reserve i, j, k, c, m, n for Nat,
  a, x, y, z, X, Y for set,
  D, E for non empty set,
  R for Relation,
  f, g for Function,
  p, q for FinSequence;

theorem Th29:
  for X being non empty set, F being Relation st rng F c= HFuncs X
  & for f,g being homogeneous Function st f in rng F & g in rng F holds
   arity f = arity g holds F is with_the_same_arity
proof
  let X be non empty set, R be Relation such that
A1: rng R c= HFuncs X and
A2: for f,g being homogeneous Function st f in rng R & g in rng R holds
  arity f = arity g;
  let f,g be Function;
  assume that
A3: f in rng R and
A4: g in rng R;
  reconsider f9 = f, g9 = g as Element of HFuncs X by A1,A3,A4;
A5: arity f9 = arity g9 by A2,A3,A4;
  hereby
    assume f is empty;
    then dom g9 c= 0-tuples_on X by A5,Th17,Th19;
    then dom g9 c= {<*>X} by FINSEQ_2:94;
    hence g is empty or dom g = {{}} by ZFMISC_1:33;
  end;
  assume that
A6: f is non empty and
  g is non empty;
  reconsider f9 as non empty Element of HFuncs X by A6;
  take arity f9, X;
  thus thesis by A5,Th19;
end;
