let X, Y, Z be non empty MetrSpace; :: thesis: for x, y, z being Element of [:the carrier of X,the carrier of Y,the carrier of Z:] holds (dist_cart3 X,Y,Z) . x,z <= ((dist_cart3 X,Y,Z) . x,y) + ((dist_cart3 X,Y,Z) . y,z)
let x, y, z be Element of [:the carrier of X,the carrier of Y,the carrier of Z:]; :: thesis: (dist_cart3 X,Y,Z) . x,z <= ((dist_cart3 X,Y,Z) . x,y) + ((dist_cart3 X,Y,Z) . 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 ;
reconsider x3 = x `3 , y3 = y `3 , z3 = z `3 as Element of Z ;
A1: ( x = [x1,x2,x3] & y = [y1,y2,y3] & z = [z1,z2,z3] ) by MCART_1:48;
set d1 = dist x1,z1;
set d2 = dist x1,y1;
set d3 = dist y1,z1;
set d4 = dist x2,z2;
set d5 = dist x2,y2;
set d6 = dist y2,z2;
set d7 = dist x3,z3;
set d8 = dist x3,y3;
set d9 = dist y3,z3;
A2: (((dist x1,y1) + (dist y1,z1)) + ((dist x2,y2) + (dist y2,z2))) + ((dist x3,y3) + (dist y3,z3)) = (((dist x1,y1) + (dist x2,y2)) + (dist x3,y3)) + (((dist y1,z1) + (dist y2,z2)) + (dist y3,z3))
.= ((dist_cart3 X,Y,Z) . x,y) + (((dist y1,z1) + (dist y2,z2)) + (dist y3,z3)) by A1, Def4
.= ((dist_cart3 X,Y,Z) . x,y) + ((dist_cart3 X,Y,Z) . y,z) by A1, Def4 ;
A3: dist x1,z1 <= (dist x1,y1) + (dist y1,z1) by METRIC_1:4;
A4: dist x2,z2 <= (dist x2,y2) + (dist y2,z2) by METRIC_1:4;
set d10 = (dist x1,z1) + (dist x2,z2);
A5: (dist x1,z1) + (dist x2,z2) <= ((dist x1,y1) + (dist y1,z1)) + ((dist x2,y2) + (dist y2,z2)) by A3, A4, XREAL_1:9;
dist x3,z3 <= (dist x3,y3) + (dist y3,z3) by METRIC_1:4;
then ((dist x1,z1) + (dist x2,z2)) + (dist x3,z3) <= (((dist x1,y1) + (dist y1,z1)) + ((dist x2,y2) + (dist y2,z2))) + ((dist x3,y3) + (dist y3,z3)) by A5, XREAL_1:9;
hence (dist_cart3 X,Y,Z) . x,z <= ((dist_cart3 X,Y,Z) . x,y) + ((dist_cart3 X,Y,Z) . y,z) by A1, A2, Def4; :: thesis: verum