let f be PartFunc of (REAL 3),REAL; :: thesis:
let u0 be Element of REAL 3; :: thesis:
let N be Neighbourhood of (proj (1,3)) . u0; :: thesis:
assume A1: ( f is_partial_differentiable_in u0,1 & N c= dom (SVF1 (1,f,u0)) ) ; :: thesis:
let h be convergent_to_0 Real_Sequence; :: thesis:
let c be constant Real_Sequence; :: thesis:
assume A2: ( rng c = {((proj (1,3)) . u0)} & rng (h + c) c= N ) ; :: thesis:
consider x0, y0, z0 being Real such that
A3: ( u0 = <*x0,y0,z0*> & ex N1 being Neighbourhood of x0 st
( N1 c= dom (SVF1 (1,f,u0)) & ex L being LINEAR ex R being REST st
for x being Real st x in N1 holds
((SVF1 (1,f,u0)) . x) - ((SVF1 (1,f,u0)) . x0) = (L . (x - x0)) + (R . (x - x0)) ) ) by A1, Th13;
consider N1 being Neighbourhood of x0 such that
A4: ( N1 c= dom (SVF1 (1,f,u0)) & ex L being LINEAR ex R being REST st
for x being Real st x in N1 holds
((SVF1 (1,f,u0)) . x) - ((SVF1 (1,f,u0)) . x0) = (L . (x - x0)) + (R . (x - x0)) ) by A3;
consider L being LINEAR, R being REST such that
A5: for x being Real st x in N1 holds
((SVF1 (1,f,u0)) . x) - ((SVF1 (1,f,u0)) . x0) = (L . (x - x0)) + (R . (x - x0)) by A4;
A6: (proj (1,3)) . u0 = x0 by A3, Th1;
then consider N2 being Neighbourhood of x0 such that
A7: ( N2 c= N & N2 c= N1 ) by RCOMP_1:17;
consider g being real number such that
A8: ( 0 < g & N2 = ].(x0 - g),(x0 + g).[ ) by RCOMP_1:def 6;
A9: x0 in N2

A10: x0 + 0 < x0 + g by A8, XREAL_1:8;
x0 - g < x0 - 0 by A8, XREAL_1:44;
hence x0 in N2 by A8, A10; :: thesis:

end;

ex n being Element of NAT st
( rng (c ^\ n) c= N2 & rng ((h + c) ^\ n) c= N2 )

proof

x0 in rng c by A2, A6, TARSKI:def 1;
then A11: lim c = x0 by SEQ_4:25;
A12: ( h is convergent & lim h = 0 ) by FDIFF_1:def 1;
then A13: lim (h + c) = 0 + x0 by A11, SEQ_2:6
.= x0 ;
h + c is convergent by A12, SEQ_2:5;
then consider n being Element of NAT such that
A14: for m being Element of NAT st n <= m holds
abs (((h + c) . m) - x0) < g by A8, A13, SEQ_2:def 7;
A15: rng (c ^\ n) = {x0} by A2, A6, VALUED_0:26;
take n ; :: thesis:
thus rng (c ^\ n) c= N2 :: thesis:

proof

let y be set ; :: according to TARSKI:def 3 :: thesis:
assume y in rng (c ^\ n) ; :: thesis:
hence y in N2 by A9, A15, TARSKI:def 1; :: thesis:

end;

let y be set ; :: according to TARSKI:def 3 :: thesis:
assume y in rng ((h + c) ^\ n) ; :: thesis:
then consider m being Element of NAT such that
A16: y = ((h + c) ^\ n) . m by FUNCT_2:113;
n + 0 <= n + m by XREAL_1:7;
then abs (((h + c) . (n + m)) - x0) < g by A14;
then ( - g < ((h + c) . (m + n)) - x0 & ((h + c) . (m + n)) - x0 < g ) by SEQ_2:1;
then ( - g < (((h + c) ^\ n) . m) - x0 & (((h + c) ^\ n) . m) - x0 < g ) by NAT_1:def 3;
then ( x0 + (- g) < ((h + c) ^\ n) . m & ((h + c) ^\ n) . m < x0 + g ) by XREAL_1:19, XREAL_1:20;
hence y in N2 by A8, A16; :: thesis:

end;

then consider n being Element of NAT such that
A17: ( rng (c ^\ n) c= N2 & rng ((h + c) ^\ n) c= N2 ) ;
A18: rng (c ^\ n) c= dom (SVF1 (1,f,u0))

proof

let y be set ; :: according to TARSKI:def 3 :: thesis:
A19: rng (c ^\ n) = rng c by VALUED_0:26;
assume y in rng (c ^\ n) ; :: thesis:
then y = x0 by A2, A6, A19, TARSKI:def 1;
then y in N by A7, A9;
hence y in dom (SVF1 (1,f,u0)) by A1; :: thesis:

end;

A20: rng ((h + c) ^\ n) c= dom (SVF1 (1,f,u0))

proof

let y be set ; :: according to TARSKI:def 3 :: thesis:
assume y in rng ((h + c) ^\ n) ; :: thesis:
then y in N2 by A17;
then y in N by A7;
hence y in dom (SVF1 (1,f,u0)) by A1; :: thesis:

end;

A21: rng c c= dom (SVF1 (1,f,u0))

proof

let y be set ; :: according to TARSKI:def 3 :: thesis:
assume y in rng c ; :: thesis:
then y = x0 by A2, A6, TARSKI:def 1;
then y in N by A7, A9;
hence y in dom (SVF1 (1,f,u0)) by A1; :: thesis:

end;

A22: rng (h + c) c= dom (SVF1 (1,f,u0))

proof

let y be set ; :: according to TARSKI:def 3 :: thesis:
assume y in rng (h + c) ; :: thesis:
then y in N by A2;
hence y in dom (SVF1 (1,f,u0)) by A1; :: thesis:

end;

A23: for x being Real st x in N2 holds
((SVF1 (1,f,u0)) . x) - ((SVF1 (1,f,u0)) . x0) = (L . (x - x0)) + (R . (x - x0)) by A5, A7;
A24: for k being Element of NAT holds ((SVF1 (1,f,u0)) . (((h + c) ^\ n) . k)) - ((SVF1 (1,f,u0)) . ((c ^\ n) . k)) = (L . ((h ^\ n) . k)) + (R . ((h ^\ n) . k))

proof

let k be Element of NAT ; :: thesis:
((h + c) ^\ n) . k in rng ((h + c) ^\ n) by VALUED_0:28;
then A25: ((h + c) ^\ n) . k in N2 by A17;
A26: (((h + c) ^\ n) . k) - ((c ^\ n) . k) = (((h ^\ n) + (c ^\ n)) . k) - ((c ^\ n) . k) by SEQM_3:15
.= (((h ^\ n) . k) + ((c ^\ n) . k)) - ((c ^\ n) . k) by SEQ_1:7
.= (h ^\ n) . k ;
A27: (c ^\ n) . k in rng (c ^\ n) by VALUED_0:28;
rng (c ^\ n) = rng c by VALUED_0:26;
then (c ^\ n) . k = x0 by A2, A6, A27, TARSKI:def 1;
hence ((SVF1 (1,f,u0)) . (((h + c) ^\ n) . k)) - ((SVF1 (1,f,u0)) . ((c ^\ n) . k)) = (L . ((h ^\ n) . k)) + (R . ((h ^\ n) . k)) by A5, A7, A25, A26; :: thesis:

end;

A28: L is total by FDIFF_1:def 3;
A29: R is total by FDIFF_1:def 2;
A30: ((SVF1 (1,f,u0)) /* ((h + c) ^\ n)) - ((SVF1 (1,f,u0)) /* (c ^\ n)) = (L /* (h ^\ n)) + (R /* (h ^\ n))

proof

now

let k be Element of NAT ; :: thesis:
thus (((SVF1 (1,f,u0)) /* ((h + c) ^\ n)) - ((SVF1 (1,f,u0)) /* (c ^\ n))) . k = (((SVF1 (1,f,u0)) /* ((h + c) ^\ n)) . k) - (((SVF1 (1,f,u0)) /* (c ^\ n)) . k) by RFUNCT_2:1
.= ((SVF1 (1,f,u0)) . (((h + c) ^\ n) . k)) - (((SVF1 (1,f,u0)) /* (c ^\ n)) . k) by A20, FUNCT_2:108
.= ((SVF1 (1,f,u0)) . (((h + c) ^\ n) . k)) - ((SVF1 (1,f,u0)) . ((c ^\ n) . k)) by A18, FUNCT_2:108
.= (L . ((h ^\ n) . k)) + (R . ((h ^\ n) . k)) by A24
.= ((L /* (h ^\ n)) . k) + (R . ((h ^\ n) . k)) by A28, FUNCT_2:115
.= ((L /* (h ^\ n)) . k) + ((R /* (h ^\ n)) . k) by A29, FUNCT_2:115
.= ((L /* (h ^\ n)) + (R /* (h ^\ n))) . k by SEQ_1:7 ; :: thesis:

end;

hence ((SVF1 (1,f,u0)) /* ((h + c) ^\ n)) - ((SVF1 (1,f,u0)) /* (c ^\ n)) = (L /* (h ^\ n)) + (R /* (h ^\ n)) by FUNCT_2:63; :: thesis:

end;

A31: ( ((L /* (h ^\ n)) + (R /* (h ^\ n))) (#) ((h ^\ n) ") is convergent & lim (((L /* (h ^\ n)) + (R /* (h ^\ n))) (#) ((h ^\ n) ")) = L . 1 )

proof

deffunc H₁( Element of NAT ) -> Element of REAL = (L . 1) + (((R /* (h ^\ n)) (#) ((h ^\ n) ")) . $1);
consider s1 being Real_Sequence such that
A32: for k being Element of NAT holds s1 . k = H₁(k) from SEQ_1:sch 1();
consider s being Real such that
A33: for p1 being Real holds L . p1 = s * p1 by FDIFF_1:def 3;
A34: L . 1 = s * 1 by A33
.= s ;

now

let m be Element of NAT ; :: thesis:
h ^\ n is non-empty by FDIFF_1:def 1;
then A35: (h ^\ n) . m <> 0 by SEQ_1:5;
thus (((L /* (h ^\ n)) + (R /* (h ^\ n))) (#) ((h ^\ n) ")) . m = (((L /* (h ^\ n)) + (R /* (h ^\ n))) . m) * (((h ^\ n) ") . m) by SEQ_1:8
.= (((L /* (h ^\ n)) . m) + ((R /* (h ^\ n)) . m)) * (((h ^\ n) ") . m) by SEQ_1:7
.= (((L /* (h ^\ n)) . m) * (((h ^\ n) ") . m)) + (((R /* (h ^\ n)) . m) * (((h ^\ n) ") . m))
.= (((L /* (h ^\ n)) . m) * (((h ^\ n) ") . m)) + (((R /* (h ^\ n)) (#) ((h ^\ n) ")) . m) by SEQ_1:8
.= (((L /* (h ^\ n)) . m) * (((h ^\ n) . m) ")) + (((R /* (h ^\ n)) (#) ((h ^\ n) ")) . m) by VALUED_1:10
.= ((L . ((h ^\ n) . m)) * (((h ^\ n) . m) ")) + (((R /* (h ^\ n)) (#) ((h ^\ n) ")) . m) by A28, FUNCT_2:115
.= ((s * ((h ^\ n) . m)) * (((h ^\ n) . m) ")) + (((R /* (h ^\ n)) (#) ((h ^\ n) ")) . m) by A33
.= (s * (((h ^\ n) . m) * (((h ^\ n) . m) "))) + (((R /* (h ^\ n)) (#) ((h ^\ n) ")) . m)
.= (s * 1) + (((R /* (h ^\ n)) (#) ((h ^\ n) ")) . m) by A35, XCMPLX_0:def 7
.= s1 . m by A32, A34 ; :: thesis:

end;

then A36: ((L /* (h ^\ n)) + (R /* (h ^\ n))) (#) ((h ^\ n) ") = s1 by FUNCT_2:63;

A37: now

let r be real number ; :: thesis:
assume A38: 0 < r ; :: thesis:
( ((h ^\ n) ") (#) (R /* (h ^\ n)) is convergent & lim (((h ^\ n) ") (#) (R /* (h ^\ n))) = 0 ) by FDIFF_1:def 2;
then consider m being Element of NAT such that
A39: for k being Element of NAT st m <= k holds
abs (((((h ^\ n) ") (#) (R /* (h ^\ n))) . k) - 0) < r by A38, SEQ_2:def 7;
take n1 = m; :: thesis:
let k be Element of NAT ; :: thesis:
assume A40: n1 <= k ; :: thesis:
abs ((s1 . k) - (L . 1)) = abs (((L . 1) + (((R /* (h ^\ n)) (#) ((h ^\ n) ")) . k)) - (L . 1)) by A32
.= abs (((((h ^\ n) ") (#) (R /* (h ^\ n))) . k) - 0) ;
hence abs ((s1 . k) - (L . 1)) < r by A39, A40; :: thesis:

end;

hence ((L /* (h ^\ n)) + (R /* (h ^\ n))) (#) ((h ^\ n) ") is convergent by A36, SEQ_2:def 6; :: thesis:
hence lim (((L /* (h ^\ n)) + (R /* (h ^\ n))) (#) ((h ^\ n) ")) = L . 1 by A36, A37, SEQ_2:def 7; :: thesis:

end;

A41: N2 c= dom (SVF1 (1,f,u0)) by A1, A7, XBOOLE_1:1;
A42: ((L /* (h ^\ n)) + (R /* (h ^\ n))) (#) ((h ^\ n) ") = ((((SVF1 (1,f,u0)) /* (h + c)) ^\ n) - ((SVF1 (1,f,u0)) /* (c ^\ n))) (#) ((h ^\ n) ") by A22, A30, VALUED_0:27
.= ((((SVF1 (1,f,u0)) /* (h + c)) ^\ n) - (((SVF1 (1,f,u0)) /* c) ^\ n)) (#) ((h ^\ n) ") by A21, VALUED_0:27
.= ((((SVF1 (1,f,u0)) /* (h + c)) - ((SVF1 (1,f,u0)) /* c)) ^\ n) (#) ((h ^\ n) ") by SEQM_3:17
.= ((((SVF1 (1,f,u0)) /* (h + c)) - ((SVF1 (1,f,u0)) /* c)) ^\ n) (#) ((h ") ^\ n) by SEQM_3:18
.= ((((SVF1 (1,f,u0)) /* (h + c)) - ((SVF1 (1,f,u0)) /* c)) (#) (h ")) ^\ n by SEQM_3:19 ;
then A43: L . 1 = lim ((h ") (#) (((SVF1 (1,f,u0)) /* (h + c)) - ((SVF1 (1,f,u0)) /* c))) by A31, SEQ_4:22;
thus (h ") (#) (((SVF1 (1,f,u0)) /* (h + c)) - ((SVF1 (1,f,u0)) /* c)) is convergent by A31, A42, SEQ_4:21; :: thesis:
thus partdiff (f,u0,1) = lim ((h ") (#) (((SVF1 (1,f,u0)) /* (h + c)) - ((SVF1 (1,f,u0)) /* c))) by A1, A3, A23, A41, A43, Th16; :: thesis: