let X be RealBanachSpace; :: thesis: for a, b, r being Real
for y0 being VECTOR of X
for G being Function of X,X st a <= b & 0 < r & ( for y1, y2 being VECTOR of X holds ||.((G /. y1) - (G /. y2)).|| <= r * ||.(y1 - y2).|| ) holds
for u, v being VECTOR of ()
for m being Element of NAT
for g, h being continuous PartFunc of REAL, the carrier of X st g = (iter ((Fredholm (G,a,b,y0)),(m + 1))) . u & h = (iter ((Fredholm (G,a,b,y0)),(m + 1))) . v holds
for t being Real st t in ['a,b'] holds
||.((g /. t) - (h /. t)).|| <= (((r * (t - a)) |^ (m + 1)) / ((m + 1) !)) * ||.(u - v).||

let a, b, r be Real; :: thesis: for y0 being VECTOR of X
for G being Function of X,X st a <= b & 0 < r & ( for y1, y2 being VECTOR of X holds ||.((G /. y1) - (G /. y2)).|| <= r * ||.(y1 - y2).|| ) holds
for u, v being VECTOR of ()
for m being Element of NAT
for g, h being continuous PartFunc of REAL, the carrier of X st g = (iter ((Fredholm (G,a,b,y0)),(m + 1))) . u & h = (iter ((Fredholm (G,a,b,y0)),(m + 1))) . v holds
for t being Real st t in ['a,b'] holds
||.((g /. t) - (h /. t)).|| <= (((r * (t - a)) |^ (m + 1)) / ((m + 1) !)) * ||.(u - v).||

let y0 be VECTOR of X; :: thesis: for G being Function of X,X st a <= b & 0 < r & ( for y1, y2 being VECTOR of X holds ||.((G /. y1) - (G /. y2)).|| <= r * ||.(y1 - y2).|| ) holds
for u, v being VECTOR of ()
for m being Element of NAT
for g, h being continuous PartFunc of REAL, the carrier of X st g = (iter ((Fredholm (G,a,b,y0)),(m + 1))) . u & h = (iter ((Fredholm (G,a,b,y0)),(m + 1))) . v holds
for t being Real st t in ['a,b'] holds
||.((g /. t) - (h /. t)).|| <= (((r * (t - a)) |^ (m + 1)) / ((m + 1) !)) * ||.(u - v).||

let G be Function of X,X; :: thesis: ( a <= b & 0 < r & ( for y1, y2 being VECTOR of X holds ||.((G /. y1) - (G /. y2)).|| <= r * ||.(y1 - y2).|| ) implies for u, v being VECTOR of ()
for m being Element of NAT
for g, h being continuous PartFunc of REAL, the carrier of X st g = (iter ((Fredholm (G,a,b,y0)),(m + 1))) . u & h = (iter ((Fredholm (G,a,b,y0)),(m + 1))) . v holds
for t being Real st t in ['a,b'] holds
||.((g /. t) - (h /. t)).|| <= (((r * (t - a)) |^ (m + 1)) / ((m + 1) !)) * ||.(u - v).|| )

assume A1: ( a <= b & 0 < r & ( for y1, y2 being VECTOR of X holds ||.((G /. y1) - (G /. y2)).|| <= r * ||.(y1 - y2).|| ) ) ; :: thesis: for u, v being VECTOR of ()
for m being Element of NAT
for g, h being continuous PartFunc of REAL, the carrier of X st g = (iter ((Fredholm (G,a,b,y0)),(m + 1))) . u & h = (iter ((Fredholm (G,a,b,y0)),(m + 1))) . v holds
for t being Real st t in ['a,b'] holds
||.((g /. t) - (h /. t)).|| <= (((r * (t - a)) |^ (m + 1)) / ((m + 1) !)) * ||.(u - v).||

set F = Fredholm (G,a,b,y0);
A2: dom G = the carrier of X by FUNCT_2:def 1;
for x1, x2 being Point of X st x1 in the carrier of X & x2 in the carrier of X holds
||.((G /. x1) - (G /. x2)).|| <= r * ||.(x1 - x2).|| by A1;
then G is_Lipschitzian_on the carrier of X by ;
then A3: G is_continuous_on dom G by ;
let u1, v1 be VECTOR of (); :: thesis: for m being Element of NAT
for g, h being continuous PartFunc of REAL, the carrier of X st g = (iter ((Fredholm (G,a,b,y0)),(m + 1))) . u1 & h = (iter ((Fredholm (G,a,b,y0)),(m + 1))) . v1 holds
for t being Real st t in ['a,b'] holds
||.((g /. t) - (h /. t)).|| <= (((r * (t - a)) |^ (m + 1)) / ((m + 1) !)) * ||.(u1 - v1).||

defpred S1[ Nat] means for g, h being continuous PartFunc of REAL, the carrier of X st g = (iter ((Fredholm (G,a,b,y0)),(\$1 + 1))) . u1 & h = (iter ((Fredholm (G,a,b,y0)),(\$1 + 1))) . v1 holds
for t being Real st t in ['a,b'] holds
||.((g /. t) - (h /. t)).|| <= (((r * (t - a)) |^ (\$1 + 1)) / ((\$1 + 1) !)) * ||.(u1 - v1).||;
reconsider Z0 = 0 as Element of NAT ;
A4: S1[ 0 ]
proof
let g, h be continuous PartFunc of REAL, the carrier of X; :: thesis: ( g = (iter ((Fredholm (G,a,b,y0)),(0 + 1))) . u1 & h = (iter ((Fredholm (G,a,b,y0)),(0 + 1))) . v1 implies for t being Real st t in ['a,b'] holds
||.((g /. t) - (h /. t)).|| <= (((r * (t - a)) |^ (0 + 1)) / ((0 + 1) !)) * ||.(u1 - v1).|| )

assume ( g = (iter ((Fredholm (G,a,b,y0)),(0 + 1))) . u1 & h = (iter ((Fredholm (G,a,b,y0)),(0 + 1))) . v1 ) ; :: thesis: for t being Real st t in ['a,b'] holds
||.((g /. t) - (h /. t)).|| <= (((r * (t - a)) |^ (0 + 1)) / ((0 + 1) !)) * ||.(u1 - v1).||

then A6: ( g = (Fredholm (G,a,b,y0)) . u1 & h = (Fredholm (G,a,b,y0)) . v1 ) by FUNCT_7:70;
for t being Real st t in ['a,b'] holds
||.((g /. t) - (h /. t)).|| <= (((r * (t - a)) |^ (Z0 + 1)) / ((Z0 + 1) !)) * ||.(u1 - v1).|| by ;
hence for t being Real st t in ['a,b'] holds
||.((g /. t) - (h /. t)).|| <= (((r * (t - a)) |^ (0 + 1)) / ((0 + 1) !)) * ||.(u1 - v1).|| ; :: thesis: verum
end;
A8: for k being Nat st S1[k] holds
S1[k + 1]
proof
let k be Nat; :: thesis: ( S1[k] implies S1[k + 1] )
assume A9: S1[k] ; :: thesis: S1[k + 1]
let g, h be continuous PartFunc of REAL, the carrier of X; :: thesis: ( g = (iter ((Fredholm (G,a,b,y0)),((k + 1) + 1))) . u1 & h = (iter ((Fredholm (G,a,b,y0)),((k + 1) + 1))) . v1 implies for t being Real st t in ['a,b'] holds
||.((g /. t) - (h /. t)).|| <= (((r * (t - a)) |^ ((k + 1) + 1)) / (((k + 1) + 1) !)) * ||.(u1 - v1).|| )

assume A10: ( g = (iter ((Fredholm (G,a,b,y0)),((k + 1) + 1))) . u1 & h = (iter ((Fredholm (G,a,b,y0)),((k + 1) + 1))) . v1 ) ; :: thesis: for t being Real st t in ['a,b'] holds
||.((g /. t) - (h /. t)).|| <= (((r * (t - a)) |^ ((k + 1) + 1)) / (((k + 1) + 1) !)) * ||.(u1 - v1).||

reconsider u = (iter ((Fredholm (G,a,b,y0)),(k + 1))) . u1, v = (iter ((Fredholm (G,a,b,y0)),(k + 1))) . v1 as VECTOR of () ;
A11: dom (iter ((Fredholm (G,a,b,y0)),(k + 1))) = the carrier of () by FUNCT_2:def 1;
A12: (iter ((Fredholm (G,a,b,y0)),((k + 1) + 1))) . u1 = ((Fredholm (G,a,b,y0)) * (iter ((Fredholm (G,a,b,y0)),(k + 1)))) . u1 by FUNCT_7:71
.= (Fredholm (G,a,b,y0)) . u by ;
A13: (iter ((Fredholm (G,a,b,y0)),((k + 1) + 1))) . v1 = ((Fredholm (G,a,b,y0)) * (iter ((Fredholm (G,a,b,y0)),(k + 1)))) . v1 by FUNCT_7:71
.= (Fredholm (G,a,b,y0)) . v by ;
consider f1, g1, Gf1 being continuous PartFunc of REAL, the carrier of X such that
A14: ( u = f1 & (Fredholm (G,a,b,y0)) . u = g1 & dom f1 = ['a,b'] & dom g1 = ['a,b'] & Gf1 = G * f1 & ( for t being Real st t in ['a,b'] holds
g1 /. t = y0 + (integral (Gf1,a,t)) ) ) by Def8, A1, A3;
consider f2, g2, Gf2 being continuous PartFunc of REAL, the carrier of X such that
A15: ( v = f2 & (Fredholm (G,a,b,y0)) . v = g2 & dom f2 = ['a,b'] & dom g2 = ['a,b'] & Gf2 = G * f2 & ( for t being Real st t in ['a,b'] holds
g2 /. t = y0 + (integral (Gf2,a,t)) ) ) by Def8, A1, A3;
set Gf12 = Gf1 - Gf2;
dom G = the carrier of X by FUNCT_2:def 1;
then ( rng f1 c= dom G & rng f2 c= dom G ) ;
then A18: ( dom Gf1 = ['a,b'] & dom Gf2 = ['a,b'] ) by ;
reconsider Gf12 = Gf1 - Gf2 as continuous PartFunc of REAL, the carrier of X ;
A20: ['a,b'] = [.a,b.] by ;
let t be Real; :: thesis: ( t in ['a,b'] implies ||.((g /. t) - (h /. t)).|| <= (((r * (t - a)) |^ ((k + 1) + 1)) / (((k + 1) + 1) !)) * ||.(u1 - v1).|| )
assume A21: t in ['a,b'] ; :: thesis: ||.((g /. t) - (h /. t)).|| <= (((r * (t - a)) |^ ((k + 1) + 1)) / (((k + 1) + 1) !)) * ||.(u1 - v1).||
then A22: ex g being Real st
( t = g & a <= g & g <= b ) by A20;
a22: ex g being Element of REAL st
( t = g & a <= g & g <= b )
proof
consider g being Real such that
H1: ( t = g & a <= g & g <= b ) by ;
reconsider g = g as Element of REAL by XREAL_0:def 1;
take g ; :: thesis: ( t = g & a <= g & g <= b )
thus ( t = g & a <= g & g <= b ) by H1; :: thesis: verum
end;
A23: dom Gf12 = (dom Gf1) /\ (dom Gf2) by VFUNCT_1:def 2
.= ['a,b'] by A18 ;
then A24: dom ||.Gf12.|| = ['a,b'] by NORMSP_0:def 2;
A27: a in ['a,b'] by ;
( min (a,t) = a & max (a,t) = t ) by ;
then A30: ( ||.Gf12.|| is_integrable_on ['a,t'] & ||.Gf12.|| | ['a,t'] is bounded & ||.(integral (Gf12,a,t)).|| <= integral (||.Gf12.||,a,t) ) by ;
['a,t'] = [.a,t.] by ;
then consider rg being PartFunc of REAL,REAL such that
A31: ( dom rg = ['a,t'] & ( for x being Real st x in ['a,t'] holds
rg . x = r * ((((r * (x - a)) |^ (k + 1)) / ((k + 1) !)) * ||.(u1 - v1).||) ) & rg is continuous & rg is_integrable_on ['a,t'] & rg | ['a,t'] is bounded & integral (rg,a,t) = (((r * (t - a)) |^ ((k + 1) + 1)) / (((k + 1) + 1) !)) * ||.(u1 - v1).|| ) by ;
A32: ['a,t'] c= ['a,b'] by ;
for x being Real st x in ['a,t'] holds
||.Gf12.|| . x <= rg . x
proof
let x be Real; :: thesis: ( x in ['a,t'] implies ||.Gf12.|| . x <= rg . x )
assume A33: x in ['a,t'] ; :: thesis: ||.Gf12.|| . x <= rg . x
then A34: Gf12 /. x = (Gf1 /. x) - (Gf2 /. x) by ;
A35: Gf1 /. x = Gf1 . x by
.= G . (f1 . x) by
.= G /. (f1 /. x) by ;
Gf2 /. x = Gf2 . x by
.= G . (f2 . x) by
.= G /. (f2 /. x) by ;
then A37: ||.((Gf1 /. x) - (Gf2 /. x)).|| <= r * ||.((f1 /. x) - (f2 /. x)).|| by ;
r * ||.((f1 /. x) - (f2 /. x)).|| <= r * ((((r * (x - a)) |^ (k + 1)) / ((k + 1) !)) * ||.(u1 - v1).||) by ;
then r * ||.((f1 /. x) - (f2 /. x)).|| <= rg . x by ;
then ||.((Gf1 /. x) - (Gf2 /. x)).|| <= rg . x by ;
hence ||.Gf12.|| . x <= rg . x by ; :: thesis: verum
end;
then A38: integral (||.Gf12.||,a,t) <= integral (rg,a,t) by ;
( g /. t = y0 + (integral (Gf1,a,t)) & h /. t = y0 + (integral (Gf2,a,t)) ) by A14, A15, A21, A12, A10, A13;
then (g /. t) - (h /. t) = ((y0 + (integral (Gf1,a,t))) - y0) - (integral (Gf2,a,t)) by RLVECT_1:27
.= ((integral (Gf1,a,t)) + (y0 - y0)) - (integral (Gf2,a,t)) by RLVECT_1:28
.= ((integral (Gf1,a,t)) + (0. X)) - (integral (Gf2,a,t)) by RLVECT_1:15
.= integral (Gf12,a,t) by ;
hence ||.((g /. t) - (h /. t)).|| <= (((r * (t - a)) |^ ((k + 1) + 1)) / (((k + 1) + 1) !)) * ||.(u1 - v1).|| by ; :: thesis: verum
end;
for k being Nat holds S1[k] from NAT_1:sch 2(A4, A8);
hence for m being Element of NAT
for g, h being continuous PartFunc of REAL, the carrier of X st g = (iter ((Fredholm (G,a,b,y0)),(m + 1))) . u1 & h = (iter ((Fredholm (G,a,b,y0)),(m + 1))) . v1 holds
for t being Real st t in ['a,b'] holds
||.((g /. t) - (h /. t)).|| <= (((r * (t - a)) |^ (m + 1)) / ((m + 1) !)) * ||.(u1 - v1).|| ; :: thesis: verum