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

theorem Th42:
  for L being complete LATTICE for mp being Function of L, InclPoset Ids L st
  mp is approximating & mp in the carrier of MonSet L
  ex AR being approximating auxiliary Relation of L st AR = (Map2Rel L).mp
proof
  let L be complete LATTICE;
  let mp be Function of L, InclPoset Ids L;
  assume that
A1: mp is approximating and
A2: mp in the carrier of MonSet L;
  consider AR be auxiliary Relation of L such that
A3: AR = (Map2Rel L).mp and
A4: for x,y be object holds [x,y] in AR iff ex x9,y9 be Element of L, s9 be
  Function of L, InclPoset Ids L st x9 = x & y9 = y & s9 = mp & x9 in s9.y9
  by A2,Def15;
  now
    let x be Element of L;
A5: ex ii be Subset of L st ( ii = mp.x)&( x = sup ii) by A1;
A6: AR-below x c= mp.x
    proof
      let a be object;
      assume a in AR-below x;
      then [a,x] in AR by Th13;
      then ex x9,y9 be Element of L, s9 be Function of L, InclPoset
      Ids L st ( x9 = a)&( y9 = x)&( s9 = mp)&( x9 in s9.y9) by A4;
      hence thesis;
    end;
    mp.x c= AR-below x
    proof
      let a be object;
      assume
A7:   a in mp.x;
      then reconsider a9 = a as Element of L by A5;
      [a9,x] in AR by A4,A7;
      hence thesis;
    end;
    hence x = sup (AR-below x) by A5,A6,XBOOLE_0:def 10;
  end;
  then reconsider AR as approximating auxiliary Relation of L by Def17;
  take AR;
  thus thesis by A3;
end;
