reserve x for object;
reserve n for Nat;
reserve D for non empty set;
reserve p,q for PartialPredicate of D;
reserve D for set;
reserve p,q for PartialPredicate of D;
reserve f,g for BinominativeFunction of D;
reserve D for non empty set;
reserve p,q for PartialPredicate of D;
reserve f,g,h for BinominativeFunction of D;

theorem
  x in dom(PP_FH(p,f,q)) implies
   x in dom p & p.x = FALSE or x in dom(q*f) & (q*f).x = TRUE
   or x in dom p & p.x = TRUE & x in dom(q*f) & (q*f).x = FALSE
  proof
    assume
A1: x in dom(PP_FH(p,f,q));
    dom(PP_FH(p,f,q)) = {d where d is Element of D:
    d in dom p & p.d = FALSE or d in dom(q*f) & (q*f).d = TRUE
    or d in dom p & p.d = TRUE & d in dom(q*f) & (q*f).d = FALSE} by Def15;
    then ex d1 being Element of D st
    d1 = x & (d1 in dom p & p.d1 = FALSE or
    d1 in dom(q*f) & (q*f).d1 = TRUE
    or d1 in dom p & p.d1 = TRUE & d1 in dom(q*f) & (q*f).d1 = FALSE) by A1;
    hence thesis;
  end;
