let m be non zero Element of NAT ; :: thesis: for I being non empty FinSequence of NAT
for X being non empty Subset of (REAL m)
for f being PartFunc of (REAL m),REAL
for d being Real st X is open & f = X --> d & rng I c= Seg m holds
( (PartDiffSeq (f,X,I)) . 0 = X --> d & ( for i being Element of NAT st 1 <= i & i <= len I holds
(PartDiffSeq (f,X,I)) . i = X --> 0 ) )

let I be non empty FinSequence of NAT ; :: thesis: for X being non empty Subset of (REAL m)
for f being PartFunc of (REAL m),REAL
for d being Real st X is open & f = X --> d & rng I c= Seg m holds
( (PartDiffSeq (f,X,I)) . 0 = X --> d & ( for i being Element of NAT st 1 <= i & i <= len I holds
(PartDiffSeq (f,X,I)) . i = X --> 0 ) )

let X be non empty Subset of (REAL m); :: thesis: for f being PartFunc of (REAL m),REAL
for d being Real st X is open & f = X --> d & rng I c= Seg m holds
( (PartDiffSeq (f,X,I)) . 0 = X --> d & ( for i being Element of NAT st 1 <= i & i <= len I holds
(PartDiffSeq (f,X,I)) . i = X --> 0 ) )

let f be PartFunc of (REAL m),REAL; :: thesis: for d being Real st X is open & f = X --> d & rng I c= Seg m holds
( (PartDiffSeq (f,X,I)) . 0 = X --> d & ( for i being Element of NAT st 1 <= i & i <= len I holds
(PartDiffSeq (f,X,I)) . i = X --> 0 ) )

let d be Real; :: thesis: ( X is open & f = X --> d & rng I c= Seg m implies ( (PartDiffSeq (f,X,I)) . 0 = X --> d & ( for i being Element of NAT st 1 <= i & i <= len I holds
(PartDiffSeq (f,X,I)) . i = X --> 0 ) ) )

assume A1: ( X is open & f = X --> d & rng I c= Seg m ) ; :: thesis: ( (PartDiffSeq (f,X,I)) . 0 = X --> d & ( for i being Element of NAT st 1 <= i & i <= len I holds
(PartDiffSeq (f,X,I)) . i = X --> 0 ) )

thus A2: (PartDiffSeq (f,X,I)) . 0 = (X --> d) | X by
.= (X /\ X) --> d by FUNCOP_1:12
.= X --> d ; :: thesis: for i being Element of NAT st 1 <= i & i <= len I holds
(PartDiffSeq (f,X,I)) . i = X --> 0

defpred S1[ Nat] means ( 1 <= \$1 & \$1 <= len I implies (PartDiffSeq (f,X,I)) . \$1 = X --> 0 );
A3: S1[ 0 ] ;
A4: for i being Nat st S1[i] holds
S1[i + 1]
proof
let i be Nat; :: thesis: ( S1[i] implies S1[i + 1] )
assume A5: S1[i] ; :: thesis: S1[i + 1]
assume A6: ( 1 <= i + 1 & i + 1 <= len I ) ; :: thesis: (PartDiffSeq (f,X,I)) . (i + 1) = X --> 0
A7: i <= i + 1 by NAT_1:12;
per cases ( i = 0 or i <> 0 ) ;
suppose A8: i = 0 ; :: thesis: (PartDiffSeq (f,X,I)) . (i + 1) = X --> 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 ; :: thesis: verum
end;
suppose A10: i <> 0 ; :: thesis: (PartDiffSeq (f,X,I)) . (i + 1) = X --> 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 ; :: thesis: verum
end;
end;
end;
for i being Nat holds S1[i] from NAT_1:sch 2(A3, A4);
hence for i being Element of NAT st 1 <= i & i <= len I holds
(PartDiffSeq (f,X,I)) . i = X --> 0 ; :: thesis: verum