reserve X, Y, Z, W for non empty MetrSpace;

theorem Th3:
  for x,y,z being Element of [:the carrier of X,the carrier of Y:]
  holds dist_cart2(X,Y).(x,z) <= dist_cart2(X,Y).(x,y) + dist_cart2(X,Y).(y,z)
proof
  let x,y,z be Element of [:the carrier of X,the carrier of Y:];
  reconsider x1 = x`1, y1 = y`1, z1 = z`1 as Element of X;
  reconsider x2 = x`2, y2 = y`2, z2 = z`2 as Element of Y;
A1: y = [y1,y2];
  set d6 = dist(y2,z2);
  set d5 = dist(x2,y2);
  set d4 = dist(x2,z2);
  set d3 = dist(y1,z1);
  set d2 = dist(x1,y1);
  set d1 = dist(x1,z1);
A2: z = [z1,z2];
A3: x = [x1,x2];
  then
A4: dist_cart2(X,Y).(x,z) = d1 + d4 by A2,Def1;
A5: d1 <= d2 + d3 & d4 <= d5 + d6 by METRIC_1:4;
  (d2 + d3) + (d5 + d6) = (d2 + d5) + (d3 + d6)
    .= dist_cart2(X,Y).(x,y) + (d3 +d6) by A3,A1,Def1
    .= dist_cart2(X,Y).(x,y) + dist_cart2(X,Y).(y,z) by A1,A2,Def1;
  hence thesis by A5,A4,XREAL_1:7;
end;
