
theorem Th26:
  for A being Universal_Algebra, a being Element of A holds
  a in Generators A iff not ex o being Element of Operations A st a in rng o
proof
  let A be Universal_Algebra;
  set Z = the set of all rng o where o is Element of Operations A;
  let a be Element of A;
  hereby
    assume a in Generators A;
    then
A1: a nin union Z by XBOOLE_0:def 5;
    given o being Element of Operations A such that
A2: a in rng o;
    rng o in Z;
    hence contradiction by A1,A2,TARSKI:def 4;
  end;
  assume
A3: not ex o being Element of Operations A st a in rng o;
  assume a nin Generators A;
  then a in union Z by XBOOLE_0:def 5;
  then consider X being set such that
A4: a in X and
A5: X in Z by TARSKI:def 4;
  ex o being Element of Operations A st X = rng o by A5;
  hence contradiction by A3,A4;
end;
