reserve X for Banach_Algebra,
  w,z,z1,z2 for Element of X,
  k,l,m,n,n1,n2 for Nat,
  seq,seq1,seq2,s,s9 for sequence of X,
  rseq for Real_Sequence;

theorem Th40:
  for z for s,t be Real holds s*z,t*z are_commutative
proof
  let z;
  let s, t be Real;
  (s*z) *(t*z) =(s*t)*(z*z) by LOPBAN_3:38
    .=(t*z)*(s*z) by LOPBAN_3:38;
  hence thesis;
end;
