let X, Y be non empty MetrSpace; 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))
let x, y, z be Element of [: the carrier of X, the carrier of Y:]; (dist_cart2 (X,Y)) . (x,z) <= ((dist_cart2 (X,Y)) . (x,y)) + ((dist_cart2 (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:
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) = (dist (x1,z1)) + (dist (x2,z2))
by A2, Def1;
A5:
( dist (x1,z1) <= (dist (x1,y1)) + (dist (y1,z1)) & dist (x2,z2) <= (dist (x2,y2)) + (dist (y2,z2)) )
by METRIC_1:4;
((dist (x1,y1)) + (dist (y1,z1))) + ((dist (x2,y2)) + (dist (y2,z2))) =
((dist (x1,y1)) + (dist (x2,y2))) + ((dist (y1,z1)) + (dist (y2,z2)))
.=
((dist_cart2 (X,Y)) . (x,y)) + ((dist (y1,z1)) + (dist (y2,z2)))
by A3, A1, Def1
.=
((dist_cart2 (X,Y)) . (x,y)) + ((dist_cart2 (X,Y)) . (y,z))
by A1, A2, Def1
;
hence
(dist_cart2 (X,Y)) . (x,z) <= ((dist_cart2 (X,Y)) . (x,y)) + ((dist_cart2 (X,Y)) . (y,z))
by A5, A4, XREAL_1:7; verum