reserve k,j,n for Nat,
  r for Real;
reserve x,x1,x2,y for Element of REAL n;
reserve f for real-valued FinSequence;

theorem Th18:
  for n being Nat holds Pitag_dist n is_metric_of REAL n
proof
  let n be Nat;
  let x,y,z be Element of REAL n;
A1: (Pitag_dist n).(y,z) = |.y-z.| by Def6;
  (Pitag_dist n).(x,y) = |.x-y.| by Def6;
  hence (Pitag_dist n).(x,y) = 0 iff x=y by Th13;
  thus (Pitag_dist n).(x,y) = |.x-y.| by Def6
    .= |.y-x .| by Th15
    .= (Pitag_dist n).(y,x) by Def6;
  (Pitag_dist n).(x,y) = |.x-y.| & (Pitag_dist n).(x,z) = |.x-z.| by Def6;
  hence (Pitag_dist n).(x,z) <= (Pitag_dist n).(x,y) + (Pitag_dist n).(y,z) by
A1,Th16;
end;
