let r, s be Real; :: thesis: ( ex x0, y0 being Real st
( z = <*x0,y0*> & ex N being Neighbourhood of x0 st
( N c= dom (SVF1 1,(pdiff1 f,2),z) & ex L being LINEAR ex R being REST st
( r = L . 1 & ( for x being Real st x in N holds
((SVF1 1,(pdiff1 f,2),z) . x) - ((SVF1 1,(pdiff1 f,2),z) . x0) = (L . (x - x0)) + (R . (x - x0)) ) ) ) ) & ex x0, y0 being Real st
( z = <*x0,y0*> & ex N being Neighbourhood of x0 st
( N c= dom (SVF1 1,(pdiff1 f,2),z) & ex L being LINEAR ex R being REST st
( s = L . 1 & ( for x being Real st x in N holds
((SVF1 1,(pdiff1 f,2),z) . x) - ((SVF1 1,(pdiff1 f,2),z) . x0) = (L . (x - x0)) + (R . (x - x0)) ) ) ) ) implies r = s )
assume that
A8: ex x0, y0 being Real st
( z = <*x0,y0*> & ex N being Neighbourhood of x0 st
( N c= dom (SVF1 1,(pdiff1 f,2),z) & ex L being LINEAR ex R being REST st
( r = L . 1 & ( for x being Real st x in N holds
((SVF1 1,(pdiff1 f,2),z) . x) - ((SVF1 1,(pdiff1 f,2),z) . x0) = (L . (x - x0)) + (R . (x - x0)) ) ) ) ) and
A9: ex x0, y0 being Real st
( z = <*x0,y0*> & ex N being Neighbourhood of x0 st
( N c= dom (SVF1 1,(pdiff1 f,2),z) & ex L being LINEAR ex R being REST st
( s = L . 1 & ( for x being Real st x in N holds
((SVF1 1,(pdiff1 f,2),z) . x) - ((SVF1 1,(pdiff1 f,2),z) . x0) = (L . (x - x0)) + (R . (x - x0)) ) ) ) ) ; :: thesis: r = s
consider x1, y1 being Real such that
A10: z = <*x1,y1*> and
A11: ex N being Neighbourhood of x1 st
( N c= dom (SVF1 1,(pdiff1 f,2),z) & ex L being LINEAR ex R being REST st
( s = L . 1 & ( for x being Real st x in N holds
((SVF1 1,(pdiff1 f,2),z) . x) - ((SVF1 1,(pdiff1 f,2),z) . x1) = (L . (x - x1)) + (R . (x - x1)) ) ) ) by A9;
consider N1 being Neighbourhood of x1 such that
N1 c= dom (SVF1 1,(pdiff1 f,2),z) and
A12: ex L being LINEAR ex R being REST st
( s = L . 1 & ( for x being Real st x in N1 holds
((SVF1 1,(pdiff1 f,2),z) . x) - ((SVF1 1,(pdiff1 f,2),z) . x1) = (L . (x - x1)) + (R . (x - x1)) ) ) by A11;
consider L1 being LINEAR, R1 being REST such that
A13: s = L1 . 1 and
A14: for x being Real st x in N1 holds
((SVF1 1,(pdiff1 f,2),z) . x) - ((SVF1 1,(pdiff1 f,2),z) . x1) = (L1 . (x - x1)) + (R1 . (x - x1)) by A12;
consider p1 being Real such that
A15: for p being Real holds L1 . p = p1 * p by FDIFF_1:def 4;
A16: s = p1 * 1 by A13, A15;
consider x0, y0 being Real such that
A17: z = <*x0,y0*> and
A18: ex N being Neighbourhood of x0 st
( N c= dom (SVF1 1,(pdiff1 f,2),z) & ex L being LINEAR ex R being REST st
( r = L . 1 & ( for x being Real st x in N holds
((SVF1 1,(pdiff1 f,2),z) . x) - ((SVF1 1,(pdiff1 f,2),z) . x0) = (L . (x - x0)) + (R . (x - x0)) ) ) ) by A8;
consider N being Neighbourhood of x0 such that
N c= dom (SVF1 1,(pdiff1 f,2),z) and
A19: ex L being LINEAR ex R being REST st
( r = L . 1 & ( for x being Real st x in N holds
((SVF1 1,(pdiff1 f,2),z) . x) - ((SVF1 1,(pdiff1 f,2),z) . x0) = (L . (x - x0)) + (R . (x - x0)) ) ) by A18;
consider L being LINEAR, R being REST such that
A20: r = L . 1 and
A21: for x being Real st x in N holds
((SVF1 1,(pdiff1 f,2),z) . x) - ((SVF1 1,(pdiff1 f,2),z) . x0) = (L . (x - x0)) + (R . (x - x0)) by A19;
consider r1 being Real such that
A22: for p being Real holds L . p = r1 * p by FDIFF_1:def 4;
A23: x0 = x1 by A17, A10, FINSEQ_1:98;
then consider N0 being Neighbourhood of x0 such that
A24: ( N0 c= N & N0 c= N1 ) by RCOMP_1:38;
consider g being real number such that
A25: 0 < g and
A26: N0 = ].(x0 - g),(x0 + g).[ by RCOMP_1:def 7;
deffunc H1( Element of NAT ) -> Element of REAL = g / ($1 + 2);
consider s1 being Real_Sequence such that
A27: for n being Element of NAT holds s1 . n = H1(n) from SEQ_1:sch 1();
now
let n be Element of NAT ; :: thesis: s1 . n <> 0
g / (n + 2) <> 0 by A25, XREAL_1:141;
hence s1 . n <> 0 by A27; :: thesis: verum
end;
then A28: s1 is non-zero by SEQ_1:7;
( s1 is convergent & lim s1 = 0 ) by A27, SEQ_4:46;
then reconsider h = s1 as convergent_to_0 Real_Sequence by A28, FDIFF_1:def 1;
A29: for n being Element of NAT ex x being Real st
( x in N & x in N1 & h . n = x - x0 )
proof
let n be Element of NAT ; :: thesis: ex x being Real st
( x in N & x in N1 & h . n = x - x0 )
take x = x0 + (g / (n + 2)); :: thesis: ( x in N & x in N1 & h . n = x - x0 )
0 + 1 < (n + 1) + 1 by XREAL_1:8;
then g / (n + 2) < g / 1 by A25, XREAL_1:78;
then A30: x0 + (g / (n + 2)) < x0 + g by XREAL_1:8;
g / (n + 2) > 0 by A25, XREAL_1:141;
then x0 + (- g) < x0 + (g / (n + 2)) by A25, XREAL_1:8;
then x0 + (g / (n + 2)) in ].(x0 - g),(x0 + g).[ by A30;
hence ( x in N & x in N1 & h . n = x - x0 ) by A24, A26, A27; :: thesis: verum
end;
A31: r = r1 * 1 by A20, A22;
A32: now
let x be Real; :: thesis: ( x in N & x in N1 implies (r * (x - x0)) + (R . (x - x0)) = (s * (x - x0)) + (R1 . (x - x0)) )
assume that
A33: x in N and
A34: x in N1 ; :: thesis: (r * (x - x0)) + (R . (x - x0)) = (s * (x - x0)) + (R1 . (x - x0))
((SVF1 1,(pdiff1 f,2),z) . x) - ((SVF1 1,(pdiff1 f,2),z) . x0) = (L . (x - x0)) + (R . (x - x0)) by A21, A33;
then (L . (x - x0)) + (R . (x - x0)) = (L1 . (x - x0)) + (R1 . (x - x0)) by A14, A23, A34;
then (r1 * (x - x0)) + (R . (x - x0)) = (L1 . (x - x0)) + (R1 . (x - x0)) by A22;
hence (r * (x - x0)) + (R . (x - x0)) = (s * (x - x0)) + (R1 . (x - x0)) by A15, A31, A16; :: thesis: verum
end;
now
dom R1 = REAL by PARTFUN1:def 4;
then A35: rng h c= dom R1 ;
let n be Nat; :: thesis: r - s = (((h " ) (#) (R1 /* h)) - ((h " ) (#) (R /* h))) . n
dom R = REAL by PARTFUN1:def 4;
then A36: rng h c= dom R ;
A37: n in NAT by ORDINAL1:def 13;
then ex x being Real st
( x in N & x in N1 & h . n = x - x0 ) by A29;
then (r * (h . n)) + (R . (h . n)) = (s * (h . n)) + (R1 . (h . n)) by A32;
then A38: ((r * (h . n)) / (h . n)) + ((R . (h . n)) / (h . n)) = ((s * (h . n)) + (R1 . (h . n))) / (h . n) by XCMPLX_1:63;
A39: (R . (h . n)) / (h . n) = (R . (h . n)) * ((h . n) " ) by XCMPLX_0:def 9
.= (R . (h . n)) * ((h " ) . n) by VALUED_1:10
.= ((R /* h) . n) * ((h " ) . n) by A37, A36, FUNCT_2:185
.= ((h " ) (#) (R /* h)) . n by VALUED_1:5 ;
h is non-zero by FDIFF_1:def 1;
then A40: h . n <> 0 by A37, SEQ_1:7;
A41: (R1 . (h . n)) / (h . n) = (R1 . (h . n)) * ((h . n) " ) by XCMPLX_0:def 9
.= (R1 . (h . n)) * ((h " ) . n) by VALUED_1:10
.= ((R1 /* h) . n) * ((h " ) . n) by A37, A35, FUNCT_2:185
.= ((h " ) (#) (R1 /* h)) . n by VALUED_1:5 ;
A42: (s * (h . n)) / (h . n) = s * ((h . n) / (h . n)) by XCMPLX_1:75
.= s * 1 by A40, XCMPLX_1:60
.= s ;
(r * (h . n)) / (h . n) = r * ((h . n) / (h . n)) by XCMPLX_1:75
.= r * 1 by A40, XCMPLX_1:60
.= r ;
then r + ((R . (h . n)) / (h . n)) = s + ((R1 . (h . n)) / (h . n)) by A38, A42, XCMPLX_1:63;
then r = s + ((((h " ) (#) (R1 /* h)) . n) - (((h " ) (#) (R /* h)) . n)) by A39, A41;
hence r - s = (((h " ) (#) (R1 /* h)) - ((h " ) (#) (R /* h))) . n by A37, VALUED_1:15; :: thesis: verum
end;
then ( ((h " ) (#) (R1 /* h)) - ((h " ) (#) (R /* h)) is V8() & (((h " ) (#) (R1 /* h)) - ((h " ) (#) (R /* h))) . 1 = r - s ) by VALUED_0:def 18;
then A43: lim (((h " ) (#) (R1 /* h)) - ((h " ) (#) (R /* h))) = r - s by SEQ_4:40;
A44: ( (h " ) (#) (R1 /* h) is convergent & lim ((h " ) (#) (R1 /* h)) = 0 ) by FDIFF_1:def 3;
( (h " ) (#) (R /* h) is convergent & lim ((h " ) (#) (R /* h)) = 0 ) by FDIFF_1:def 3;
then r - s = 0 - 0 by A43, A44, SEQ_2:26;
hence r = s ; :: thesis: verum