let Y be RealNormSpace; :: thesis:
let I be Function of REAL,(REAL-NS 1); :: thesis:
let x0 be Point of (REAL-NS 1); :: thesis:
let y0 be Real; :: thesis:
let g be PartFunc of REAL,Y; :: thesis:
let f be PartFunc of (REAL-NS 1),Y; :: 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;
thus ( f is_differentiable_in x0 implies g is_differentiable_in y0 ) by A2, A3, A4, A5, FTh42, A1; :: thesis:
assume g is_differentiable_in y0 ; :: thesis:
then consider N0 being Neighbourhood of y0 such that
A6: N0 c= dom (f * I) and
A7: ex L being LinearFunc of Y ex R being RestFunc of Y st
for y being Real st y in N0 holds
((f * I) /. y) - ((f * I) /. y0) = (L /. (y - y0)) + (R /. (y - y0)) by A5;
reconsider y0 = y0 as Element of REAL by XREAL_0:def 1;
A8: I * J = id (REAL-NS 1) by Lm1, A1, Lm2, FUNCT_1:39;
A9: g * J = f * (id (REAL-NS 1)) by A5, A8, RELAT_1:36
.= f by FUNCT_2:17 ;
consider e0 being Real such that
A10: 0 < e0 and
A11: N0 = ].(y0 - e0),(y0 + e0).[ by RCOMP_1:def 6;
reconsider e = e0 as Element of REAL by XREAL_0:def 1;
set N = { z where z is Point of (REAL-NS 1) : ||.(z - x0).|| < e } ;
consider L being LinearFunc of Y, R being RestFunc of Y such that
A12: for y1 being Real st y1 in N0 holds
((f * I) /. y1) - ((f * I) /. y0) = (L /. (y1 - y0)) + (R /. (y1 - y0)) by A7;
reconsider R0 = R * J as RestFunc of (REAL-NS 1),Y by FTh41;
reconsider L0 = L * J as Lipschitzian LinearOperator of (REAL-NS 1),Y by FTh41;

now :: thesis:

let z be object ; :: thesis:

hereby :: thesis:

assume z in { z where z is Point of (REAL-NS 1) : ||.(z - x0).|| < e } ; :: thesis:
then consider w being Point of (REAL-NS 1) such that
A13: z = w and
A14: ||.(w - x0).|| < e ;
reconsider y2 = J . w as Element of REAL ;
J . x0 = y0 by A4, Lm2;
then J . (w - x0) = y2 - y0 by PDIFF_1:4;
then |.(y2 - y0).| < e by A14, PDIFF_1:4;
then A15: y2 in N0 by A11, RCOMP_1:1;
w in the carrier of (REAL-NS 1) ;
then w in dom J by Lm2, REAL_NS1:def 4;
then w = I . y2 by A1, FUNCT_1:34;
hence z in I .: N0 by A13, A15, FUNCT_2:35; :: thesis:

end;

assume z in I .: N0 ; :: thesis:
then consider yy being object such that
A16: yy in REAL and
A17: yy in N0 and
A18: z = I . yy by FUNCT_2:64;
reconsider y3 = yy as Element of REAL by A16;
set w = I . y3;
I . y0 = x0 by A4, A1, Lm2;
then A19: (I . y3) - x0 = I . (y3 - y0) by A1, Lm1, PDIFF_1:3;
|.(y3 - y0).| < e by A11, A17, RCOMP_1:1;
then ||.((I . y3) - x0).|| < e by A19, A1, Lm1, PDIFF_1:3;
hence z in { z where z is Point of (REAL-NS 1) : ||.(z - x0).|| < e } by A18; :: thesis:

end;

then A20: { z where z is Point of (REAL-NS 1) : ||.(z - x0).|| < e } = I .: N0 by TARSKI:2;
I .: (dom g) = I .: (I " (dom f)) by A5, RELAT_1:147;
then A211: I .: (dom g) = dom f by A1, Lm1, FUNCT_1:77, PDIFF_1:2;
then A21: { z where z is Point of (REAL-NS 1) : ||.(z - x0).|| < e } c= dom f by A5, A6, A20, RELAT_1:123;
{ z where z is Point of (REAL-NS 1) : ||.(z - x0).|| < e } c= the carrier of (REAL-NS 1)

proof

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 A22: { z where z is Point of (REAL-NS 1) : ||.(z - x0).|| < e } is Neighbourhood of x0 by A10, NFCONT_1:def 1;
J * I = id REAL by A1, Lm2, FUNCT_1:39;
then (J * I) .: N0 = N0 by FRECHET:13;
then A23: J .: { z where z is Point of (REAL-NS 1) : ||.(z - x0).|| < e } = N0 by A20, RELAT_1:126;
A24: for y being Point of (REAL-NS 1) st y in { z where z is Point of (REAL-NS 1) : ||.(z - x0).|| < e } holds
(f /. y) - (f /. x0) = (L0 . (y - x0)) + (R0 /. (y - x0))

proof

x0 in the carrier of (REAL-NS 1) ;
then x0 in dom J by Lm2, REAL_NS1:def 4;
then g . (J . x0) = f . x0 by A9, FUNCT_1:13;
then A25: g . (J . x0) = f /. x0 by A2, PARTFUN1:def 6;
A26: J . x0 = y0 by A4, Lm2;
let y be Point of (REAL-NS 1); :: thesis:
assume A27: y in { z where z is Point of (REAL-NS 1) : ||.(z - x0).|| < e } ; :: thesis:
set y3 = J . y;
reconsider p1 = g /. (J . y), p2 = g /. y0, q1 = L /. ((J . y) - y0), q2 = R /. ((J . y) - y0) as VECTOR of Y ;
A28: J . x0 = y0 by A4, Lm2;
(g /. (J . y)) - (g /. y0) = q1 + q2 by A5, A12, A23, A27, FUNCT_2:35;
then (g /. (J . y)) - (g /. (J . x0)) = (L /. (J . (y - x0))) + (R /. ((J . y) - y0)) by A28, PDIFF_1:4;
then A29: (g /. (J . y)) - (g /. (J . x0)) = (L /. (J . (y - x0))) + (R /. (J . (y - x0))) by A28, PDIFF_1:4;
A30: dom L0 = the carrier of (REAL-NS 1) by FUNCT_2:def 1;
y - x0 in the carrier of (REAL-NS 1) ;
then y - x0 in dom J by Lm2, REAL_NS1:def 4;
then A31: R . (J . (y - x0)) = (R * J) . (y - x0) by FUNCT_1:13;
R0 is total by NDIFF_1:def 5;
then A32: dom (R * J) = the carrier of (REAL-NS 1) by PARTFUN1:def 2;
A33: R . (J . (y - x0)) = R0 /. (y - x0) by A31, A32, PARTFUN1:def 6;
J . (y - x0) in dom R by A32, FUNCT_1:11;
then A34: R /. (J . (y - x0)) = R0 /. (y - x0) by A33, PARTFUN1:def 6;
y in the carrier of (REAL-NS 1) ;
then A35: y in dom J by Lm2, REAL_NS1:def 4;
g . (J . y) = f . y by A9, A35, FUNCT_1:13;
then A36: g . (J . y) = f /. y by A21, A27, PARTFUN1:def 6;
J . y in dom g by A21, A27, A9, FUNCT_1:11;
then g /. (J . y) = f /. y by A36, PARTFUN1:def 6;
then (g /. (J . y)) - (g /. (J . x0)) = (f /. y) - (f /. x0) by A3, A26, A25, PARTFUN1:def 6;
hence (f /. y) - (f /. x0) = (L0 . (y - x0)) + (R0 /. (y - x0)) by A29, A34, A30, FUNCT_1:12; :: thesis:

end;

reconsider L0 = L0 as Point of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),Y)) by LOPBAN_1:def 9;
for x being Point of (REAL-NS 1) st x in { z where z is Point of (REAL-NS 1) : ||.(z - x0).|| < e } holds
(f /. x) - (f /. x0) = (L0 . (x - x0)) + (R0 /. (x - x0)) by A24;
hence f is_differentiable_in x0 by A22, A211, A5, A6, A20, RELAT_1:123; :: thesis: