now :: thesis: for x, y, z being Element of MetrStruct(# [: the carrier of X, the carrier of Y, the carrier of Z:],(dist_cart3S (X,Y,Z)) #) holds

( ( 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)) )

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( ( 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)) )

let x, y, z be Element of MetrStruct(# [: the carrier of X, the carrier of Y, the carrier of Z:],(dist_cart3S (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)) )

reconsider x9 = x, y9 = y, z9 = 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)) . (x9,y9) by METRIC_1:def 1;

hence ( dist (x,y) = 0 iff x = y ) by Th14; :: thesis: ( dist (x,y) = dist (y,x) & dist (x,z) <= (dist (x,y)) + (dist (y,z)) )

dist (y,x) = (dist_cart3S (X,Y,Z)) . (y9,x9) by METRIC_1:def 1;

hence dist (x,y) = dist (y,x) by A1, Th15; :: thesis: dist (x,z) <= (dist (x,y)) + (dist (y,z))

( dist (x,z) = (dist_cart3S (X,Y,Z)) . (x9,z9) & dist (y,z) = (dist_cart3S (X,Y,Z)) . (y9,z9) ) by METRIC_1:def 1;

hence dist (x,z) <= (dist (x,y)) + (dist (y,z)) by A1, Th18; :: thesis: verum

end;reconsider x9 = x, y9 = y, z9 = 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)) . (x9,y9) by METRIC_1:def 1;

hence ( dist (x,y) = 0 iff x = y ) by Th14; :: thesis: ( dist (x,y) = dist (y,x) & dist (x,z) <= (dist (x,y)) + (dist (y,z)) )

dist (y,x) = (dist_cart3S (X,Y,Z)) . (y9,x9) by METRIC_1:def 1;

hence dist (x,y) = dist (y,x) by A1, Th15; :: thesis: dist (x,z) <= (dist (x,y)) + (dist (y,z))

( dist (x,z) = (dist_cart3S (X,Y,Z)) . (x9,z9) & dist (y,z) = (dist_cart3S (X,Y,Z)) . (y9,z9) ) by METRIC_1:def 1;

hence dist (x,z) <= (dist (x,y)) + (dist (y,z)) by A1, Th18; :: thesis: verum