reserve a, b for Real,
  r for Real,
  rr for Real,
  i, j, n for Nat,
  M for non empty MetrSpace,
  p, q, s for Point of TOP-REAL 2,
  e for Point of Euclid 2,
  w for Point of Euclid n,
  z for Point of M,
  A, B for Subset of TOP-REAL n,
  P for Subset of TOP-REAL 2,
  D for non empty Subset of TOP-REAL 2;
reserve a, b for Real;
reserve a, b for Real;
reserve r for Real;
reserve r1,r2,s1,s2 for Real;

theorem Th84:
  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]| and
A2: v = |[r2,s2]|;
A3: |[r1,s1]|`1 = r1;
A4: |[r2,s2]|`2 = s2;
A5: |[r2,s2]|`1 = r2;
A6: |[r1,s1]|`2 = s1;
  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,A6,A5,A4,TOPREAL3:7;
end;
