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,1)),z)) & ex L being LinearFunc ex R being RestFunc st
( r = L . 1 & ( for x being Real st x in N holds
((SVF1 (1,(pdiff1 (f,1)),z)) . x) - ((SVF1 (1,(pdiff1 (f,1)),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,1)),z)) & ex L being LinearFunc ex R being RestFunc st
( s = L . 1 & ( for x being Real st x in N holds
((SVF1 (1,(pdiff1 (f,1)),z)) . x) - ((SVF1 (1,(pdiff1 (f,1)),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,1)),z)) & ex L being LinearFunc ex R being RestFunc st
( r = L . 1 & ( for x being Real st x in N holds
((SVF1 (1,(pdiff1 (f,1)),z)) . x) - ((SVF1 (1,(pdiff1 (f,1)),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,1)),z)) & ex L being LinearFunc ex R being RestFunc st
( s = L . 1 & ( for x being Real st x in N holds
((SVF1 (1,(pdiff1 (f,1)),z)) . x) - ((SVF1 (1,(pdiff1 (f,1)),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,1)),z)) & ex L being LinearFunc ex R being RestFunc st
( s = L . 1 & ( for x being Real st x in N holds
((SVF1 (1,(pdiff1 (f,1)),z)) . x) - ((SVF1 (1,(pdiff1 (f,1)),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,1)),z)) and
A12: ex L being LinearFunc ex R being RestFunc st
( s = L . 1 & ( for x being Real st x in N1 holds
((SVF1 (1,(pdiff1 (f,1)),z)) . x) - ((SVF1 (1,(pdiff1 (f,1)),z)) . x1) = (L . (x - x1)) + (R . (x - x1)) ) ) by A11;
consider L1 being LinearFunc, R1 being RestFunc such that
A13: s = L1 . 1 and
A14: for x being Real st x in N1 holds
((SVF1 (1,(pdiff1 (f,1)),z)) . x) - ((SVF1 (1,(pdiff1 (f,1)),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 3;
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,1)),z)) & ex L being LinearFunc ex R being RestFunc st
( r = L . 1 & ( for x being Real st x in N holds
((SVF1 (1,(pdiff1 (f,1)),z)) . x) - ((SVF1 (1,(pdiff1 (f,1)),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,1)),z)) and
A19: ex L being LinearFunc ex R being RestFunc st
( r = L . 1 & ( for x being Real st x in N holds
((SVF1 (1,(pdiff1 (f,1)),z)) . x) - ((SVF1 (1,(pdiff1 (f,1)),z)) . x0) = (L . (x - x0)) + (R . (x - x0)) ) ) by A18;
consider L being LinearFunc, R being RestFunc such that
A20: r = L . 1 and
A21: for x being Real st x in N holds
((SVF1 (1,(pdiff1 (f,1)),z)) . x) - ((SVF1 (1,(pdiff1 (f,1)),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 3;
A23: x0 = x1 by A17, A10, FINSEQ_1:77;
then consider N0 being Neighbourhood of x0 such that
A24: ( N0 c= N & N0 c= N1 ) by RCOMP_1:17;
consider g being Real such that
A25: 0 < g and
A26: N0 = ].(x0 - g),(x0 + g).[ by RCOMP_1:def 6;
deffunc H1( Nat) -> set = g / ($1 + 2);
consider s1 being Real_Sequence such that
A27: for n being Nat holds s1 . n = H1(n) from SEQ_1:sch 1();
now :: thesis: for n being Nat holds s1 . n <> 0
let n be Nat; :: thesis: s1 . n <> 0
g / (n + 2) <> 0 by A25, XREAL_1:139;
hence s1 . n <> 0 by A27; :: thesis: verum
end;
then A28: s1 is non-zero by SEQ_1:5;
( s1 is convergent & lim s1 = 0 ) by A27, SEQ_4:31;
then reconsider h = s1 as non-zero 0 -convergent 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 x0 + (g / (n + 2)) ; :: thesis: ( x0 + (g / (n + 2)) in N & x0 + (g / (n + 2)) in N1 & h . n = (x0 + (g / (n + 2))) - x0 )
0 + 1 < (n + 1) + 1 by XREAL_1:6;
then g / (n + 2) < g / 1 by A25, XREAL_1:76;
then A30: x0 + (g / (n + 2)) < x0 + g by XREAL_1:6;
g / (n + 2) > 0 by A25, XREAL_1:139;
then x0 + (- g) < x0 + (g / (n + 2)) by A25, XREAL_1:6;
then x0 + (g / (n + 2)) in ].(x0 - g),(x0 + g).[ by A30;
hence ( x0 + (g / (n + 2)) in N & x0 + (g / (n + 2)) in N1 & h . n = (x0 + (g / (n + 2))) - x0 ) by A24, A26, A27; :: thesis: verum
end;
A31: r = r1 * 1 by A20, A22;
A32: now :: thesis: for x being Real st x in N & x in N1 holds
(r * (x - x0)) + (R . (x - x0)) = (s * (x - x0)) + (R1 . (x - x0))
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,1)),z)) . x) - ((SVF1 (1,(pdiff1 (f,1)),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;
A35: r - s in REAL by XREAL_0:def 1;
for n being Nat holds r - s = (((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h))) . n
proof
dom R1 = REAL by PARTFUN1:def 2;
then A36: 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 2;
then A37: rng h c= dom R ;
A38: n in NAT by ORDINAL1:def 12;
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 A39: ((r * (h . n)) / (h . n)) + ((R . (h . n)) / (h . n)) = ((s * (h . n)) + (R1 . (h . n))) / (h . n) by XCMPLX_1:62;
A40: (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 A38, A37, FUNCT_2:108
.= ((h ") (#) (R /* h)) . n by VALUED_1:5 ;
A41: h . n <> 0 by SEQ_1:5;
A42: (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 A38, A36, FUNCT_2:108
.= ((h ") (#) (R1 /* h)) . n by VALUED_1:5 ;
A43: (s * (h . n)) / (h . n) = s * ((h . n) / (h . n)) by XCMPLX_1:74
.= s * 1 by A41, XCMPLX_1:60
.= s ;
(r * (h . n)) / (h . n) = r * ((h . n) / (h . n)) by XCMPLX_1:74
.= r * 1 by A41, XCMPLX_1:60
.= r ;
then r + ((R . (h . n)) / (h . n)) = s + ((R1 . (h . n)) / (h . n)) by A39, A43, XCMPLX_1:62;
then r = s + ((((h ") (#) (R1 /* h)) . n) - (((h ") (#) (R /* h)) . n)) by A40, A42;
hence r - s = (((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h))) . n by RFUNCT_2:1; :: thesis: verum
end;
then ( ((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h)) is constant & (((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h))) . 1 = r - s ) by VALUED_0:def 18, A35;
then A44: lim (((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h))) = r - s by SEQ_4:25;
A45: ( (h ") (#) (R1 /* h) is convergent & lim ((h ") (#) (R1 /* h)) = 0 ) by FDIFF_1:def 2;
( (h ") (#) (R /* h) is convergent & lim ((h ") (#) (R /* h)) = 0 ) by FDIFF_1:def 2;
then r - s = 0 - 0 by A44, A45, SEQ_2:12;
hence r = s ; :: thesis: verum