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;
