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;

theorem
  for p, q being Point of Euclid 2, z being Point of TOP-REAL 2 st p =
  0.REAL 2 & q = z holds dist(p,q) = |.z.|
proof
  let p, q be Point of Euclid 2, z be Point of TOP-REAL 2 such that
A1: p = 0.REAL 2 and
A2: q = z;
  reconsider P = p as Point of TOP-REAL 2 by TOPREAL3:8;
A3: 0.REAL 2 = 0.TOP-REAL 2 by EUCLID:66;
  then
A4: P`1 = 0 by A1,Th22;
A5: P`2 = 0 by A1,A3,Th22;
  dist(p,q) = (Pitag_dist 2).(p,q) by METRIC_1:def 1
    .= sqrt ((P`1 - z`1)^2 + (P`2 - z`2)^2) by A2,TOPREAL3:7
    .= sqrt ((z`1)^2 + (z`2)^2) by A4,A5;
  hence thesis by JGRAPH_1:30;
end;
