reserve X for TopStruct,
  A for Subset of X;
reserve X for TopSpace,
  A,B for Subset of X;
reserve X for non empty TopSpace,
  A for Subset of X;
reserve X for TopSpace,
  A,B for Subset of X;
reserve X for non empty TopSpace,
  A, B for Subset of X;

theorem Th41:
  A is everywhere_dense iff ex C being Subset of X st C c= A & C
  is open & C is dense
proof
  thus A is everywhere_dense implies ex C being Subset of X st C c= A & C is
  open & C is dense
  proof
    assume
A1: A is everywhere_dense;
    take Int A;
    thus thesis by A1,TOPS_1:16;
  end;
  given C being Subset of X such that
A2: C c= A & C is open & C is dense;
  Int A is dense by A2,TOPS_1:24,44;
  hence thesis;
end;
