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 Th18:
  x in dom p & p.x = TRUE & x in dom q & q.x = TRUE implies
   PP_and(p,q).x = TRUE
  proof
    assume that
A1: x in dom p and
A2: p.x = TRUE and
A3: x in dom q and
A4: q.x = TRUE;
A5: PP_not(p).x = FALSE & PP_not(q).x = FALSE by A1,A3,A2,A4,Def2;
A6: dom PP_not(p) = dom p & dom PP_not(q) = dom q by Def2;
    dom PP_or(PP_not(p),PP_not(q)) = {d where d is Element of D:
    d in dom PP_not(p) & PP_not(p).d = TRUE or
    d in dom PP_not(q) & PP_not(q).d = TRUE
    or d in dom PP_not(p) & PP_not(p).d = FALSE &
    d in dom PP_not(q) & PP_not(q).d = FALSE} by Def4;
    then
A7: x in dom PP_or(PP_not(p),PP_not(q)) by A1,A3,A5,A6;
    (PP_or(PP_not(p),PP_not(q))).x = FALSE by A1,A3,A5,A6,Def4;
    hence thesis by A7,Def2;
  end;
