
theorem Th6:
  for D being set,p being FinSequence of D,i,j,k being Element of
NAT st i in dom p & j in dom p & i <= j & i <= k & k <= len p - j + i - 1 holds
  Del(p,i,j).k = p.(j -'i + k + 1)
proof
  let D be set;
  let p be FinSequence of D;
  let i,j,k be Element of NAT;
  assume that
A1: i in dom p and
A2: j in dom p and
A3: i <= j and
A4: i <= k and
A5: k <= len p - j + i - 1;
A6: i -' 1 <= i by NAT_D:35;
  i -' 1 <= i by NAT_D:35;
  then k >= i -' 1 by A4,XXREAL_0:2;
  then k - (i -' 1) >= (i -' 1) - (i -' 1) by XREAL_1:9;
  then
A7: k - (i -' 1) = k -' (i -' 1) by XREAL_0:def 2;
A8: 1 <=i by A1,FINSEQ_3:25;
  then
A9: i -' 1 + 1 <= k by A4,XREAL_1:235;
  i-1 >= 1-1 by A8,XREAL_1:9;
  then
A10: i -' 1 = i-1 by XREAL_0:def 2;
  j-i >= i-i by A3,XREAL_1:9;
  then
A11: j -' i = j-i by XREAL_0:def 2;
A12: j <= len p by A2,FINSEQ_3:25;
  then
A13: len (p/^j) = len p - j by RFINSEQ:def 1;
  k <= len p - j + (i - 1) by A5;
  then
A14: k - (i -' 1) <= len p - j by A10,XREAL_1:20;
  1 <= k - (i -' 1) by A9,XREAL_1:19;
  then
A15: k - (i -' 1) in dom (p/^j) by A7,A13,A14,FINSEQ_3:25;
  i <= len p by A1,FINSEQ_3:25;
  then len (p|(i -' 1)) = i -' 1 by A6,FINSEQ_1:59,XXREAL_0:2;
  hence Del(p,i,j).k = (p/^j).(k - (i -' 1)) by A9,Th5
    .= p.(j + (k + (1 - i))) by A12,A10,A15,RFINSEQ:def 1
    .= p.(j -'i + k + 1) by A11;
end;
