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
( f is_partial_differentiable_on X,I & f `partial| (X,I) is_continuous_on X )
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
( f is_partial_differentiable_on X,I & f `partial| (X,I) is_continuous_on X )
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
( f is_partial_differentiable_on X,I & f `partial| (X,I) is_continuous_on X )
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
( f is_partial_differentiable_on X,I & f `partial| (X,I) is_continuous_on X )
let d be Real; :: thesis: ( X is open & f = X --> d & rng I c= Seg m implies ( f is_partial_differentiable_on X,I & f `partial| (X,I) is_continuous_on X ) )
assume A1: ( X is open & f = X --> d & rng I c= Seg m ) ; :: thesis: ( f is_partial_differentiable_on X,I & f `partial| (X,I) is_continuous_on X )
for i being Element of NAT st i <= (len I) - 1 holds
(PartDiffSeq (f,X,I)) . i is_partial_differentiable_on X,I /. (i + 1)

proof

let i be Element of NAT ; :: thesis: ( i <= (len I) - 1 implies (PartDiffSeq (f,X,I)) . i is_partial_differentiable_on X,I /. (i + 1) )
assume A2: i <= (len I) - 1 ; :: thesis: (PartDiffSeq (f,X,I)) . i is_partial_differentiable_on X,I /. (i + 1)
(len I) - 1 <= (len I) - 0 by XREAL_1:10;
then A3: i <= len I by A2, XXREAL_0:2;

per cases ( i = 0 or i <> 0 ) ;

suppose i = 0 ; :: thesis: (PartDiffSeq (f,X,I)) . i is_partial_differentiable_on X,I /. (i + 1)

then A4: (PartDiffSeq (f,X,I)) . i = X --> d by A1, Th17;
( 1 <= I /. (i + 1) & I /. (i + 1) <= m ) by Lm1, A1, A2;
hence (PartDiffSeq (f,X,I)) . i is_partial_differentiable_on X,I /. (i + 1) by A4, A1, Th15; :: thesis: verum

end;

suppose i <> 0 ; :: thesis: (PartDiffSeq (f,X,I)) . i is_partial_differentiable_on X,I /. (i + 1)

then 1 <= i by NAT_1:14;
then A5: (PartDiffSeq (f,X,I)) . i = X --> 0 by A1, A3, Th17;
( 1 <= I /. (i + 1) & I /. (i + 1) <= m ) by Lm1, A1, A2;
hence (PartDiffSeq (f,X,I)) . i is_partial_differentiable_on X,I /. (i + 1) by A5, A1, Th15; :: thesis: verum

end;

end;

end;

hence f is_partial_differentiable_on X,I ; :: thesis: f `partial| (X,I) is_continuous_on X
reconsider k = (len I) - 1 as Element of NAT by INT_1:5, FINSEQ_1:20;
A6: f `partial| (X,I) = ((PartDiffSeq (f,X,I)) . k) `partial| (X,(I /. (k + 1))) by PDIFF_9:def 7;
A7: (len I) - 1 <= (len I) - 0 by XREAL_1:10;

per cases ( k = 0 or k <> 0 ) ;

suppose k = 0 ; :: thesis: f `partial| (X,I) is_continuous_on X

then A8: (PartDiffSeq (f,X,I)) . k = X --> d by A1, Th17;
( 1 <= I /. (k + 1) & I /. (k + 1) <= m ) by A1, Lm1;
hence f `partial| (X,I) is_continuous_on X by A1, A6, A8, Th15; :: thesis: verum

end;

suppose k <> 0 ; :: thesis: f `partial| (X,I) is_continuous_on X

then 1 <= k by NAT_1:14;
then A9: (PartDiffSeq (f,X,I)) . k = X --> 0 by A1, A7, Th17;
( 1 <= I /. (k + 1) & I /. (k + 1) <= m ) by A1, Lm1;
hence f `partial| (X,I) is_continuous_on X by A1, A6, A9, Th15; :: thesis: verum

end;

end;