reserve n for Nat,
  i,j for Nat,
  r,s,r1,s1,r2,s2,r9,s9 for Real,
  p,q for Point of TOP-REAL 2,
  G for Go-board,
  x,y for set,
  v for Point of Euclid 2;

theorem Th6: :: TOPREAL3:12
  for u,v being Point of Euclid 2 st u = |[r1,s1]| & v = |[r2,s2]|
  holds dist(u,v) =sqrt ((r1 - r2)^2 + (s1 - s2)^2)
proof
  let u,v be Point of Euclid 2 such that
A1: u = |[r1,s1]| & v = |[r2,s2]|;
A2: |[r1,s1]|`1 = r1 & |[r1,s1]|`2 = s1 by EUCLID:52;
A3: |[r2,s2]|`1 = r2 & |[r2,s2]|`2 = s2 by EUCLID:52;
  thus dist(u,v) = (Pitag_dist 2).(u,v) by METRIC_1:def 1
    .= sqrt ((r1 - r2)^2 + (s1 - s2)^2) by A1,A2,A3,TOPREAL3:7;
end;
