reserve X, Y, Z, W for non empty MetrSpace;
reserve X,Y for non empty MetrSpace;
reserve Z for non empty MetrSpace;

theorem Th26:
  for x,y being Element of [:REAL,REAL,REAL:] holds taxi_dist3.(x,
  y) = taxi_dist3.(y,x)
proof
  let x,y be Element of [:REAL,REAL,REAL:];
  reconsider x1 = x`1_3, x2 = x`2_3, x3 = x`3_3,
   y1 = y`1_3, y2 = y`2_3, y3 = y`3_3 as
  Element of REAL;
A1: x = [x1,x2,x3] & y = [y1,y2,y3];
  then
  taxi_dist3.(x,y) = real_dist.(x1,y1) + real_dist.(x2,y2) + real_dist.(x3
  ,y3) by Def20
    .= real_dist.(y1,x1) + real_dist.(x2,y2) + real_dist.(x3,y3) by METRIC_1:9
    .= real_dist.(y1,x1) + real_dist.(y2,x2) + real_dist.(x3,y3) by METRIC_1:9
    .= real_dist.(y1,x1) + real_dist.(y2,x2) + real_dist.(y3,x3) by METRIC_1:9
    .= taxi_dist3.(y,x) by A1,Def20;
  hence thesis;
end;
