theorem Th1:
  for X be non empty set,t be FinSequence of X,k be Nat st k+1 <= len t
  holds t/^k = <*t.(k+1)*>^(t/^(k+1))
  proof
    let X be non empty set,t be FinSequence of X,k be Nat;
A1: k+1-'1 = k+1-1 by XREAL_0:def 2 .= k;
    assume k+1 <= len t;then
    t = (t|(k+1-'1)) ^ <*t.(k+1)*> ^ (t/^(k+1)) by NAT_1:11,FINSEQ_5:84
    .= (t|k) ^ (<*t.(k+1)*> ^ (t/^(k+1))) by A1,FINSEQ_1:32;
    then (t|k)^(t/^k) = (t|k) ^ (<*t.(k+1)*> ^ (t/^(k+1))) by RFINSEQ:8;
    hence t/^k = <*t.(k+1)*>^(t/^(k+1)) by FINSEQ_1:33;
  end;
