
theorem Th10:
  for p being FinSequence, n being Nat st n in dom p & n+2 <= len p
  holds mid(p,n,n+2) = <*p.n, p.(n+1),p.(n+2)*>
proof
  let p be FinSequence, n be Nat;
  assume A1: n in dom p & n+2 <= len p;
  then A2: 1 <= n & (n+1)+1 <= len p by FINSEQ_3:25;
  A3: 1 <= n+1 by XREAL_1:31;
  n+1+1-1 <= len p - 0 by A1, XREAL_1:13;
  then A5: n+1 in dom p by A3, FINSEQ_3:25;
  A6: mid(p,n+1,n+2) = <*p.(n+1), p.(n+2)*> by A2, A5, Th9;
  thus mid(p,n,n+2) = <*p.n*> ^ mid(p,n+1,n+2) by A1, Th8, XREAL_1:29
    .= <*p.n*> ^ (<*p.(n+1)*>^<*p.(n+2)*>) by A6, FINSEQ_1:def 9
    .= <*p.n*> ^ <*p.(n+1)*> ^ <*p.(n+2)*> by FINSEQ_1:32
    .= <*p.n, p.(n+1),p.(n+2)*> by FINSEQ_1:def 10;
end;
