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 Th15:
  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}
  proof
    set X = {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};
    set Y = {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};
    X = Y
    proof
      thus X c= Y
      proof
        let x;
        assume x in X;
        then ex d being TypeSCNominativeData of V,A st d = x &
        (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);
        hence thesis;
      end;
      let x;
      assume x in Y;
      then consider d being Element of ND(V,A) such that
A1:   d = x &
      (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);
      d is TypeSCNominativeData of V,A by NOMIN_1:39;
      hence thesis by A1;
    end;
    hence thesis by PARTPR_1:def 4;
  end;
