reserve a,b,c,d,m,x,n,j,k,l for Nat,
  t,u,v,z for Integer,
  f,F for FinSequence of NAT;
reserve p,q,r,s for real number;
reserve a,b,c,d,m,x,n,k,l for Nat,
  t,z for Integer,
  f,F,G for FinSequence of REAL;
reserve q,r,s for real number;
reserve D for set;

theorem Th29:
  for n st n in dom f &
    m in dom ((f|n)/^1) holds ((f|n)/^1).m = f.(m+1)
  proof
    let n such that
    A0: n in dom f;
    set F = (f|n);
    assume
    A1: m in dom ((f|n)/^1);
    A1a:  m+1 >= 0+1 by XREAL_1:6;
    (f|n) <> {} by A1; then
    A2: 1 in dom (f|n) by FINSEQ_5:6; then
    m+1 <= len (f|n) by A1,Th15,Th3; then
    A4: m+1 <= n by A0,Th10;
    (((f|n))/^1).m = (((f|n))/^1)/.m by A1, PARTFUN1:def 6
    .= (f|n)/.(1+m) by A1,FINSEQ_5:27
    .= (f|n).(1+m) by A1,A2,Th15,PARTFUN1:def 6
    .= f.(1+m) by A0,Th7,A1a,A4;
    hence thesis;
  end;
