reserve x for object;
reserve D for set;
reserve p for PartialPredicate of D;
reserve D for non empty set;
reserve p,q,r for PartialPredicate of D;

theorem Th62:
  for a,b being Element of PartialPredConnectivesLatt(D)
   st a = p & b = PP_not(p) holds b is_a_partial_complement_of a
  proof
    set L = PartialPredConnectivesLatt(D);
    let a,b be Element of L such that
A1: a = p & b = PP_not(p);
    Top L = PP_True(D) by Th60;
    hence Top L | dom a = PP_or(PP_not(p),p) by A1,Th49
    .= b"\/"a by A1,Def12;
    hence a"\/"b = Top L | dom a;
    Bottom L = PP_False(D) by Th61;
    hence Bottom L | dom a = PP_and(PP_not(p),p) by A1,Th51
    .= b"/\"a by A1,Def11;
    hence a"/\"b = Bottom L | dom a;
  end;
