reserve i,j,k,n for Nat;
reserve D for non empty set,
  p for Element of D,
  f,g for FinSequence of D;

theorem
  n <= len f implies (Ins(f,n,p)).(n+1) = p
proof
A1: 1 <= n+1 by NAT_1:11;
  assume n <= len f;
  then
A2: len(f|n) = n by FINSEQ_1:59;
  then len((f|n)^<*p*>) = n+1 by FINSEQ_2:16;
  then n+1 in dom((f|n)^<*p*>) by A1,FINSEQ_3:25;
  hence (Ins(f,n,p)).(n+1) = ((f|n)^<*p*>).(n+1) by Lm1
    .= p by A2,FINSEQ_1:42;
end;
