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;

theorem Th72:
  the carrier of X1 = the carrier of X2 & (for C1 being Subset of
X1, C2 being Subset of X2 st C1 = C2 holds (C1 is open iff C2 is open)) implies
  the TopStruct of X1 = the TopStruct of X2
proof
  assume
A1: the carrier of X1 = the carrier of X2;
  assume
A2: for C1 being Subset of X1, C2 being Subset of X2 st C1 = C2 holds (
  C1 is open iff C2 is open);
  now
    let D be object;
    assume
A3: D in the topology of X2;
    then reconsider C2 = D as Subset of X2;
    reconsider C1 = C2 as Subset of X1 by A1;
    C2 is open by A3;
    then C1 is open by A2;
    hence D in the topology of X1;
  end;
  then
A4: the topology of X2 c= the topology of X1 by TARSKI:def 3;
  now
    let D be object;
    assume
A5: D in the topology of X1;
    then reconsider C1 = D as Subset of X1;
    reconsider C2 = C1 as Subset of X2 by A1;
    C1 is open by A5;
    then C2 is open by A2;
    hence D in the topology of X2;
  end;
  then the topology of X1 c= the topology of X2 by TARSKI:def 3;
  hence thesis by A1,A4,XBOOLE_0:def 10;
end;
