let X, Y be non empty MetrSpace; for x, y, z being Element of [: the carrier of X, the carrier of Y:] holds (dist_cart2S (X,Y)) . (x,z) <= ((dist_cart2S (X,Y)) . (x,y)) + ((dist_cart2S (X,Y)) . (y,z))
let x, y, z be Element of [: the carrier of X, the carrier of Y:]; (dist_cart2S (X,Y)) . (x,z) <= ((dist_cart2S (X,Y)) . (x,y)) + ((dist_cart2S (X,Y)) . (y,z))
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:
x = [x1,x2]
;
set d5 = dist (x2,y2);
set d3 = dist (y1,z1);
set d1 = dist (x1,z1);
A2:
y = [y1,y2]
;
set d6 = dist (y2,z2);
set d4 = dist (x2,z2);
set d2 = dist (x1,y1);
A3:
z = [z1,z2]
;
( 0 <= (dist (x1,z1)) ^2 & 0 <= (dist (x2,z2)) ^2 )
by XREAL_1:63;
then A4:
0 + 0 <= ((dist (x1,z1)) ^2) + ((dist (x2,z2)) ^2)
by XREAL_1:7;
( dist (x2,z2) <= (dist (x2,y2)) + (dist (y2,z2)) & 0 <= dist (x2,z2) )
by METRIC_1:4, METRIC_1:5;
then A5:
(dist (x2,z2)) ^2 <= ((dist (x2,y2)) + (dist (y2,z2))) ^2
by SQUARE_1:15;
( dist (x1,z1) <= (dist (x1,y1)) + (dist (y1,z1)) & 0 <= dist (x1,z1) )
by METRIC_1:4, METRIC_1:5;
then
(dist (x1,z1)) ^2 <= ((dist (x1,y1)) + (dist (y1,z1))) ^2
by SQUARE_1:15;
then
((dist (x1,z1)) ^2) + ((dist (x2,z2)) ^2) <= (((dist (x1,y1)) + (dist (y1,z1))) ^2) + (((dist (x2,y2)) + (dist (y2,z2))) ^2)
by A5, XREAL_1:7;
then A6:
sqrt (((dist (x1,z1)) ^2) + ((dist (x2,z2)) ^2)) <= sqrt ((((dist (x1,y1)) + (dist (y1,z1))) ^2) + (((dist (x2,y2)) + (dist (y2,z2))) ^2))
by A4, SQUARE_1:26;
A7:
( 0 <= dist (x2,y2) & 0 <= dist (y2,z2) )
by METRIC_1:5;
( 0 <= dist (x1,y1) & 0 <= dist (y1,z1) )
by METRIC_1:5;
then
sqrt ((((dist (x1,y1)) + (dist (y1,z1))) ^2) + (((dist (x2,y2)) + (dist (y2,z2))) ^2)) <= (sqrt (((dist (x1,y1)) ^2) + ((dist (x2,y2)) ^2))) + (sqrt (((dist (y1,z1)) ^2) + ((dist (y2,z2)) ^2)))
by A7, Th12;
then
sqrt (((dist (x1,z1)) ^2) + ((dist (x2,z2)) ^2)) <= (sqrt (((dist (x1,y1)) ^2) + ((dist (x2,y2)) ^2))) + (sqrt (((dist (y1,z1)) ^2) + ((dist (y2,z2)) ^2)))
by A6, XXREAL_0:2;
then
(dist_cart2S (X,Y)) . (x,z) <= (sqrt (((dist (x1,y1)) ^2) + ((dist (x2,y2)) ^2))) + (sqrt (((dist (y1,z1)) ^2) + ((dist (y2,z2)) ^2)))
by A1, A3, Def10;
then
(dist_cart2S (X,Y)) . (x,z) <= ((dist_cart2S (X,Y)) . (x,y)) + (sqrt (((dist (y1,z1)) ^2) + ((dist (y2,z2)) ^2)))
by A1, A2, Def10;
hence
(dist_cart2S (X,Y)) . (x,z) <= ((dist_cart2S (X,Y)) . (x,y)) + ((dist_cart2S (X,Y)) . (y,z))
by A2, A3, Def10; verum