
theorem Th28:
  for I being non empty set
  for J being RelStr-yielding non-Empty ManySortedSet of I
  for x,y being Element of product J holds
  x <= y iff for i being Element of I holds x.i <= y.i
proof
  let I be non empty set;
  let J be RelStr-yielding non-Empty ManySortedSet of I;
  set L = product J;
  let x,y be Element of L;
A1: the carrier of L = product Carrier J by YELLOW_1:def 4;
  hereby
    assume
A2: x <= y;
    let i be Element of I;
    ex f,g being Function st f = x & g = y &
for i be object st i in I
    ex R being RelStr, xi,yi being Element of R
    st R = J.i & xi = f.i & yi = g.i & xi <= yi by A1,A2,YELLOW_1:def 4;
    then ex R being RelStr, xi,yi being Element of R
    st R = J.i & xi = x.i & yi = y.i & xi <= yi;
    hence x.i <= y.i;
  end;
  assume
A3: for i being Element of I holds x.i <= y.i;
  ex f,g being Function st f = x & g = y &
for i be object st i in I
  ex R being RelStr, xi,yi being Element of R
  st R = J.i & xi = f.i & yi = g.i & xi <= yi
  proof
    take f = x, g = y;
    thus f = x & g = y;
    let i be object;
    assume i in I;
    then reconsider j = i as Element of I;
    take J.j, x.j, y.j;
    thus thesis by A3;
  end;
  hence thesis by A1,YELLOW_1:def 4;
end;
