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