
theorem Th17:
  for m be non zero Element of NAT, I be non empty FinSequence of NAT,
      X be non empty Subset of REAL m, f be PartFunc of REAL m,REAL,
      d be Real st X is open & f = X --> d & rng I c= Seg m
           holds
    (PartDiffSeq(f,X,I)).0 = X --> d
      &
    for i be Element of NAT st 1<=i & i <= len I
      holds (PartDiffSeq(f,X,I)).i = X --> 0
proof
  let m be non zero Element of NAT,
     I be non empty FinSequence of NAT,
     X be non empty Subset of REAL m,
     f be PartFunc of REAL m,REAL,
     d be Real;
  assume
A1:X is open & f = X --> d & rng I c= Seg m;
  thus
A2: (PartDiffSeq(f,X,I)).0 = (X --> d) | X  by A1,PDIFF_9:def 7
                           .= (X /\ X)--> d by FUNCOP_1:12
                           .=  X --> d;
  defpred P[Nat] means
    1<=$1 & $1 <= (len I) implies (PartDiffSeq(f,X,I)).$1 = X --> 0;
A3:P[0];
A4:for i be Nat st P[i] holds P[i+1]
  proof
    let i be Nat;
    assume
A5:  P[i];
    assume
A6:   1<=i+1 & i+1 <= len I;
A7: i <= i + 1 by NAT_1:12;
    per cases;
    suppose A8: i=0;
A9:     1 <= I/.(i+1) & I/.(i+1) <= m by Lm2,A1,A6;
      thus (PartDiffSeq(f,X,I)).(i+1)
          = ((PartDiffSeq(f,X,I)).i)`partial|(X,I/.(i+1)) by PDIFF_9:def 7
         .= X --> 0 by A8,A9,A1,A2,Th16;
    end;
    suppose A10: i <> 0;
A11:     1 <= I/.(i+1) & I/.(i+1) <= m by Lm2,A1,A6;
      thus (PartDiffSeq(f,X,I)).(i+1)
          = ((PartDiffSeq(f,X,I)).i)`partial|(X,I/.(i+1)) by PDIFF_9:def 7
         .= X --> 0 by A10,A11,A1,Th16,A7,A5,NAT_1:14,XXREAL_0:2,A6;
    end;
  end;
  for i be Nat holds P[i] from NAT_1:sch 2(A3,A4);
  hence thesis;
end;
