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

theorem Th14:
  for e1, e2, e3, e4, e5, e6 being Point of Euclid n, p1, p2, p3,
p4 being Point of TOP-REAL n st e1 = p1 & e2 = p2 & e3 = p3 & e4 = p4 & e5 = x*
p1+y*p3 & e6 = x*p2+y*p4 & dist(e1,e2) < p & dist(e3,e4) < q & x <> 0 & y <> 0
  holds dist(e5,e6) < |.x.|*p + |.y.|*q
proof
  let e1, e2, e3, e4, e5, e6 be Point of Euclid n, p1, p2, p3, p4 be Point of
  TOP-REAL n such that
A1: e1 = p1 and
A2: e2 = p2 and
A3: e3 = p3 and
A4: e4 = p4 and
A5: e5 = x*p1+y*p3 and
A6: e6 = x*p2+y*p4 and
A7: dist(e1,e2) < p and
A8: dist(e3,e4) < q and
A9: x <> 0 and
A10: y <> 0;
  reconsider f1 = e1, f2 = e2, f3 = e3, f4 = e4, f5 = e5, f6 = e6 as Element
  of REAL n by A1,A2,A3,A4,A5,A6,EUCLID:22;
A11: x*f2 = x*p2 & y*f4 = y*p4 by A2,A4;
  x*f1 = x*p1 & y*f3 = y*p3 by A1,A3;
  then
A12: f5 = x*f1+y*f3 by A5;
A13: 0 < |.y.| by A10,COMPLEX1:47;
  dist(e3,e4) = |. f3-f4 .| by SPPOL_1:5;
  then
A14: |.y.|*|.f3-f4.| < |.y.|*q by A8,A13,XREAL_1:68;
A15: 0 < |.x.| by A9,COMPLEX1:47;
  dist(e1,e2) = |. f1-f2 .| by SPPOL_1:5;
  then |.x.|*|.f1-f2.| < |.x.|*p by A7,A15,XREAL_1:68;
  then
A16: |.x.|*|.f1-f2.| + |.y.|*|.f3-f4.| < |.x.|*p + |.y.|*q by A14,XREAL_1:8;
  |. x*(f1-f2) + y*(f3-f4) .| <= |. x*(f1-f2) .| + |. y*(f3-f4) .| by EUCLID:12
;
  then |. x*(f1-f2) + y*(f3-f4) .| <= |.x*(f1-f2).| + |.y.|*|.(f3-f4).| by
EUCLID:11;
  then
A17: |. x*(f1-f2) + y*(f3-f4) .| <= |.x.|*|.f1-f2.| + |.y.|*|.f3-f4 .| by
EUCLID:11;
  dist(e5,e6) = |.f5-f6.| by SPPOL_1:5
    .= |. x*f1+y*f3 - (x*f2+y*f4) .| by A6,A12,A11
    .= |. (x*f1-x*f2) + (y*f3-y*f4) .| by Th9
    .= |. x*(f1-f2) + (y*f3-y*f4) .| by Th7
    .= |. x*(f1-f2) + y*(f3-f4) .| by Th7;
  hence thesis by A17,A16,XXREAL_0:2;
end;
