let x, y be Element of [:REAL ,REAL :]; :: thesis: ( taxi_dist2 . x,y = 0 iff x = y )
reconsider x1 = x `1 , x2 = x `2 , y1 = y `1 , y2 = y `2 as Element of REAL by MCART_1:10;
A1: ( x = [x1,x2] & y = [y1,y2] ) by MCART_1:24;
thus ( taxi_dist2 . x,y = 0 implies x = y ) :: thesis: ( x = y implies taxi_dist2 . x,y = 0 )

proof

assume A2: taxi_dist2 . x,y = 0 ; :: thesis: x = y
set d1 = real_dist . x1,y1;
set d2 = real_dist . x2,y2;
A3: (real_dist . x1,y1) + (real_dist . x2,y2) = 0 by A1, A2, Def7;
real_dist . x1,y1 = abs (x1 - y1) by METRIC_1:def 13;
then A4: 0 <= real_dist . x1,y1 by COMPLEX1:132;
real_dist . x2,y2 = abs (x2 - y2) by METRIC_1:def 13;
then A5: 0 <= real_dist . x2,y2 by COMPLEX1:132;
then A6: ( real_dist . x1,y1 = 0 & real_dist . x2,y2 = 0 ) by A3, A4, XREAL_1:29;
real_dist . x1,y1 = 0 by A3, A4, A5, XREAL_1:29;
then x1 = y1 by METRIC_1:9;
hence x = y by A1, A6, METRIC_1:9; :: thesis: verum

end;

assume A7: x = y ; :: thesis: taxi_dist2 . x,y = 0
then A8: real_dist . x2,y2 = 0 by METRIC_1:9;
taxi_dist2 . x,y = (real_dist . x1,y1) + (real_dist . x2,y2) by A1, Def7
.= 0 by A7, A8, METRIC_1:9 ;
hence taxi_dist2 . x,y = 0 ; :: thesis: verum