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 Th21:
  (z rExpSeq).0 = 1.X & Expan(0,z,w).0 = 1.X
proof
  thus (z rExpSeq).0 = 1/(0!)*(z #N 0) by Def2
    .= 1/1 *1.X by LOPBAN_3:def 9,NEWTON:12
    .= 1.X by LOPBAN_3:38;
A1: 0-'0=0 by XREAL_1:232;
  hence Expan(0,z,w).0 = (Coef(0)).0 * (z #N 0) * (w #N 0) by Def6
    .= 1/(1 * 1) * z #N 0 * w #N 0 by A1,Def3,NEWTON:12
    .= (z GeoSeq).0 * w #N 0 by LOPBAN_3:38
    .= 1.X * (w GeoSeq).0 by LOPBAN_3:def 9
    .= 1.X * 1.X by LOPBAN_3:def 9
    .= 1.X by LOPBAN_3:38;
end;
