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 Th73:
  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 closed iff C2 is closed))
  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 closed iff C2 is closed);
  now
    let C1 be Subset of X1, C2 be Subset of X2;
    assume
A3: C1 = C2;
    thus C1 is open implies C2 is open
    proof
      assume C1 is open;
      then C1` is closed by TOPS_1:4;
      then C2` is closed by A1,A2,A3;
      hence thesis by TOPS_1:4;
    end;
    thus C2 is open implies C1 is open
    proof
      assume C2 is open;
      then C2` is closed by TOPS_1:4;
      then C1` is closed by A1,A2,A3;
      hence thesis by TOPS_1:4;
    end;
  end;
  hence thesis by A1,Th72;
end;
