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_imp(p,q)) =
    {d where d is TypeSCNominativeData of V,A:
             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 D = {d where d is TypeSCNominativeData of V,A:
             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};
A1: dom F = {d where d is Element of ND(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 PARTPR_1:def 4;
A2: dom P = dom p by PARTPR_1:def 2;
    thus dom F c= D
    proof
      let x;
      assume x in dom F;
      then consider d being Element of ND(V,A) 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;
      reconsider d as TypeSCNominativeData of V,A by NOMIN_1:39;
      per cases by A4;
      suppose that
A5:     d in dom P and
A6:     P.d = TRUE;
        p.d = FALSE by A2,A5,A6,PARTPR_1:5;
        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,PARTPR_1:4;
        hence thesis by A2,A3,A7,A9;
      end;
    end;
    let x;
    assume x in D;
    then consider d being TypeSCNominativeData of V,A 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,PARTPR_1:def 2;
      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,PARTPR_1:def 2;
      hence thesis by A1,A2,A10,A14,A15,A17;
    end;
  end;
