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 Th15:
  for X being non empty ZeroStr, seq being sequence of X holds 0 <
  k implies (Shift(seq)).k=seq.(k -' 1)
proof
  let X be non empty ZeroStr, seq be sequence of X;
A1: k in NAT by ORDINAL1:def 12;
  assume
A2: 0 < k;
  then
A3: 0+1 <= k by INT_1:7,A1;
  consider m being Nat such that
A4: m+1=k by A2,NAT_1:6;
A5: m=k-1 by A4;
  thus (Shift(seq)).k=seq.m by A4,Def5
    .=seq.(k -' 1) by A3,A5,XREAL_1:233;
end;
