
theorem Th31:
  for A being Universal_Algebra for B being Subset of A
  for G being GeneratorSet of A st G c= B holds B is GeneratorSet of A
proof
  let A be Universal_Algebra;
  let B be Subset of A;
  let G be GeneratorSet of A such that
A1: G c= B;
  now
    let a be Element of A;
    consider n being Nat such that
A2: a in G|^n by Th30;
    take n;
    G|^n c= B|^n by A1,Th22;
    hence a in B|^n by A2;
  end;
  hence thesis by Th30;
end;
