reserve x, y, i for object,
  L for up-complete Semilattice;
reserve L for complete LATTICE,
  a, b, c for Element of L,
  J for non empty set,
  K for non-empty ManySortedSet of J;
reserve J, K, D for non empty set,
  j for Element of J,
  k for Element of K;
reserve J for non empty set,
  K for non-empty ManySortedSet of J;

theorem Th26:
  for L being completely-distributive LATTICE for X being Subset
  of L for x being Element of L holds x "/\" sup X = "\/"({x"/\"y where y is
  Element of L: y in X}, L)
proof
  let L be completely-distributive LATTICE;
  let X be Subset of L;
  let x be Element of L;
  set A = {x"/\"y where y is Element of L: y in X};
  per cases;
  suppose
A1: X is empty;
    now
      set z = the Element of A;
      assume A <> {};
      then z in A;
      then ex y being Element of L st z = x"/\"y & y in X;
      hence contradiction by A1;
    end;
    then
A2: "\/"(A, L) = Bottom L by YELLOW_0:def 11;
    sup X = Bottom L by A1,YELLOW_0:def 11;
    hence thesis by A2,WAYBEL_1:3;
  end;
  suppose
    X is non empty;
    hence thesis by Lm15;
  end;
end;
