
theorem Th4:
  for D being set,p being FinSequence of D,i,j,k being Element of
  NAT st i in dom p & 1 <= k & k <= i-1 holds Del(p,i,j).k = p.k
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: 1 <= k and
A3: k <= i-1;
A4: i <= len p by A1,FINSEQ_3:25;
A5: k <= i -' 1 by A3,XREAL_0:def 2;
A6: i -' 1 <= i by NAT_D:35;
  then len (p|(i -' 1)) = i -' 1 by A4,FINSEQ_1:59,XXREAL_0:2;
  then
A7: k in dom (p|(i -' 1)) by A2,A5,FINSEQ_3:25;
  i -' 1 <= len p by A4,A6,XXREAL_0:2;
  then k <= len p by A5,XXREAL_0:2;
  then
A8: k in dom p by A2,FINSEQ_3:25;
  thus Del(p,i,j).k = (p|(i -' 1)).k by A7,FINSEQ_1:def 7
    .= (p|(i -' 1))/.k by A7,PARTFUN1:def 6
    .= p/.k by A7,FINSEQ_4:70
    .= p.k by A8,PARTFUN1:def 6;
end;
