reserve X for set,
  x,y,z for Element of BooleLatt X,
  s for set;
reserve y for Element of BooleLatt X;
reserve L for Lattice,
  p,q for Element of L;
reserve A for RelStr,
  a,b,c for Element of A;
reserve A for non empty RelStr,
  a,b,c,c9 for Element of A;
reserve V for with_suprema antisymmetric RelStr,
  u1,u2,u3,u4 for Element of V;
reserve N for with_infima antisymmetric RelStr,
  n1,n2,n3,n4 for Element of N;
reserve K for with_suprema with_infima reflexive antisymmetric RelStr,
  k1,k2,k3 for Element of K;
reserve p9,q9 for Element of LattPOSet L;

theorem Th30:
  X is_less_than p iff X is_<=_than p%
proof
  thus X is_less_than p implies X is_<=_than p%
  proof
    assume
A1: for q st q in X holds q [= p;
    let p9;
A2: (%p9)% = %p9;
    assume p9 in X;
    then %p9 [= p by A1;
    hence thesis by A2,Th7;
  end;
  assume
A3: for q9 st q9 in X holds q9 <= p%;
  let q;
  assume q in X;
  then q% <= p% by A3;
  hence thesis by Th7;
end;
