
theorem Th11:
  for X being set, L being non empty RelStr for f,g being Function
  of X, the carrier of L for x,y being Element of L|^X st x = f & y = g holds x
  <= y iff f <= g
proof
  let X be set, L be non empty RelStr;
  let f,g be Function of X, the carrier of L;
  let x,y be Element of L|^X such that
A1: x = f and
A2: y = g;
  set J = X --> L;
A3: L|^X = product J by YELLOW_1:def 5;
A4: the carrier of product J = product Carrier J by YELLOW_1:def 4;
  then
A5: x <= y iff ex f,g being Function st f = x & g = y &
for i be object st i in
X ex R being RelStr, xi,yi being Element of R st R = J.i & xi = f.i & yi = g.i
  & xi <= yi by A3,YELLOW_1:def 4;
  hereby
    assume
A6: x <= y;
    thus f <= g
    proof
      let i be set;
      assume
A7:   i in X;
      then
A8:   J.i = L by FUNCOP_1:7;
      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,A5,A6,A7;
      hence thesis by A8;
    end;
  end;
  assume
A9: for j being set st j in X ex a, b being Element of L st a = f.j & b
  = g.j & a <= b;
  now
    reconsider f9 = f, g9 = g as Function;
    take f9, g9;
    thus f9 = x & g9 = y by A1,A2;
    let i be object;
    assume
A10: i in X;
    then
A11: J.i = L by FUNCOP_1:7;
    ex a, b being Element of L st a = f.i & b = g.i & a <= b by A9,A10;
    hence ex R being RelStr, xi,yi being Element of R st R = J.i & xi = f9.i &
    yi = g9.i & xi <= yi by A11;
  end;
  hence thesis by A4,A3,YELLOW_1:def 4;
end;
