
theorem Th32:
  for A being Universal_Algebra for G being GeneratorSet of A
  for a being Element of A
  st not ex o being Element of Operations A st a in rng o holds a in G
proof
  let A be Universal_Algebra;
  let G be GeneratorSet of A;
  let a be Element of A;
  assume
A1: for o being Element of Operations A holds a nin rng o;
  defpred P[Nat] means a nin G|^$1;
  assume a nin G;
  then
A2: P[0] by Th18;
A3: now
    let n be Nat;
    assume
A4: P[n];
    thus P[n+1]
    proof
      assume a in G|^(n+1);
      then a in (G|^n) \/ {Den(o,A).p
      where o is (Element of dom the charact of A),
      p is Element of (the carrier of A)*: p in dom Den(o,A) & rng p c= G|^n}
      by Th19;
      then a in {Den(o,A).p where o is (Element of dom the charact of A),
      p is Element of (the carrier of A)*: p in dom Den(o,A) & rng p c= G|^n}
      by A4,XBOOLE_0:def 3;
      then consider o being (Element of dom the charact of A), p being
      Element of (the carrier of A)* such that
A5:   a = Den(o,A).p and
A6:   p in dom Den(o,A) and rng p c= G|^n;
      a in rng Den(o,A) by A5,A6,FUNCT_1:def 3;
      hence contradiction by A1;
    end;
  end;
  for n being Nat holds P[n] from NAT_1:sch 2(A2,A3);
  hence contradiction by Th30;
end;
