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

theorem Th9:
  for x,y,z being Element of [:the carrier of X,the carrier of Y,
  the carrier of Z,the carrier of W:] holds dist_cart4(X,Y,Z,W).(x,z) <=
  dist_cart4(X,Y,Z,W).(x,y) + dist_cart4(X,Y,Z,W).(y,z)
proof
  let x,y,z be Element of [:the carrier of X,the carrier of Y,the carrier of Z
  ,the carrier of W:];
  reconsider x1 = x`1_4, y1 = y`1_4, z1 = z`1_4 as Element of X;
  reconsider x2 = x`2_4, y2 = y`2_4, z2 = z`2_4 as Element of Y;
  reconsider x3 = x`3_4, y3 = y`3_4, z3 = z`3_4 as Element of Z;
  reconsider x4 = x`4_4, y4 = y`4_4, z4 = z`4_4 as Element of W;
A1: x = [x1,x2,x3,x4];
  set d7 = dist(x3,z3), d8 = dist(x3,y3), d9 = dist(y3,z3);
  set d1 = dist(x1,z1), d2 = dist(x1,y1), d3 = dist(y1,z1);
A2: y = [y1,y2,y3,y4];
  set d10 = dist(x4,z4), d11 = dist(x4,y4), d12 = dist(y4,z4);
  set d4 = dist(x2,z2), d5 = dist(x2,y2), d6 = dist(y2,z2);
A3: z = [z1,z2,z3,z4];
  set d16 = d7 + d10;
  set d14 = d1 + d4;
  set d17 = (d8 + d9) + (d11 + d12), d15 = (d2 + d3) + (d5 + d6);
  d7 <= d8 + d9 & d10 <= d11 + d12 by METRIC_1:4;
  then
A4: d16 <= d17 by XREAL_1:7;
  d1 <= d2 + d3 & d4 <= d5 + d6 by METRIC_1:4;
  then d14 <= d15 by XREAL_1:7;
  then
A5: d14 + d16 <= d15 + d17 by A4,XREAL_1:7;
  (d15 + d17) = ((d2 + d5) + (d8 + d11)) + ((d3 + d6) + (d9 + d12))
    .= dist_cart4(X,Y,Z,W).(x,y) + ((d3 +d6) + (d9 + d12)) by A1,A2,Def7
    .= dist_cart4(X,Y,Z,W).(x,y) + dist_cart4(X,Y,Z,W).(y,z) by A2,A3,Def7;
  hence thesis by A1,A3,A5,Def7;
end;
