reserve a for set;
reserve L for lower-bounded sup-Semilattice;
reserve x for Element of L;
reserve L for complete LATTICE;

theorem Th45:
  L is continuous iff for R being approximating auxiliary Relation of L holds
  L-waybelow c= R & L-waybelow is approximating
proof
  set AR = L-waybelow;
  hereby
    assume
A1: L is continuous;
    then reconsider L9 = L as lower-bounded meet-continuous LATTICE;
    thus for R be approximating auxiliary Relation of L holds AR c= R
    & AR is approximating
    proof
      let R be approximating auxiliary Relation of L;
      reconsider R9 = R as approximating auxiliary Relation of L9;
      for a,b be object st [a,b] in AR holds [a,b] in R
      proof
        let a,b be object;
        assume
A2:     [a,b] in AR;
        then reconsider a9 = a, b9 = b as Element of L9 by ZFMISC_1:87;
        a9 << b9 by A2,Def1;
        then
A3:     a9 in waybelow b9 by WAYBEL_3:7;
A4:     meet { A1-below b9 where A1 is auxiliary Relation of L9 :
        A1 in App L9 } = waybelow b9 by Th44;
        R9 in App L9 by Def19;
        then R9-below b9 in
        { A1-below b9 where A1 is auxiliary Relation of L9 : A1 in App L9 };
        then waybelow b9 c= R9-below b9 by A4,SETFAM_1:3;
        hence thesis by A3,Th13;
      end;
      hence AR c= R;
      thus thesis by A1;
    end;
  end;
  assume
A5: for R be approximating auxiliary Relation of L holds AR c= R
  & AR is approximating;
  for x being Element of L holds x = sup waybelow x
  proof
    let x be Element of L;
    x = sup (AR-below x) by A5,Def17;
    hence thesis by Th40;
  end;
  then L is satisfying_axiom_of_approximation by WAYBEL_3:def 5;
  hence thesis;
end;
