
theorem
  for L being Semilattice, X being Subset of L, x being Element of L st
  ex_sup_of {x} "/\" X,L & ex_sup_of X,L holds sup ({x} "/\" X) <= x "/\" sup X
proof
  let L be Semilattice, X be Subset of L, x be Element of L such that
A1: ex_sup_of {x} "/\" X,L and
A2: ex_sup_of X,L;
A3: {x} "/\" X = {x "/\" y where y is Element of L : y in X} by Th42;
  {x} "/\" X is_<=_than x "/\" sup X
  proof
    let q be Element of L;
    assume q in {x} "/\" X;
    then consider y being Element of L such that
A4: q = x "/\" y and
A5: y in X by A3;
    x <= x & y <= sup X by A2,A5,Th1;
    hence q <= x "/\" sup X by A4,YELLOW_3:2;
  end;
  hence thesis by A1,YELLOW_0:def 9;
end;
