
theorem Th63:
  for S being Semilattice st for x being Element of S holds x "/\"
  is lower_adjoint for X being Subset of S st ex_sup_of X,S for 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 Semilattice such that
A1: for x being Element of S holds x "/\" is lower_adjoint;
  let X be Subset of S such that
A2: ex_sup_of X,S;
  let x be Element of S;
  x "/\" is sups-preserving by A1,Th13;
  then x "/\" preserves_sup_of X;
  then sup ((x "/\").:X) = (x "/\").(sup X) by A2;
  hence x "/\" "\/"(X,S) = sup ((x "/\").:X) by Def18
    .= "\/"({x"/\" y where y is Element of S: y in X},S) by Th61;
end;
