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

theorem Th8:
  for x,y being Element of [:the carrier of X,the carrier of Y,the
carrier of Z,the carrier of W:] holds dist_cart4(X,Y,Z,W).(x,y) = dist_cart4(X,
  Y,Z,W).(y,x)
proof
  let x,y be Element of [:the carrier of X,the carrier of Y,the carrier of Z,
  the carrier of W:];
  reconsider x1 = x`1_4, y1 = y`1_4 as Element of X;
  reconsider x2 = x`2_4, y2 = y`2_4 as Element of Y;
  reconsider x3 = x`3_4, y3 = y`3_4 as Element of Z;
  reconsider x4 = x`4_4, y4 = y`4_4 as Element of W;
A1: x = [x1,x2,x3,x4] & y = [y1,y2,y3,y4];
  then
  dist_cart4(X,Y,Z,W).(x,y) = (dist(y1,x1) + dist(y2,x2)) + (dist(y3,x3) +
  dist(x4,y4)) by Def7
    .= dist_cart4(X,Y,Z,W).(y,x) by A1,Def7;
  hence thesis;
end;
