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

theorem
  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 = p1+p3
  & e6 = p2+p4 & dist(e1,e2) < x & dist(e3,e4) < y holds dist(e5,e6) < x+y
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 = p1+p3 and
A6: e6 = p2+p4 and
A7: dist(e1,e2) < x & dist(e3,e4) < y;
  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;
A8: |. f1-f2 + (f3-f4) .| <= |. f1-f2 .| + |. f3-f4 .| & dist(e1,e2) + dist
  (e3, e4) < x + y by A7,EUCLID:12,XREAL_1:8;
  reconsider u1 = f1, u2 = f2, u3 = f3, u4 = f4, u6 = f6 as Element of n
  -tuples_on REAL by EUCLID:def 1;
  u2+u4 = u6 by A2,A4,A6;
  then
A9: f1+f3-f6 = (u1-u2)+(u3-u4) by Th9
    .= (f1-f2) + (f3-f4);
A10: dist(e1,e2) = |. f1-f2 .| & dist(e3,e4) = |. f3-f4 .| by SPPOL_1:5;
  dist(e5,e6) = |.f5-f6.| by SPPOL_1:5
    .= |. (f1-f2) + (f3-f4) .| by A1,A3,A5,A9;
  hence thesis by A10,A8,XXREAL_0:2;
end;
