reserve p, q, x, y for Real,
  n for Nat;

theorem Th13:
  for e1, e2, e5, e6 being Point of Euclid n, p1, p2 being Point
of TOP-REAL n st e1 = p1 & e2 = p2 & e5 = y*p1 & e6 = y*p2 & dist(e1,e2) < x &
  y <> 0 holds dist(e5,e6) < |.y.|*x
proof
  let e1, e2, e5, e6 be Point of Euclid n, p1, p2 be Point of TOP-REAL n such
  that
A1: e1 = p1 and
A2: e2 = p2 and
A3: e5 = y*p1 and
A4: e6 = y*p2 and
A5: dist(e1,e2) < x and
A6: y <> 0;
  reconsider f1 = e1, f2 = e2, f5= e5, f6 = e6 as Element of REAL n by A1,A2,A3
,A4,EUCLID:22;
A7: dist(e1,e2) = |. f1-f2 .| by SPPOL_1:5;
A8: 0 < |.y.| by A6,COMPLEX1:47;
  dist(e5,e6) = |.f5-f6.| by SPPOL_1:5
    .= |. y*f1-f6 .| by A1,A3
    .= |. y*f1-y*f2 .| by A2,A4
    .= |. y*(f1-f2) .| by Th7
    .= |.y.| * |. f1-f2 .| by EUCLID:11;
  hence thesis by A5,A7,A8,XREAL_1:68;
end;
