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 Th19:
  x in dom p & p.x = FALSE implies PP_and(p,q).x = FALSE
  proof
    assume that
A1: x in dom p and
A2: p.x = FALSE;
A3: PP_not(p).x = TRUE by A1,A2,Def2;
A4: dom PP_not(p) = dom p by Def2;
A5: 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;
A6: x in dom PP_or(PP_not(p),PP_not(q)) by A1,A3,A4,A5;
    (PP_or(PP_not(p),PP_not(q))).x = TRUE by A1,A3,A4,Def4;
    hence thesis by A6,Def2;
  end;
