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;
reserve f for Choice_Function of BOOL(the carrier of A);
reserve fC,fC1,fC2 for Chain of f;

theorem Th43:
  union(Chains(f)) <> {}
proof
  {f.(the carrier of A)} is Chain of f by Th36;
  then {f.(the carrier of A)} in Chains(f) by Def13;
  hence thesis by ORDERS_1:6;
end;
