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

theorem Th10:
  for x,y being Element of [:the carrier of X,the carrier of Y:]
  holds dist_cart2S(X,Y).(x,y) = 0 iff x = y
proof
  let x,y be Element of [:the carrier of X,the carrier of Y:];
  reconsider x1 = x`1, y1 = y`1 as Element of X;
  reconsider x2 = x`2, y2 = y`2 as Element of Y;
A1: x = [x1,x2] & y = [y1,y2];
  thus dist_cart2S(X,Y).(x,y) = 0 implies x = y
  proof
    set d2 = dist(x2,y2);
    set d1 = dist(x1,y1);
    assume dist_cart2S(X,Y).(x,y) = 0;
    then
A2: sqrt(d1^2 + d2^2) = 0 by A1,Def10;
A3: 0 <= d1^2 & 0 <= d2^2 by XREAL_1:63;
    then d1 = 0 by A2,Lm1;
    then
A4: x1 = y1 by METRIC_1:2;
    d2 = 0 by A2,A3,Lm1;
    hence thesis by A1,A4,METRIC_1:2;
  end;
  assume x = y;
  then
A5: (dist(x1,y1))^2 = 0^2 & (dist(x2,y2))^2 = 0^2 by METRIC_1:1;
  dist_cart2S(X,Y).(x,y) = sqrt((dist(x1,y1))^2 + (dist(x2,y2))^2) by A1,Def10
    .= 0 by A5;
  hence thesis;
end;
