reserve q,th,r for Real,
  a,b,p for Real,
  w,z for Complex,
  k,l,m,n,n1,n2 for Nat,
  seq,seq1,seq2,cq1 for Complex_Sequence,
  rseq,rseq1,rseq2 for Real_Sequence,
  rr for set,
  hy1 for 0-convergent non-zero Real_Sequence;

theorem Th10:
  (z ExpSeq).0 = 1 & Expan(0,z,w).0 = 1
proof
  thus (z ExpSeq).0 = (z |^ 0)/(0! ) by Def4
    .= 1 by Th1,COMSEQ_3:def 1;
A1: 0-'0=0 by XREAL_1:232;
  hence Expan(0,z,w).0 = (Coef(0)).0 * (z |^ 0) * (w |^ 0) by Def9
    .= 1/(1 * 1) * z |^ 0 * w |^ 0 by A1,Def6,Th1
    .= 1r * (w GeoSeq).0 by COMSEQ_3:def 1
    .= 1 by COMSEQ_3:def 1;
end;
