set M = MetrStruct(# [:the carrier of X,the carrier of Y,the carrier of Z:],(dist_cart3 X,Y,Z) #);

now

let x, y, z be Element of MetrStruct(# [:the carrier of X,the carrier of Y,the carrier of Z:],(dist_cart3 X,Y,Z) #); :: thesis: ( ( dist x,y = 0 implies x = y ) & ( x = y implies dist x,y = 0 ) & dist x,y = dist y,x & dist x,z <= (dist x,y) + (dist y,z) )
A1: dist x,y = (dist_cart3 X,Y,Z) . x,y by METRIC_1:def 1;
hence ( dist x,y = 0 iff x = y ) by Th12; :: thesis: ( dist x,y = dist y,x & dist x,z <= (dist x,y) + (dist y,z) )
dist y,x = (dist_cart3 X,Y,Z) . y,x by METRIC_1:def 1;
hence dist x,y = dist y,x by A1, Th13; :: thesis: dist x,z <= (dist x,y) + (dist y,z)
( dist x,z = (dist_cart3 X,Y,Z) . x,z & dist y,z = (dist_cart3 X,Y,Z) . y,z ) by METRIC_1:def 1;
hence dist x,z <= (dist x,y) + (dist y,z) by A1, Th14; :: thesis: verum

end;

hence MetrStruct(# [:the carrier of X,the carrier of Y,the carrier of Z:],(dist_cart3 X,Y,Z) #) is non empty strict MetrSpace by METRIC_1:6; :: thesis: verum