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
  SC_exists(PP_BottomPred(ND(V,A)),v) = PP_BottomPred(ND(V,A))
  proof
    set B = PP_BottomPred(ND(V,A));
    set T = PP_True(ND(V,A));
    set o = SC_exists(B,v);
    thus dom o = dom B
    proof
      thus dom o c= dom B
      proof
        let x;
        assume x in dom o;
        then (ex d1 being TypeSCNominativeData of V,A st
        local_overlapping(V,A,x,d1,v) in dom B &
        B.local_overlapping(V,A,x,d1,v) = TRUE) or
        (for d1 being TypeSCNominativeData of V,A holds
        local_overlapping(V,A,x,d1,v) in dom B &
        B.local_overlapping(V,A,x,d1,v) = FALSE) by Th18;
        hence thesis;
      end;
      thus dom B c= dom o;
    end;
    let x;
    assume x in dom o;
    then (ex d1 being TypeSCNominativeData of V,A st
    local_overlapping(V,A,x,d1,v) in dom B &
    B.local_overlapping(V,A,x,d1,v) = TRUE) or
    (for d1 being TypeSCNominativeData of V,A holds
    local_overlapping(V,A,x,d1,v) in dom B &
    B.local_overlapping(V,A,x,d1,v) = FALSE) by Th18;
    hence thesis;
  end;
