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;

theorem Th57:
  for Y0 being non empty SubSpace of X, C, A being Subset of X, B
being Subset of Y0 st C is open & C c= the carrier of Y0 & A c= C & A = B holds
  Int A = Int B
proof
  let Y0 be non empty SubSpace of X, C, A be Subset of X, B be Subset of Y0;
  assume
A1: C is open;
  assume
A2: C c= the carrier of Y0;
  assume
A3: A c= C;
  assume
A4: A = B;
A5: Int B c= B by TOPS_1:16;
  then reconsider D = Int B as Subset of X by A4,XBOOLE_1:1;
  Int B c= C by A3,A4,A5,XBOOLE_1:1;
  then D is open by A1,A2,TSEP_1:9;
  then
A6: D c= Int A by A4,TOPS_1:16,24;
  Int A c= Int B by A4,Th56;
  hence thesis by A6;
end;
