reserve T for TopSpace,
  B for Subset of T;

theorem Th7:
  T^alpha /\ D(c,alpha)(T) = the topology of T
proof
  thus T^alpha /\ D(c,alpha)(T) c= the topology of T
  proof
    let x be object;
    assume
A1: x in T^alpha /\ D(c,alpha)(T);
    then x in T^alpha by XBOOLE_0:def 4;
    then consider A being Subset of T such that
A2: x = A and
A3: A is alpha-set of T;
    x in D(c,alpha)(T) by A1,XBOOLE_0:def 4;
    then consider Z being Subset of T such that
A4: x = Z and
A5: Int Z = alphaInt Z;
    A = alphaInt A by A3,Th2;
    then Z is open by A2,A4,A5;
    hence thesis by A4,PRE_TOPC:def 2;
  end;
  let x be object;
  assume
A6: x in the topology of T;
  then reconsider K = x as Subset of T;
  K is open by A6,PRE_TOPC:def 2;
  then
A7: K = Int K by TOPS_1:23;
  then K c= Int Cl Int K by PRE_TOPC:18,TOPS_1:19;
  then
A8: K is alpha-set of T by Def1;
  then Int K = alphaInt K by A7,Th2;
  then
A9: K in {B: Int B = alphaInt B};
  K in T^alpha by A8;
  hence thesis by A9,XBOOLE_0:def 4;
end;
