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;
reserve D for Subset of X;
reserve Y0 for SubSpace of X;
reserve X0 for SubSpace of X;
reserve X0 for non empty SubSpace of X;
reserve X1,X2 for TopStruct;
reserve X1,X2 for TopSpace;
reserve D1 for Subset of X1,
  D2 for Subset of X2;

theorem Th81:
  D1 c= D2 & the TopStruct of X1 = the TopStruct of X2 implies Cl D1 c= Cl D2
proof
  assume
A1: D1 c= D2;
  assume
A2: the TopStruct of X1 = the TopStruct of X2;
  then reconsider C2 = D1 as Subset of X2;
  Cl D1 = Cl C2 by A2,Th80;
  hence thesis by A1,PRE_TOPC:19;
end;
