reserve X for Complex_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 Th20:
  (z ExpSeq).0 = 1.X & Expan(0,z,w).0 = 1.X
proof
  thus (z ExpSeq).0 = 1r/(0!)*(z #N 0) by Def1
    .= 1r/(0!)*(z GeoSeq).0 by CLOPBAN3:def 8
    .= 1r * 1.X by CLOPBAN3:def 7,SIN_COS:1
    .= 1.X by CLVECT_1:def 5;
A1: 0-'0=0 by XREAL_1:232;
  hence Expan(0,z,w).0 = (Coef(0)).0 * (z #N 0) * (w #N 0) by Def2
    .= 1r/(1r * 1r) * z #N 0 * w #N 0 by A1,COMPLEX1:def 4,SIN_COS:1,def 6
    .= z #N 0 * w #N 0 by CLVECT_1:def 5,COMPLEX1:def 4
    .= (z GeoSeq).0 * w #N 0 by CLOPBAN3:def 8
    .= (z GeoSeq).0 * (w GeoSeq).0 by CLOPBAN3:def 8
    .= 1.X * (w GeoSeq).0 by CLOPBAN3:def 7
    .= 1.X * 1.X by CLOPBAN3:def 7
    .= 1.X by VECTSP_1:def 4;
end;
