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 , y1 = y `1 as Element of X ;
reconsider x2 = x `2 , y2 = y `2 as Element of Y ;
reconsider x3 = x `3 , y3 = y `3 as Element of Z ;
A1: ( x = [x1,x2,x3] & y = [y1,y2,y3] ) by MCART_1:48;
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