reserve T for TopSpace,
  B for Subset of T;
reserve X,Y for non empty TopSpace;
reserve f for Function of X,Y;

theorem
  f is continuous iff f is alpha-continuous (c,alpha)-continuous
proof
A1: [#]Y <> {};
  hereby
    assume
A2: f is continuous;
    thus f is alpha-continuous
    proof
      let V be Subset of Y;
      assume V is open;
      then f"V is open by A1,A2,TOPS_2:43;
      then f"V in the topology of X by PRE_TOPC:def 2;
      then f"V in X^alpha /\ D(c,alpha)(X) by Th7;
      hence thesis by XBOOLE_0:def 4;
    end;
    thus f is (c,alpha)-continuous
    proof
      let G be Subset of Y;
      assume G is open;
      then f"G is open by A1,A2,TOPS_2:43;
      then f"G in the topology of X by PRE_TOPC:def 2;
      then f"G in X^alpha /\ D(c,alpha)(X) by Th7;
      hence thesis by XBOOLE_0:def 4;
    end;
  end;
  assume
A3: f is alpha-continuous (c,alpha)-continuous;
  now
    let V be Subset of Y;
    assume V is open;
    then f"V in X^alpha & f"V in D(c,alpha)(X) by A3;
    then f"V in X^alpha /\ D(c,alpha)(X) by XBOOLE_0:def 4;
    then f"V in the topology of X by Th7;
    hence f"V is open by PRE_TOPC:def 2;
  end;
  hence thesis by A1,TOPS_2:43;
end;
