;
  end;
