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

theorem Th2:
  for x,y being Element of [:the carrier of X,the carrier of Y:]
  holds dist_cart2(X,Y).(x,y) = dist_cart2(X,Y).(y,x)
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];
  then dist_cart2(X,Y).(x,y) = dist(y1,x1) + dist(x2,y2) by Def1
    .= dist_cart2(X,Y).(y,x) by A1,Def1;
  hence thesis;
end;
