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 Th8:
  x in dom(PP_or(p,q)) implies
   x in dom p & p.x = TRUE or x in dom q & q.x = TRUE
   or x in dom p & p.x = FALSE & x in dom q & q.x = FALSE
  proof
    assume
A1: x in dom(PP_or(p,q));
    dom(PP_or(p,q)) = {d where d is Element of D:
    d in dom p & p.d = TRUE or d in dom q & q.d = TRUE
    or d in dom p & p.d = FALSE & d in dom q & q.d = FALSE} by Def4;
    then ex d1 being Element of D st
    d1 = x & (d1 in dom p & p.d1 = TRUE
    or d1 in dom q & q.d1 = TRUE
    or d1 in dom p & p.d1 = FALSE & d1 in dom q & q.d1 = FALSE) by A1;
    hence thesis;
  end;
