theorem
  |- X iff X c= TAUT(A)
proof
  hereby
    assume
A1: |- X;
    now
      let p;
      assume p in X;
      then p is valid by A1;
      hence p in TAUT(A) by CQC_THE1:def 10;
    end;
    hence X c= TAUT(A);
  end;
  thus thesis by CQC_THE1:def 10;
end;
