theorem Th18:
  for f being Homomorphism of 1L,L2 st f is onto holds
  L2 is upper-bounded & f preserves_top
proof
  let f be Homomorphism of 1L,L2;
  set r = f.(Top 1L);
  assume
A1: f is onto;
A2: now
    let a2 be Element of L2;
    consider a1 be Element of 1L such that
A3: f.a1 = a2 by A1,Th6;
    thus r"\/"a2 = f.(Top 1L "\/" a1) by A3,D1
      .= r;
    hence a2"\/"r = r;
  end;
  thus L2 is upper-bounded
  by A2;
  then Top L2=r by A2,LATTICES:def 17;
  hence thesis;
end;
