reserve T for TopSpace,
  B for Subset of T;

theorem Th1:
  alphaInt B = pInt B iff sInt B = psInt B
proof
  hereby
    Cl Int B c= Cl B by PRE_TOPC:19,TOPS_1:16;
    then Int Cl Int B c= Int Cl B by TOPS_1:19;
    then Cl Int Cl Int B c= Cl Int Cl B by PRE_TOPC:19;
    then Cl Int B c= Cl Int Cl B by TOPS_1:26;
    then
A1: sInt B c= B /\ Cl Int Cl B by XBOOLE_1:26;
    assume alphaInt B = pInt B;
    then Cl (B /\ Int Cl B) c= Cl( Int Cl Int B) by PRE_TOPC:19,XBOOLE_1:17;
    then
A2: Cl (B /\ Int Cl B) c= Cl Int B by TOPS_1:26;
    Cl(B /\ Int Cl B) = Cl(Cl B /\ Int Cl B) by TOPS_1:14;
    then Cl(B /\ Int Cl B) = Cl Int Cl B by TOPS_1:16,XBOOLE_1:28;
    then B /\ Cl Int Cl B c= B /\ Cl Int B by A2,XBOOLE_1:26;
    hence sInt B = psInt B by A1;
  end;
A3: Int Cl B c= Cl Int Cl B by PRE_TOPC:18;
  assume psInt B = sInt B;
  then
A4: psInt B c= Cl Int B by XBOOLE_1:17;
  B /\ Int Cl B c= B /\ Cl Int Cl B by PRE_TOPC:18,XBOOLE_1:26;
  then B /\ Int Cl B c= Cl Int B by A4;
  then
A5: Cl(B /\ Int Cl B) c= Cl Cl Int B by PRE_TOPC:19;
  Cl(B /\ Int Cl B) = Cl(Cl B /\ Int Cl B) by TOPS_1:14;
  then Cl Int Cl B c= Cl Int B by A5,TOPS_1:16,XBOOLE_1:28;
  then Int Cl B c= Cl Int B by A3;
  then Int Int Cl B c= Int Cl Int B by TOPS_1:19;
  then
A6: B /\ Int Cl B c= B /\ Int Cl Int B by XBOOLE_1:26;
  Cl Int B c= Cl B by PRE_TOPC:19,TOPS_1:16;
  then Int Cl Int B c= Int Cl B by TOPS_1:19;
  then alphaInt B c= B /\ Int Cl B by XBOOLE_1:26;
  hence thesis by A6;
end;
