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;

theorem
  the carrier of X1 = the carrier of X2 & (for C1 being Subset of X1, C2
being Subset of X2 st C1 = C2 holds Cl C1 = Cl C2) 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 Cl
  C1 = Cl C2;
  now
    let C1 be Subset of X1, C2 be Subset of X2;
    assume
A3: C1 = C2;
    thus C1 is closed implies C2 is closed
    proof
      assume C1 is closed;
      then C1 = Cl C1 by PRE_TOPC:22;
      then C2 = Cl C2 by A2,A3;
      hence thesis;
    end;
    thus C2 is closed implies C1 is closed
    proof
      assume C2 is closed;
      then C2 = Cl C2 by PRE_TOPC:22;
      then C1 = Cl C1 by A2,A3;
      hence thesis;
    end;
  end;
  hence thesis by A1,Th73;
end;
