now

let x, y, z be Element of ; :: 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) )
reconsider x' = x, y' = y, z' = z as Element of [:the carrier of X,the carrier of Y,the carrier of Z:] ;
A1: dist x,y = (dist_cart3S X,Y,Z) . x',y' by METRIC_1:def 1;
hence ( dist x,y = 0 iff x = y ) by Th10; :: thesis: ( dist x,y = dist y,x & dist x,z <= (dist x,y) + (dist y,z) )
dist y,x = (dist_cart3S X,Y,Z) . y',x' by METRIC_1:def 1;
hence dist x,y = dist y,x by A1, Th11; :: thesis: dist x,z <= (dist x,y) + (dist y,z)
( dist x,z = (dist_cart3S X,Y,Z) . x',z' & dist y,z = (dist_cart3S X,Y,Z) . y',z' ) by METRIC_1:def 1;
hence dist x,z <= (dist x,y) + (dist y,z) by A1, Th16; :: thesis: verum

end;

hence MetrStruct(# [:the carrier of X,the carrier of Y,the carrier of Z:],(dist_cart3S X,Y,Z) #) is non empty strict MetrSpace by METRIC_1:6; :: thesis: verum