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

theorem Th5:
  for x,y being Element of [:the carrier of X,the carrier of Y,the
  carrier of Z:] holds dist_cart3(X,Y,Z).(x,y) = dist_cart3(X,Y,Z).(y,x)
proof
  let x,y be Element of [:the carrier of X,the carrier of Y,the carrier of Z:];
  reconsider x1 = x`1_3, y1 = y`1_3 as Element of X;
  reconsider x2 = x`2_3, y2 = y`2_3 as Element of Y;
  reconsider x3 = x`3_3, y3 = y`3_3 as Element of Z;
A1: x = [x1,x2,x3] & y = [y1,y2,y3];
  then dist_cart3(X,Y,Z).(x,y) = (dist(y1,x1) + dist(y2,x2)) + dist(y3,x3) by
Def4
    .= dist_cart3(X,Y,Z).(y,x) by A1,Def4;
  hence thesis;
end;
