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

theorem Th30:
  for x,y,z being Element of [:REAL,REAL,REAL:] holds Eukl_dist3.(
  x,z) <= Eukl_dist3.(x,y) + Eukl_dist3.(y,z)
proof
  let x,y,z be Element of [:REAL,REAL,REAL:];
  reconsider x1 = x`1_3, x2 = x`2_3, x3 = x`3_3,
     y1 = y`1_3, y2 = y`2_3, y3 = y`3_3, z1 =
  z`1_3, z2 = z`2_3, z3 = z`3_3 as Element of REAL;
A1: x = [x1,x2,x3];
  set d9 = real_dist.(y3,z3);
  set d5 = real_dist.(x2,y2), d6 = real_dist.(y2,z2);
  set d1 = real_dist.(x1,z1), d2 = real_dist.(x1,y1);
A2: y = [y1,y2,y3];
  d9 = |.y3 - z3.| by METRIC_1:def 12;
  then
A3: 0 <= d9 by COMPLEX1:46;
  d6 = |.y2 - z2.| by METRIC_1:def 12;
  then
A4: 0 <= d6 by COMPLEX1:46;
  d5 = |.x2 - y2.| by METRIC_1:def 12;
  then
A5: 0 <= d5 by COMPLEX1:46;
  set d7 = real_dist.(x3,z3), d8 = real_dist.(x3,y3);
  set d3 = real_dist.(y1,z1), d4 = real_dist.(x2,z2);
A6: z = [z1,z2,z3];
  d7 = |.x3 - z3.| by METRIC_1:def 12;
  then 0 <= d7 by COMPLEX1:46;
  then
A7: d7^2 <= (d8 + d9)^2 by METRIC_1:10,SQUARE_1:15;
  d4 = |.x2 - z2.| by METRIC_1:def 12;
  then 0 <= d4 by COMPLEX1:46;
  then
A8: d4^2 <= (d5 + d6)^2 by METRIC_1:10,SQUARE_1:15;
  d1 = |.x1 - z1.| by METRIC_1:def 12;
  then 0 <= d1 by COMPLEX1:46;
  then d1^2 <= (d2 + d3)^2 by METRIC_1:10,SQUARE_1:15;
  then d1^2 + d4^2 <= (d2 + d3)^2 + (d5 + d6)^2 by A8,XREAL_1:7;
  then
A9: d1^2 + d4^2 + d7^2 <= (d2 + d3)^2 + (d5 + d6)^2 + (d8 + d9)^2 by A7,
XREAL_1:7;
  0 <= d1^2 & 0 <= d4^2 by XREAL_1:63;
  then
A10: 0 + 0 <= d1^2 + d4^2 by XREAL_1:7;
  0 <= d7^2 by XREAL_1:63;
  then 0 + 0 <= (d1^2 + d4^2) + d7^2 by A10,XREAL_1:7;
  then
A11: sqrt(d1^2 + d4^2 + d7^2) <= sqrt((d2 + d3)^2 + (d5 + d6)^2 + (d8 + d9)
  ^2) by A9,SQUARE_1:26;
  d8 = |.x3 - y3.| by METRIC_1:def 12;
  then
A12: 0 <= d8 by COMPLEX1:46;
  d3 = |.y1 - z1.| by METRIC_1:def 12;
  then
A13: 0 <= d3 by COMPLEX1:46;
  d2 = |.x1 - y1.| by METRIC_1:def 12;
  then 0 <= d2 by COMPLEX1:46;
  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 A13,A5,A4,A12,A3,Lm2;
  then
  sqrt(d1^2 + d4^2 + d7^2) <= sqrt(d2^2 + d5^2 + d8^2) + sqrt(d3^2 + d6^2
  + d9^2) by A11,XXREAL_0:2;
  then
  Eukl_dist3.(x,z) <= sqrt(d2^2 + d5^2 + d8^2) + sqrt(d3^2 + d6^2 + d9^2)
  by A1,A6,Def22;
  then Eukl_dist3.(x,z) <= Eukl_dist3.(x,y) + sqrt((d3)^2 + (d6)^2 + d9^2) by
A1,A2,Def22;
  hence thesis by A2,A6,Def22;
end;
