reserve U0 for Universal_Algebra,
  U1 for SubAlgebra of U0,
  o for operation of U0;

theorem Th9:
  for H be non empty Subset of Sub(U0) holds ((Carr U0).:H) is non empty
proof
  let H be non empty Subset of Sub(U0);
  consider u being Element of Sub(U0) such that
A1: u in H by SUBSET_1:4;
  (Carr U0).u in ((Carr U0).:H) by A1,FUNCT_2:35;
  hence thesis;
end;
