
theorem LM060:
  for x being REAL-valued FinSequence,
      i being Nat,
      I0 being set
  st I0 c= Seg i & Seg (i+1) c= dom x
  holds
  Seq (x | (i+1), I0 \/ {i+1}) = Seq (x | i, I0) ^ <* x . (i+1) *>
  proof
    let x be REAL-valued FinSequence,
    i be Nat,
    I0 be set;
    assume A1: I0 c= Seg i & Seg (i+1) c= dom x;
    A4: Seg i c= Seg (i+1) by NAT_1:11,FINSEQ_1:5; then
    A17: I0 c= dom x by A1;
A6: dom (x | (i+1)) = Seg (i+1) by A1,RELAT_1:62; then
    A7: {i+1} c= dom (x | (i+1)) by FINSEQ_1:4,ZFMISC_1:31;
    A8: I0 c= Seg (i+1) by A4,A1; then
    A9: I0 \/ {i+1} c= Seg (i+1) by A7,A6,XBOOLE_1:8;
    A10: Seq (x | (i+1),I0 \/ {i+1}) = Seq (x | (I0 \/ {i+1}))
      by A8,A7,A6,XBOOLE_1:8,RELAT_1:74
    .= (x | (I0 \/ {i+1})) * Sgm (dom (x | (I0 \/ {i+1})));
    A11: I0 \/ {i+1} c= dom x by A1,A9; then
    A12: dom (x | (I0 \/ {i+1})) = I0 \/ {i+1} by RELAT_1:62;
    i < i + 1 & i+1 in Seg(i+1) by FINSEQ_1:4,NAT_1:16;
    then
A13: Sgm (I0 \/ {i+1}) = Sgm (I0) ^ <* i+1 *> by A1,LM040;
    A14: dom (x | (I0 \/ {i+1})) = (I0 \/ {i+1}) by RELAT_1:62,A11;
    rng (x | (I0 \/ {i+1})) c= REAL; then
    reconsider f = x | (I0 \/ {i+1})
    as Function of (I0 \/ {i+1}), REAL by A14,FUNCT_2:2;
    I0 is included_in_Seg by A1,FINSEQ_1:def 13;
    then
A19: rng Sgm (I0) = I0 by FINSEQ_1:def 14; then
    reconsider p = Sgm (I0) as FinSequence of (I0 \/ {i+1})
    by FINSEQ_1:def 4,XBOOLE_1:10;
    i+1 in {i+1} by TARSKI:def 1; then
    reconsider d = i+1 as Element of (I0 \/ {i+1}) by XBOOLE_0:def 3;
    A15: Seq (x | (i+1), I0 \/ {i+1})
      = (f * p) ^ <* f . d *> by FINSEQOP:8,A13,A10,A12;
    (f * p) = (x | I0) * (Sgm (I0)) by LM050,A19
    .= Seq ((x | I0) | I0) by A17,RELAT_1:62
    .= Seq (x | i, I0) by A1,RELAT_1:74;
    hence thesis by A15,FUNCT_1:49;
  end;
