
theorem Th7:
  for X being MetrSpace,
      x, y, v,w being Element of X holds
    |. dist(x,y) - dist(v,w) .| <= dist (x,v) + dist (y,w)
proof
  let X be MetrSpace,
      x, z, v,w be Element of X;
  dist (x,z)- dist (v,w) = dist (x,z) - dist (v,z)
               + (dist (v,z) - dist (v,w)); then
A1: |. dist (x,z) - dist (v,w).|
        <= |.dist (x,z) - dist (v,z).|
         + |.dist (v,z) - dist (v,w).| by COMPLEX1:56;
A2: |.dist (x,z) - dist (v,z).| <= dist(x,v) by METRIC_6:1;
   |.dist (v,z) - dist (v,w).| <= dist(z,w) by METRIC_6:1; then
   |.dist (x,z) - dist (v,z).| + |.dist (v,z)- dist (v,w).|
      <= dist(x,v) + dist(z,w) by XREAL_1:7,A2;
  hence thesis by A1,XXREAL_0:2;
end;
