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 Th77:
  D1 = D2 & the TopStruct of X1 = the TopStruct of X2 implies Int D1 = Int D2
proof
  assume
A1: D1 = D2;
A2: Int D1 c= D1 by TOPS_1:16;
  then reconsider C2 = Int D1 as Subset of X2 by A1,XBOOLE_1:1;
  assume
A3: the TopStruct of X1 = the TopStruct of X2;
  then
A4: C2 c= Int D2 by A1,A2,Th76,TOPS_1:24;
A5: Int D2 c= D2 by TOPS_1:16;
  then reconsider C1 = Int D2 as Subset of X1 by A1,XBOOLE_1:1;
  C1 c= Int D1 by A1,A3,A5,Th76,TOPS_1:24;
  hence thesis by A4;
end;
