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

theorem Th41:
  for L being LATTICE holds IntRel L is approximating
proof
  let L be LATTICE;
  set AR = IntRel L;
  let x be Element of L;
  set A = {y where y is Element of L: [y,x] in AR};
  set E = { u where u is Element of L : u <= x };
A1: A c= E
  proof
    let a be object;
    assume a in A;
    then consider v being Element of L such that
A2: a = v and
A3: [v,x] in AR;
    v <= x by A3,ORDERS_2:def 5;
    hence thesis by A2;
  end;
  E c= A
  proof
    let a be object;
    assume a in E;
    then consider v being Element of L such that
A4: a = v and
A5: v <= x;
    [v,x] in AR by A5,ORDERS_2:def 5;
    hence thesis by A4;
  end;
  then
A6: {y where y is Element of L: [y,x] in AR} = E by A1;
  {y where y is Element of L: y <= x } c= the carrier of L
  proof
    let z be object;
    assume z in {y where y is Element of L: y <= x };
    then ex y being Element of L st z = y & y <= x;
    hence thesis;
  end;
  then reconsider E as Subset of L;
A7: x is_>=_than E
  proof
    let b be Element of L;
    assume b in E;
    then ex b9 be Element of L st ( b9 = b)&( b9 <= x);
    hence b <= x;
  end;
  now
    let b be Element of L;
    assume
A8: b is_>=_than E;
    x in E;
    hence x <= b by A8;
  end;
  hence thesis by A6,A7,YELLOW_0:30;
end;
