
theorem Th2:
  for p being FinSequence, m, n, a being Nat st a in dom
(m, n)-cut p ex k being Nat st k in dom p & p.k = ((m,n)-cut p).a &
  k+1 = m+a & m <= k & k <= n
proof
  let p be FinSequence, m, n, a be Nat such that
A1: a in dom (m, n)-cut p;
  set cp = (m,n)-cut p;
A2: a <= len cp by A1,FINSEQ_3:25;
  per cases;
  suppose
A3: 1<=m & m<=n & n<=len p;
    0+1<= a by A1,FINSEQ_3:25;
    then consider i1 being Nat such that
    0<=i1 and
A4: i1<len cp and
A5: a=i1+1 by A2,FINSEQ_6:127;
    take k = m+i1;
    m <= k by NAT_1:11;
    then
A6: 1 <= k by A3,XXREAL_0:2;
A7: len cp +m = n+1 by A3,FINSEQ_6:def 4;
A8: m+i1 < m + len cp by A4,XREAL_1:6;
    then m+i1 <= n by A7,NAT_1:13;
    then k <= len p by A3,XXREAL_0:2;
    hence k in dom p by A6,FINSEQ_3:25;
    thus p.k = cp.a by A3,A4,A5,FINSEQ_6:def 4;
    thus k+1 = m+a by A5;
    thus thesis by A7,A8,NAT_1:11,13;
  end;
  suppose
    not (1<=m & m<=n & n<=len p);
    hence thesis by A1,FINSEQ_6:def 4,RELAT_1:38;
  end;
end;
