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 Th25:
  for L being non empty antisymmetric transitive with_infima
  RelStr for x being Element of L for X, Y being Subset of L st ex_sup_of X,L &
ex_sup_of Y,L & Y = {x"/\"y where y is Element of L: y in X} holds x "/\" sup X
  >= sup Y
proof
  let L be non empty antisymmetric transitive with_infima RelStr;
  let x be Element of L;
  let X, Y be Subset of L such that
A1: ex_sup_of X,L and
A2: ex_sup_of Y,L and
A3: Y = {x"/\"y where y is Element of L: y in X};
  Y is_<=_than (x "/\" sup X)
  proof
    let y be Element of L;
    assume y in Y;
    then consider z being Element of L such that
A4: y = x "/\" z and
A5: z in X by A3;
A6: y <= z by A4,YELLOW_0:23;
    X is_<=_than sup X by A1,YELLOW_0:30;
    then z <= sup X by A5;
    then
A7: y <= sup X by A6,YELLOW_0:def 2;
    y <= x by A4,YELLOW_0:23;
    hence thesis by A7,YELLOW_0:23;
  end;
  hence thesis by A2,YELLOW_0:30;
end;
