
theorem
  for S being complete non empty Poset holds (for x being Element of S
holds x "/\" is lower_adjoint) iff for X being Subset of S, x being Element of
  S holds x "/\" "\/"(X,S) = "\/"({x"/\"y where y is Element of S: y in X},S)
proof
  let S be complete non empty Poset;
  thus (for x being Element of S holds x "/\" is lower_adjoint) implies for X
  being Subset of S, x being Element of S holds x "/\" "\/"(X,S) = "\/"({x"/\"y
  where y is Element of S: y in X},S) by Th63,YELLOW_0:17;
  assume
A1: for X being Subset of S, x being Element of S holds x "/\" "\/"(X,S)
  = "\/"({x"/\"y where y is Element of S: y in X},S);
  let x be Element of S;
  x "/\" is sups-preserving
  proof
    let X be Subset of S;
    assume ex_sup_of X,S;
    thus ex_sup_of (x "/\").:X,S by YELLOW_0:17;
    thus (x "/\").(sup X) = x "/\" "\/"(X,S) by Def18
      .= "\/"({x"/\" y where y is Element of S: y in X},S) by A1
      .= sup ((x "/\").:X) by Th61;
  end;
  hence thesis by Th17;
end;
