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 Th31:
  dom(PP_imp(p,q)) =
    {d where d is Element of D:
             d in dom p & p.d = FALSE or d in dom q & q.d = TRUE
          or d in dom p & p.d = TRUE & d in dom q & q.d = FALSE}
  proof
    set F = PP_imp(p,q);
    set P = PP_not(p);
    set Dimp = {d1 where d1 is Element of D:
             d1 in dom p & p.d1 = FALSE or d1 in dom q & q.d1 = TRUE
          or d1 in dom p & p.d1 = TRUE & d1 in dom q & q.d1 = FALSE};
A1: dom(F) = {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;
A2: dom P = dom p by Def2;
    thus dom F c= Dimp
    proof
      let x;
      assume x in dom F;
      then consider d being Element of D such that
A3:   x = d and
A4:   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 A1;
      per cases by A4;
      suppose that
A5:     d in dom P and
A6:     P.d = TRUE;
        p.d = FALSE by A2,A5,A6,Th5;
        hence thesis by A2,A3,A5;
      end;
      suppose d in dom q & q.d = TRUE;
        hence thesis by A3;
      end;
      suppose that
A7:     d in dom P & d in dom q and
A8:     P.d = FALSE and
A9:     q.d = FALSE;
        p.d = TRUE by A2,A7,A8,Th4;
        hence thesis by A2,A3,A7,A9;
      end;
    end;
    let x;
    assume x in Dimp;
    then consider d being Element of D such that
A10: x = d and
A11: d in dom p & p.d = FALSE or d in dom q & q.d = TRUE
    or d in dom p & p.d = TRUE & d in dom q & q.d = FALSE;
    per cases by A11;
    suppose that
A12:  d in dom p and
A13:  p.d = FALSE;
      P.d = TRUE by A12,A13,Def2;
      hence thesis by A1,A2,A10,A12;
    end;
    suppose d in dom q & q.d = TRUE;
      hence thesis by A1,A10;
    end;
    suppose that
A14:  d in dom p and
A15:  d in dom q and
A16:  p.d = TRUE and
A17:  q.d = FALSE;
      P.d = FALSE by A14,A16,Def2;
      hence thesis by A1,A2,A10,A14,A15,A17;
    end;
  end;
