let x, y be Element of [:REAL,REAL,REAL:]; :: thesis: taxi_dist3 . (x,y) = taxi_dist3 . (y,x)
reconsider x1 = x `1 , x2 = x `2 , x3 = x `3 , y1 = y `1 , y2 = y `2 , y3 = y `3 as Element of REAL ;
A1: ( x = [x1,x2,x3] & y = [y1,y2,y3] ) by MCART_1:48;
then taxi_dist3 . (x,y) = ((real_dist . (x1,y1)) + (real_dist . (x2,y2))) + (real_dist . (x3,y3)) by Def11
.= ((real_dist . (y1,x1)) + (real_dist . (x2,y2))) + (real_dist . (x3,y3)) by METRIC_1:10
.= ((real_dist . (y1,x1)) + (real_dist . (y2,x2))) + (real_dist . (x3,y3)) by METRIC_1:10
.= ((real_dist . (y1,x1)) + (real_dist . (y2,x2))) + (real_dist . (y3,x3)) by METRIC_1:10
.= taxi_dist3 . (y,x) by A1, Def11 ;
hence taxi_dist3 . (x,y) = taxi_dist3 . (y,x) ; :: thesis: verum