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 Th4:
  0 < k implies (Shift(seq)).k=seq.(k-'1)
proof
  assume
A1: 0 < k;
A2: 0+1 <= k by INT_1:7,A1;
  consider m be Nat such that
A3: m+1=k by A1,NAT_1:6;
A4: m=k-1 by A3;
  thus (Shift seq).k=seq.m by A3,Def8
    .=seq.(k-'1) by A2,A4,XREAL_1:233;
end;
