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

theorem Th18:
  for x,y,z being Element of [:the carrier of X,the carrier of Y,
the carrier of Z:] holds dist_cart3S(X,Y,Z).(x,z) <= dist_cart3S(X,Y,Z).(x,y) +
  dist_cart3S(X,Y,Z).(y,z)
proof
  let x,y,z be Element of [:the carrier of X,the carrier of Y,the carrier of Z
  :];
  reconsider x1 = x`1_3, y1 = y`1_3, z1 = z`1_3 as Element of X;
  reconsider x2 = x`2_3, y2 = y`2_3, z2 = z`2_3 as Element of Y;
  reconsider x3 = x`3_3, y3 = y`3_3, z3 = z`3_3 as Element of Z;
A1: x = [x1,x2,x3];
  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];
  d7 <= d8 + d9 & 0 <= d7 by METRIC_1:4,5;
  then
A3: d7^2 <= (d8 + d9)^2 by SQUARE_1:15;
A4: 0 <= d8 & 0 <= d9 by METRIC_1:5;
  set d4 = dist(x2,z2), d5 = dist(x2,y2), d6 = dist(y2,z2);
A5: z = [z1,z2,z3];
  d4 <= d5 + d6 & 0 <= d4 by METRIC_1:4,5;
  then
A6: d4^2 <= (d5 + d6)^2 by SQUARE_1:15;
A7: 0 <= d5 & 0 <= d6 by METRIC_1:5;
  0 <= d1^2 & 0 <= d4^2 by XREAL_1:63;
  then
A8: 0 + 0 <= d1^2 + d4^2 by XREAL_1:7;
  d1 <= d2 + d3 & 0 <= d1 by METRIC_1:4,5;
  then d1^2 <= (d2 + d3)^2 by SQUARE_1:15;
  then d1^2 + d4^2 <= (d2 + d3)^2 + (d5 + d6)^2 by A6,XREAL_1:7;
  then
A9: d1^2 + d4^2 + d7^2 <= (d2 + d3)^2 + (d5 + d6)^2 + (d8 + d9)^2 by A3,
XREAL_1:7;
  0 <= d7^2 by XREAL_1:63;
  then 0 + 0 <= (d1^2 + d4^2) + d7^2 by A8,XREAL_1:7;
  then
A10: sqrt(d1^2 + d4^2 + d7^2) <= sqrt((d2 + d3)^2 + (d5 + d6) ^2 + (d8 + d9)
  ^2) by A9,SQUARE_1:26;
  0 <= d2 & 0 <= d3 by METRIC_1:5;
  then sqrt((d2 + d3)^2 + (d5 + d6)^2 + (d8 + d9)^2) <= sqrt(d2^2 + d5^2 + d8
  ^2) + sqrt(d3^2 + d6^2 + d9^2) by A7,A4,Lm2;
  then
  sqrt(d1^2 + d4^2 + d7^2) <= sqrt(d2^2 + d5^2 + d8^2) + sqrt(d3^2 + d6^2
  + d9^2) by A10,XXREAL_0:2;
  then dist_cart3S(X,Y,Z).(x,z) <= sqrt(d2^2 + d5^2 + d8^2) + sqrt((d3)^2 + (
  d6)^2 + d9^2 ) by A1,A5,Def13;
  then dist_cart3S(X,Y,Z).(x,z) <= dist_cart3S(X,Y,Z).(x,y) + sqrt((d3)^2 + (
  d6)^2 + d9^2) by A1,A2,Def13;
  hence thesis by A2,A5,Def13;
end;
