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 Th80:
  D1 = D2 & the TopStruct of X1 = the TopStruct of X2 implies Cl D1 = Cl D2
proof
  assume
A1: D1 = D2;
  assume
A2: the TopStruct of X1 = the TopStruct of X2;
  then reconsider C2 = Cl D1 as Subset of X2;
  D1 c= Cl D1 by PRE_TOPC:18;
  then
A3: Cl D2 c= C2 by A1,A2,Th79,TOPS_1:5;
  reconsider C1 = Cl D2 as Subset of X1 by A2;
  D2 c= Cl D2 by PRE_TOPC:18;
  then Cl D1 c= C1 by A1,A2,Th79,TOPS_1:5;
  hence thesis by A3;
end;
