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

theorem Th25:
  for x,y being Element of [:REAL,REAL,REAL:] holds
  taxi_dist3.(x,y) = 0 iff x = y
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];
  thus taxi_dist3.(x,y) = 0 implies x = y
  proof
    set d3 = real_dist.(x3,y3);
    set d2 = real_dist.(x2,y2);
    set d1 = real_dist.(x1,y1);
    set d4 = d1 + d2;
    d3 = |.x3 - y3.| by METRIC_1:def 12;
    then
A2: 0 <= d3 by COMPLEX1:46;
    d1 = |.x1 - y1.| by METRIC_1:def 12;
    then
A3: 0 <= d1 by COMPLEX1:46;
    assume taxi_dist3.(x,y) = 0;
    then
A4: d4 + d3 = 0 by A1,Def20;
    d2 = |.x2 - y2.| by METRIC_1:def 12;
    then
A5: 0 <= d2 by COMPLEX1:46;
    then
A6: 0 + 0 <= d1 + d2 by A3,XREAL_1:7;
    then
A7: d4 = 0 by A4,A2,XREAL_1:27;
    then d1 = 0 by A5,A3,XREAL_1:27;
    then
A8: x1 = y1 by METRIC_1:8;
    d3 = 0 by A4,A2,A6,XREAL_1:27;
    then
A9: x3 = y3 by METRIC_1:8;
    d2 = 0 by A5,A3,A7,XREAL_1:27;
    hence thesis by A1,A9,A8,METRIC_1:8;
  end;
  assume
A10: x = y;
  then
A11: real_dist.(x1,y1) = 0 & real_dist.(x2,y2) = 0 by METRIC_1:8;
  taxi_dist3.(x,y) = real_dist.(x1,y1) + real_dist.(x2,y2) + real_dist.(
  x3,y3) by A1,Def20
    .= 0 by A10,A11,METRIC_1:8;
  hence thesis;
end;
