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 Th18:
  x in dom SC_exists(p,v) implies
   (ex d1 being TypeSCNominativeData of V,A st
      local_overlapping(V,A,x,d1,v) in dom p &
      p.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 p &
      p.local_overlapping(V,A,x,d1,v) = FALSE)
  proof
    assume
A1: x in dom SC_exists(p,v);
    dom(SC_exists(p,v)) = {d where d is TypeSCNominativeData of V,A:
    (ex d1 being TypeSCNominativeData of V,A st
      local_overlapping(V,A,d,d1,v) in dom p &
      p.local_overlapping(V,A,d,d1,v) = TRUE) or
    (for d1 being TypeSCNominativeData of V,A holds
      local_overlapping(V,A,d,d1,v) in dom p &
      p.local_overlapping(V,A,d,d1,v) = FALSE)} by Def1;
    then ex d2 being TypeSCNominativeData of V,A st x = d2 &
    ((ex d1 being TypeSCNominativeData of V,A st
      local_overlapping(V,A,d2,d1,v) in dom p &
      p.local_overlapping(V,A,d2,d1,v) = TRUE) or
    (for d1 being TypeSCNominativeData of V,A holds
      local_overlapping(V,A,d2,d1,v) in dom p &
      p.local_overlapping(V,A,d2,d1,v) = FALSE)) by A1;
    hence thesis;
  end;
