reserve n,i,j,k,l for Nat;
reserve D for non empty set;
reserve c,d for Element of D;
reserve p,q,q9,r for FinSequence of D;

theorem
  p = q^<*c*>^r & i = len q + 1 implies (for l st 1 <= l & l <= len q
holds p.l = q.l) & p.i = c & for l st i + 1 <= l & l <= len p holds p.l = r.(l-
  i)
proof
  set q9 = q^<*c*>;
  assume that
A1: p = q9^r and
A2: i = len q + 1;
A3: p = q^(<*c*>^r) by A1,FINSEQ_1:32;
  thus for l st 1 <= l & l <= len q holds p.l = q.l
  proof
    let l;
    assume 1 <= l & l <= len q;
    then l in dom q by FINSEQ_3:25;
    hence thesis by A3,FINSEQ_1:def 7;
  end;
A4: len q9 = i by A2,FINSEQ_2:16;
  i in Seg(i) by A2,FINSEQ_1:3;
  then i in dom q9 by A4,FINSEQ_1:def 3;
  hence p.i = q9.i by A1,FINSEQ_1:def 7
    .= c by A2,FINSEQ_1:42;
  len p = len q9 + len r by A1,FINSEQ_1:22;
  hence thesis by A1,A4,FINSEQ_1:23;
end;
