let n be non zero Element of NAT ; :: thesis:
let I be Function of REAL,(REAL-NS 1); :: thesis:
let x0 be Point of (REAL-NS 1); :: thesis:
let y0 be Element of REAL ; :: thesis:
let g be PartFunc of REAL,(REAL-NS n); :: thesis:
let f be PartFunc of (REAL-NS 1),(REAL-NS n); :: thesis:
assume that
A1: I = (proj (1,1)) " and
A2: x0 in dom f and
A3: y0 in dom g and
A4: x0 = <*y0*> and
A5: f * I = g ; :: thesis:
reconsider J = proj (1,1) as Function of (REAL-NS 1),REAL by Lm1;
assume A6: f is_differentiable_in x0 ; :: thesis:
then consider NN being Neighbourhood of x0 such that
A7: NN c= dom f and
A8: ex L being Point of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS n))) ex R being RestFunc of (REAL-NS 1),(REAL-NS n) st
for x being Point of (REAL-NS 1) st x in NN holds
(f /. x) - (f /. x0) = (L . (x - x0)) + (R /. (x - x0)) by NDIFF_1:def 6;
A9: I * J = id (REAL-NS 1) by Lm1, A1, Lm2, FUNCT_1:39;
A10: g * J = f * (id (REAL-NS 1)) by A9, A5, RELAT_1:36
.= f by FUNCT_2:17 ;
consider e being Real such that
A11: 0 < e and
A12: { z where z is Point of (REAL-NS 1) : ||.(z - x0).|| < e } c= NN by NFCONT_1:def 1;
consider L being Point of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS n))), R being RestFunc of (REAL-NS 1),(REAL-NS n) such that
A13: for x9 being Point of (REAL-NS 1) st x9 in NN holds
(f /. x9) - (f /. x0) = (L . (x9 - x0)) + (R /. (x9 - x0)) by A8;
reconsider R0 = R * I as RestFunc of (REAL-NS n) by A1, Th40;
L is LinearOperator of (REAL-NS 1),(REAL-NS n) by LOPBAN_1:def 9;
then reconsider L0 = L * I as LinearFunc of (REAL-NS n) by A1, Th40;
set N = { z where z is Point of (REAL-NS 1) : ||.(z - x0).|| < e } ;
A14: { z where z is Point of (REAL-NS 1) : ||.(z - x0).|| < e } c= the carrier of (REAL-NS 1)

let y be object ; :: according to TARSKI:def 3 :: thesis:
assume y in { z where z is Point of (REAL-NS 1) : ||.(z - x0).|| < e } ; :: thesis:
then ex z being Point of (REAL-NS 1) st
( y = z & ||.(z - x0).|| < e ) ;
hence y in the carrier of (REAL-NS 1) ; :: thesis:

end;

then reconsider N = { z where z is Point of (REAL-NS 1) : ||.(z - x0).|| < e } as Neighbourhood of x0 by A11, NFCONT_1:def 1;
set N0 = { z where z is Element of REAL : |.(z - y0).| < e } ;
A15: N c= dom f by A7, A12;

now :: thesis:

let z be object ; :: thesis:

hereby :: thesis:

assume z in { z where z is Element of REAL : |.(z - y0).| < e } ; :: thesis:
then consider y9 being Element of REAL such that
A16: z = y9 and
A17: |.(y9 - y0).| < e ;
reconsider w = I . y9 as Point of (REAL-NS 1) ;
x0 = I . y0 by A1, A4, Lm2;
then w - x0 = I . (y9 - y0) by A1, Lm1, PDIFF_1:3;
then ||.(w - x0).|| = |.(y9 - y0).| by A1, Lm1, PDIFF_1:3;
then A18: w in { z0 where z0 is Point of (REAL-NS 1) : ||.(z0 - x0).|| < e } by A17;
J . (I . y9) = y9 by A1, Lm2, FUNCT_1:35;
hence z in J .: N by A18, A16, FUNCT_2:35; :: thesis:

end;

assume z in J .: N ; :: thesis:
then consider ww being object such that
ww in REAL 1 and
A19: ww in N and
A20: z = J . ww by FUNCT_2:64;
consider w being Point of (REAL-NS 1) such that
A21: ww = w and
A22: ||.(w - x0).|| < e by A19;
reconsider y9 = J . w as Element of REAL ;
J . x0 = y0 by A4, Lm2;
then J . (w - x0) = y9 - y0 by PDIFF_1:4;
then |.(y9 - y0).| < e by A22, PDIFF_1:4;
hence z in { z where z is Element of REAL : |.(z - y0).| < e } by A20, A21; :: thesis:

end;

then A23: { z where z is Element of REAL : |.(z - y0).| < e } = J .: N by TARSKI:2;
J .: (dom f) = J .: (J " (dom g)) by A10, RELAT_1:147;
then A24: J .: (dom f) = dom (f * I) by A5, Lm2, FUNCT_1:77;
A25: I * J = id (REAL 1) by A1, Lm2, FUNCT_1:39;
N c= dom f by A7, A12;
then A26: { z where z is Element of REAL : |.(z - y0).| < e } c= dom (f * I) by A24, A23, RELAT_1:123;
A27: ].(y0 - e),(y0 + e).[ c= { z where z is Element of REAL : |.(z - y0).| < e }

proof

now :: thesis:

let d be object ; :: thesis:
assume A28: d in ].(y0 - e),(y0 + e).[ ; :: thesis:
reconsider y1 = d as Element of REAL by A28;
|.(y1 - y0).| < e by A28, RCOMP_1:1;
hence d in { z where z is Element of REAL : |.(z - y0).| < e } ; :: thesis:

end;

hence ].(y0 - e),(y0 + e).[ c= { z where z is Element of REAL : |.(z - y0).| < e } ; :: thesis:

end;

{ z where z is Element of REAL : |.(z - y0).| < e } c= ].(y0 - e),(y0 + e).[

proof

let d be object ; :: according to TARSKI:def 3 :: thesis:
assume d in { z where z is Element of REAL : |.(z - y0).| < e } ; :: thesis:
then ex r being Element of REAL st
( d = r & |.(r - y0).| < e ) ;
hence d in ].(y0 - e),(y0 + e).[ by RCOMP_1:1; :: thesis:

end;

then { z where z is Element of REAL : |.(z - y0).| < e } = ].(y0 - e),(y0 + e).[ by A27, XBOOLE_0:def 10;
then A29: { z where z is Element of REAL : |.(z - y0).| < e } is Neighbourhood of y0 by A11, RCOMP_1:def 6;
N c= REAL 1 by A14, REAL_NS1:def 4;
then (I * J) .: N = N by A25, FRECHET:13;
then A30: I .: { z where z is Element of REAL : |.(z - y0).| < e } = N by A23, RELAT_1:126;
A31: for y1 being Real st y1 in { z where z is Element of REAL : |.(z - y0).| < e } holds
((f * I) /. y1) - ((f * I) /. y0) = (L0 /. (y1 - y0)) + (R0 /. (y1 - y0))

proof

let y1 be Real; :: thesis:
reconsider yy = y1 as Element of REAL by XREAL_0:def 1;
reconsider y9 = I . yy as Point of (REAL-NS 1) ;
R is total by NDIFF_1:def 5;
then A32: dom R = the carrier of (REAL-NS 1) by PARTFUN1:def 2;
R0 is total by NDIFF_3:def 1;
then A33: dom (R * I) = REAL by PARTFUN1:def 2;
reconsider dy1y0 = y1 - y0 as Element of REAL by XREAL_0:def 1;
I . dy1y0 in dom R by A32;
then R /. (I . (y1 - y0)) = R . (I . (y1 - y0)) by PARTFUN1:def 6;
then R /. (I . dy1y0) = R0 . dy1y0 by A33, FUNCT_1:12;
then A34: R /. (I . (y1 - y0)) = R0 /. dy1y0 by A33, PARTFUN1:def 6;
L0 is total ;
then A35: dom (L * I) = REAL by PARTFUN1:def 2;
assume A36: y1 in { z where z is Element of REAL : |.(z - y0).| < e } ; :: thesis:
then A37: I . yy in N by A30, FUNCT_2:35;
then A38: f /. (I . y1) = f . (I . y1) by A15, PARTFUN1:def 6;
f /. (I . y1) = (f * I) . y1 by A38, A26, A36, FUNCT_1:12;
then A39: f /. (I . y1) = (f * I) /. y1 by A26, A36, PARTFUN1:def 6;
A40: x0 = I . y0 by A4, A1, Lm2;
then f /. (I . y0) = f . (I . y0) by A2, PARTFUN1:def 6;
then f /. (I . y0) = (f * I) . y0 by A3, A5, FUNCT_1:12;
then A41: f /. (I . y0) = (f * I) /. y0 by A3, A5, PARTFUN1:def 6;
reconsider d = y1 - y0 as Element of REAL by XREAL_0:def 1;
(f /. y9) - (f /. x0) = (L . (y9 - x0)) + (R /. (y9 - x0)) by A13, A12, A37;
then A42: (f /. (I . y1)) - (f /. (I . y0)) = (L . (I . d)) + (R /. (y9 - x0)) by A1, A40, Lm1, PDIFF_1:3;
L . (I . d) = L0 /. dy1y0 by A35, FUNCT_1:12;
hence ((f * I) /. y1) - ((f * I) /. y0) = (L0 /. (y1 - y0)) + (R0 /. (y1 - y0)) by A40, A42, A34, A39, A41, A1, Lm1, PDIFF_1:3; :: thesis:

end;

then A43: f * I is_differentiable_in y0 by A29, A26, NDIFF_3:def 3;
thus g is_differentiable_in y0 by A5, A31, A29, A26, NDIFF_3:def 3; :: thesis:
A44: diff ((f * I),y0) = L0 /. jj by A31, A43, A29, A26, NDIFF_3:def 4;
thus A45: diff (g,y0) = ((diff (f,x0)) * I) . 1 by A5, A44, A6, A13, A7, NDIFF_1:def 7
.= (diff (f,x0)) . (I . jj) by A1, FUNCT_1:13, PDIFF_1:2
.= (diff (f,x0)) . <*1*> by A1, Lm2 ; :: thesis:
let r be Element of REAL ; :: thesis:
A46: <*jj*> is Element of REAL 1 by FINSEQ_2:98;
then reconsider x = <*1*> as Point of (REAL-NS 1) by REAL_NS1:def 4;
A47: diff (f,x0) is LinearOperator of (REAL-NS 1),(REAL-NS n) by LOPBAN_1:def 9;
thus (diff (f,x0)) . <*r*> = (diff (f,x0)) . <*(r * 1)*>
.= (diff (f,x0)) . (r * <*1*>) by RVSUM_1:47
.= (diff (f,x0)) . (r * x) by A46, REAL_NS1:3
.= r * (diff (g,y0)) by A45, A47, LOPBAN_1:def 5 ; :: thesis: