reserve X,Y,x,y for set;
reserve A for non empty Poset;
reserve a,a1,a2,a3,b,c for Element of A;
reserve S,T for Subset of A;

theorem Th20:
  a in S implies not a in LowerCone(S)
proof
  assume that
A1: a in S and
A2: a in LowerCone(S);
  ex a1 st a1 = a & for a2 st a2 in S holds a1 < a2 by A2;
  hence thesis by A1;
end;
