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

theorem Th19:
  for x,y being Element of [:REAL,REAL:] holds taxi_dist2.(x,y) = 0 iff x = y
proof
  let x,y be Element of [:REAL,REAL:];
  reconsider x1 = x`1, x2 = x`2, y1 = y`1, y2 = y`2 as Element of REAL;
A1: x = [x1,x2] & y = [y1,y2];
  thus taxi_dist2.(x,y) = 0 implies x = y
  proof
    set d2 = real_dist.(x2,y2);
    set d1 = real_dist.(x1,y1);
    d1 = |.x1 - y1.| by METRIC_1:def 12;
    then
A2: 0 <= d1 by COMPLEX1:46;
    d2 = |.x2 - y2.| by METRIC_1:def 12;
    then
A3: 0 <= d2 by COMPLEX1:46;
    assume taxi_dist2.(x,y) = 0;
    then
A4: d1 + d2 = 0 by A1,Def16;
    then d1 = 0 by A2,A3,XREAL_1:27;
    then
A5: x1 = y1 by METRIC_1:8;
    d2 = 0 by A4,A2,A3,XREAL_1:27;
    hence thesis by A1,A5,METRIC_1:8;
  end;
  assume
A6: x = y;
  then
A7: real_dist.(x2,y2) = 0 by METRIC_1:8;
  taxi_dist2.(x,y) = real_dist.(x1,y1) + real_dist.(x2,y2) by A1,Def16
    .= 0 by A6,A7,METRIC_1:8;
  hence thesis;
end;
