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

theorem
  L is continuous iff (L is meet-continuous &
  ex R being approximating auxiliary Relation of L st
  for R9 being approximating auxiliary Relation of L holds R c= R9)
proof
  hereby
    assume
A1: L is continuous;
    hence L is meet-continuous;
    reconsider R = L-waybelow as approximating auxiliary Relation of L
    by A1;
    take R;
    thus for R9 be approximating auxiliary Relation of L holds R c= R9
    by A1,Th45;
  end;
  assume
A2: L is meet-continuous;
  given R be approximating auxiliary Relation of L such that
A3: for R9 be approximating auxiliary Relation of L holds R c= R9;
  for x being Element of L holds x = sup waybelow x
  proof
    let x be Element of L;
    set K = {AR-below x where AR is auxiliary Relation of L : AR in App L};
A4: meet K = waybelow x by A2,Th44;
    R in App L by Def19;
    then
A5: R-below x in K;
    then
A6: waybelow x c= R-below x by A4,SETFAM_1:3;
A7: sup (R-below x) = x by Def17;
    for a st a in K holds R-below x c= a
    proof
      let a;
      assume a in K;
      then consider AA be auxiliary Relation of L such that
A8:   a = AA-below x and
A9:   AA in App L;
      reconsider AA as approximating auxiliary Relation of L by A9,Def19;
      let b be object;
      assume
A10:  b in R-below x;
      R-below x c= AA-below x by A3,Th29;
      hence thesis by A8,A10;
    end;
    then R-below x c= meet K by A5,SETFAM_1:5;
    hence thesis by A4,A6,A7,XBOOLE_0:def 10;
  end;
  then L is satisfying_axiom_of_approximation by WAYBEL_3:def 5;
  hence thesis;
end;
