let r, s be Real; :: thesis: ( ex x0, y0 being Real st
( z = <*x0,y0*> & ex N being Neighbourhood of y0 st
( N c= dom (SVF1 (2,(pdiff1 (f,2)),z)) & ex L being LinearFunc ex R being RestFunc st
( r = L . 1 & ( for y being Real st y in N holds
((SVF1 (2,(pdiff1 (f,2)),z)) . y) - ((SVF1 (2,(pdiff1 (f,2)),z)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) ) ) & ex x0, y0 being Real st
( z = <*x0,y0*> & ex N being Neighbourhood of y0 st
( N c= dom (SVF1 (2,(pdiff1 (f,2)),z)) & ex L being LinearFunc ex R being RestFunc st
( s = L . 1 & ( for y being Real st y in N holds
((SVF1 (2,(pdiff1 (f,2)),z)) . y) - ((SVF1 (2,(pdiff1 (f,2)),z)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) ) ) implies r = s )
assume that
A8: ex x0, y0 being Real st
( z = <*x0,y0*> & ex N being Neighbourhood of y0 st
( N c= dom (SVF1 (2,(pdiff1 (f,2)),z)) & ex L being LinearFunc ex R being RestFunc st
( r = L . 1 & ( for y being Real st y in N holds
((SVF1 (2,(pdiff1 (f,2)),z)) . y) - ((SVF1 (2,(pdiff1 (f,2)),z)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) ) ) and
A9: ex x0, y0 being Real st
( z = <*x0,y0*> & ex N being Neighbourhood of y0 st
( N c= dom (SVF1 (2,(pdiff1 (f,2)),z)) & ex L being LinearFunc ex R being RestFunc st
( s = L . 1 & ( for y being Real st y in N holds
((SVF1 (2,(pdiff1 (f,2)),z)) . y) - ((SVF1 (2,(pdiff1 (f,2)),z)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) ) ) ; :: thesis: r = s
consider x1, y1 being Real such that
A10: z = <*x1,y1*> and
A11: ex N being Neighbourhood of y1 st
( N c= dom (SVF1 (2,(pdiff1 (f,2)),z)) & ex L being LinearFunc ex R being RestFunc st
( s = L . 1 & ( for y being Real st y in N holds
((SVF1 (2,(pdiff1 (f,2)),z)) . y) - ((SVF1 (2,(pdiff1 (f,2)),z)) . y1) = (L . (y - y1)) + (R . (y - y1)) ) ) ) by A9;
consider N1 being Neighbourhood of y1 such that
N1 c= dom (SVF1 (2,(pdiff1 (f,2)),z)) and
A12: ex L being LinearFunc ex R being RestFunc st
( s = L . 1 & ( for y being Real st y in N1 holds
((SVF1 (2,(pdiff1 (f,2)),z)) . y) - ((SVF1 (2,(pdiff1 (f,2)),z)) . y1) = (L . (y - y1)) + (R . (y - y1)) ) ) by A11;
consider L1 being LinearFunc, R1 being RestFunc such that
A13: s = L1 . 1 and
A14: for y being Real st y in N1 holds
((SVF1 (2,(pdiff1 (f,2)),z)) . y) - ((SVF1 (2,(pdiff1 (f,2)),z)) . y1) = (L1 . (y - y1)) + (R1 . (y - y1)) 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 y0 st
( N c= dom (SVF1 (2,(pdiff1 (f,2)),z)) & ex L being LinearFunc ex R being RestFunc st
( r = L . 1 & ( for y being Real st y in N holds
((SVF1 (2,(pdiff1 (f,2)),z)) . y) - ((SVF1 (2,(pdiff1 (f,2)),z)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) ) by A8;
consider N being Neighbourhood of y0 such that
N c= dom (SVF1 (2,(pdiff1 (f,2)),z)) and
A19: ex L being LinearFunc ex R being RestFunc st
( r = L . 1 & ( for y being Real st y in N holds
((SVF1 (2,(pdiff1 (f,2)),z)) . y) - ((SVF1 (2,(pdiff1 (f,2)),z)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) by A18;
consider L being LinearFunc, R being RestFunc such that
A20: r = L . 1 and
A21: for y being Real st y in N holds
((SVF1 (2,(pdiff1 (f,2)),z)) . y) - ((SVF1 (2,(pdiff1 (f,2)),z)) . y0) = (L . (y - y0)) + (R . (y - y0)) by A19;
consider r1 being Real such that
A22: for p being Real holds L . p = r1 * p by FDIFF_1:def 3;
A23: y0 = y1 by A17, A10, FINSEQ_1:77;
then consider N0 being Neighbourhood of y0 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 = ].(y0 - g),(y0 + 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 y being Real st
( y in N & y in N1 & h . n = y - y0 )
proof
let n be Element of NAT ; :: thesis: ex y being Real st
( y in N & y in N1 & h . n = y - y0 )
take y0 + (g / (n + 2)) ; :: thesis: ( y0 + (g / (n + 2)) in N & y0 + (g / (n + 2)) in N1 & h . n = (y0 + (g / (n + 2))) - y0 )
0 + 1 < (n + 1) + 1 by XREAL_1:6;
then g / (n + 2) < g / 1 by A25, XREAL_1:76;
then A30: y0 + (g / (n + 2)) < y0 + g by XREAL_1:6;
g / (n + 2) > 0 by A25, XREAL_1:139;
then y0 + (- g) < y0 + (g / (n + 2)) by A25, XREAL_1:6;
then y0 + (g / (n + 2)) in ].(y0 - g),(y0 + g).[ by A30;
hence ( y0 + (g / (n + 2)) in N & y0 + (g / (n + 2)) in N1 & h . n = (y0 + (g / (n + 2))) - y0 ) by A24, A26, A27; :: thesis: verum
end;
A31: r = r1 * 1 by A20, A22;
A32: now :: thesis: for y being Real st y in N & y in N1 holds
(r * (y - y0)) + (R . (y - y0)) = (s * (y - y0)) + (R1 . (y - y0))
let y be Real; :: thesis: ( y in N & y in N1 implies (r * (y - y0)) + (R . (y - y0)) = (s * (y - y0)) + (R1 . (y - y0)) )
assume that
A33: y in N and
A34: y in N1 ; :: thesis: (r * (y - y0)) + (R . (y - y0)) = (s * (y - y0)) + (R1 . (y - y0))
((SVF1 (2,(pdiff1 (f,2)),z)) . y) - ((SVF1 (2,(pdiff1 (f,2)),z)) . y0) = (L . (y - y0)) + (R . (y - y0)) by A21, A33;
then (L . (y - y0)) + (R . (y - y0)) = (L1 . (y - y0)) + (R1 . (y - y0)) by A14, A23, A34;
then (r1 * (y - y0)) + (R . (y - y0)) = (L1 . (y - y0)) + (R1 . (y - y0)) by A22;
hence (r * (y - y0)) + (R . (y - y0)) = (s * (y - y0)) + (R1 . (y - y0)) by A15, A31, A16; :: thesis: verum
end;
A35: r - s in REAL by XREAL_0:def 1;
now :: thesis: for n being Nat holds r - s = (((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h))) . n
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 y being Real st
( y in N & y in N1 & h . n = y - y0 ) 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