reserve a,b,c,v,v1,x,y for object;
reserve V,A for set;
reserve d for TypeSCNominativeData of V,A;
reserve p,q,r for SCPartialNominativePredicate of V,A;

theorem
  dom(PP_and(p,q)) =
    {d where d is TypeSCNominativeData of V,A:
             d in dom p & p.d = FALSE or d in dom q & q.d = FALSE
          or d in dom p & p.d = TRUE & d in dom q & q.d = TRUE}
  proof
    set F = PP_and(p,q);
    set P = PP_not(p);
    set Q = PP_not(q);
    set D = {d where d is TypeSCNominativeData of V,A:
             d in dom p & p.d = FALSE or d in dom q & q.d = FALSE
          or d in dom p & p.d = TRUE & d in dom q & q.d = TRUE};
A1: dom F = dom PP_or(P,Q) by PARTPR_1:def 2;
A2: dom PP_or(P,Q) =
      {d where d is TypeSCNominativeData of V,A:
      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 Th15;
A3: dom P = dom p by PARTPR_1:def 2;
A4: dom Q = dom q by PARTPR_1:def 2;
    thus dom F c= D
    proof
      let x;
      assume x in dom F;
      then consider d being TypeSCNominativeData of V,A such that
A5:   x = d and
A6:   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,A2;
      per cases by A6;
      suppose that
A7:     d in dom P and
A8:     P.d = TRUE;
        p.d = FALSE by A3,A7,A8,PARTPR_1:5;
        hence thesis by A3,A5,A7;
      end;
      suppose that
A9:     d in dom Q and
A10:    Q.d = TRUE;
        q.d = FALSE by A4,A9,A10,PARTPR_1:5;
        hence thesis by A4,A5,A9;
      end;
      suppose that
A11:    d in dom P & d in dom Q and
A12:    P.d = FALSE & Q.d = FALSE;
        p.d = TRUE & q.d = TRUE by A3,A4,A11,A12,PARTPR_1:4;
        hence thesis by A3,A4,A5,A11;
      end;
    end;
    let x;
    assume x in D;
    then consider d being TypeSCNominativeData of V,A such that
A13: x = d and
A14: d in dom p & p.d = FALSE or d in dom q & q.d = FALSE
    or d in dom p & p.d = TRUE & d in dom q & q.d = TRUE;
    per cases by A14;
    suppose that
A15:  d in dom p and
A16:  p.d = FALSE;
      P.d = TRUE by A15,A16,PARTPR_1:def 2;
      hence thesis by A1,A2,A3,A13,A15;
    end;
    suppose that
A17:  d in dom q and
A18:  q.d = FALSE;
      Q.d = TRUE by A17,A18,PARTPR_1:def 2;
      hence thesis by A1,A2,A4,A13,A17;
    end;
    suppose that
A19:  d in dom p & d in dom q and
A20:  p.d = TRUE & q.d = TRUE;
      P.d = FALSE & Q.d = FALSE by A19,A20,PARTPR_1:def 2;
      hence thesis by A1,A2,A3,A4,A13,A19;
    end;
  end;
