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_IF(p,f,g)) implies
   x in dom p & p.x = TRUE & x in dom f or
   x in dom p & p.x = FALSE & x in dom g
  proof
    assume
A1: x in dom(PP_IF(p,f,g));
    dom(PP_IF(p,f,g)) = {d where d is Element of D:
    d in dom p & p.d = TRUE & d in dom f
    or d in dom p & p.d = FALSE & d in dom g} by Def13;
    then ex d1 being Element of D st
    d1 = x & (d1 in dom p & p.d1 = TRUE & d1 in dom f
    or d1 in dom p & p.d1 = FALSE & d1 in dom g) by A1;
    hence thesis;
  end;
