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 Th59:
  (Infty_dist n).(x,y) = 0 iff x = y
  proof
    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 n and
A2: (Infty_dist n).(x,y) = sup S by Th57;
    hereby
      assume
A3:   (Infty_dist n).(x,y) = 0;
A4:   dom x = Seg n & dom y = Seg n by FINSEQ_2:124;
      now
        let i be object;
        assume
A5:     i in dom x; then
A6:     i in Seg n by FINSEQ_2:124;
        set AXY = abs(x-y).i;
        reconsider f = (x-y) as complex-valued Function;
A7:     AXY = |.(x-y).i.| by VALUED_1:18; then
A8:     0 <= AXY by COMPLEX1:46;
        AXY = |.x.i - y.i.| by A5,A7,RVSUM_1:27;
        then AXY in S by A1,A6;
        then abs(x-y).i = 0 by A8,A3,A2,XXREAL_2:4; then
A9:     (x-y).i = 0 by A7,ABSVALUE:2;
        reconsider rx = x, ry = y as Element of n-tuples_on REAL;
        (rx-ry).i = rx.i - ry.i by A5,RVSUM_1:27;
        hence x.i = y.i by A9;
      end;
      hence x = y by A4,FUNCT_1:def 11;
    end;
    assume
A10: x = y;
    S = {0}
    proof
      for t be object st t in S holds t in {0}
      proof
        let t be object;
        assume t in S;
        then consider i be Element of Seg n such that
A11:    t = |.x.i-y.i.| by A1;
        t = 0 by A11,A10,ABSVALUE:2;
        hence t in {0} by TARSKI:def 1;
      end; then
A12:  S c= {0};
      for t be object st t in {0} holds t in S
      proof
        let t be object;
        assume
A13:    t in {0};
        1 <= 1 & 1 <= n by NAT_1:53;
        then 1 is Element  of Seg n by FINSEQ_1:1;
        then |.x.1 - y.1.| in S & |.x.1 - y.1.| = 0 by A10,ABSVALUE:2,A1;
        hence t in S by A13,TARSKI:def 1;
      end;
      then {0} c= S;
      hence thesis by A12;
    end;
    hence (Infty_dist n).(x,y) = 0 by A2,XXREAL_2:11;
  end;
