let X, Y, Z be non empty MetrSpace; :: thesis: for x, y being Element of [: the carrier of X, the carrier of Y, the carrier of Z:] holds (dist_cart3 (X,Y,Z)) . (x,y) = (dist_cart3 (X,Y,Z)) . (y,x)
let x, y be Element of [: the carrier of X, the carrier of Y, the carrier of Z:]; :: thesis: (dist_cart3 (X,Y,Z)) . (x,y) = (dist_cart3 (X,Y,Z)) . (y,x)
reconsider x1 = x `1_3 , y1 = y `1_3 as Element of X ;
reconsider x2 = x `2_3 , y2 = y `2_3 as Element of Y ;
reconsider x3 = x `3_3 , y3 = y `3_3 as Element of Z ;
A1: ( x = [x1,x2,x3] & y = [y1,y2,y3] ) ;
then (dist_cart3 (X,Y,Z)) . (x,y) = ((dist (y1,x1)) + (dist (y2,x2))) + (dist (y3,x3)) by Def4
.= (dist_cart3 (X,Y,Z)) . (y,x) by A1, Def4 ;
hence (dist_cart3 (X,Y,Z)) . (x,y) = (dist_cart3 (X,Y,Z)) . (y,x) ; :: thesis: verum