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 g, h being continuous PartFunc of REAL, the carrier of X st g = (Fredholm (G,a,b,y0)) . u & h = (Fredholm (G,a,b,y0)) . v holds
for t being Real st t in ['a,b'] holds
||.((g /. t) - (h /. t)).|| <= (r * (t - a)) * ||.(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 g, h being continuous PartFunc of REAL, the carrier of X st g = (Fredholm (G,a,b,y0)) . u & h = (Fredholm (G,a,b,y0)) . v holds
for t being Real st t in ['a,b'] holds
||.((g /. t) - (h /. t)).|| <= (r * (t - a)) * ||.(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 g, h being continuous PartFunc of REAL, the carrier of X st g = (Fredholm (G,a,b,y0)) . u & h = (Fredholm (G,a,b,y0)) . v holds
for t being Real st t in ['a,b'] holds
||.((g /. t) - (h /. t)).|| <= (r * (t - a)) * ||.(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 g, h being continuous PartFunc of REAL, the carrier of X st g = (Fredholm (G,a,b,y0)) . u & h = (Fredholm (G,a,b,y0)) . v holds
for t being Real st t in ['a,b'] holds
||.((g /. t) - (h /. t)).|| <= (r * (t - a)) * ||.(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 g, h being continuous PartFunc of REAL, the carrier of X st g = (Fredholm (G,a,b,y0)) . u & h = (Fredholm (G,a,b,y0)) . v holds
for t being Real st t in ['a,b'] holds
||.((g /. t) - (h /. t)).|| <= (r * (t - a)) * ||.(u - v).||

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 u, v be VECTOR of (); :: thesis: for g, h being continuous PartFunc of REAL, the carrier of X st g = (Fredholm (G,a,b,y0)) . u & h = (Fredholm (G,a,b,y0)) . v holds
for t being Real st t in ['a,b'] holds
||.((g /. t) - (h /. t)).|| <= (r * (t - a)) * ||.(u - v).||

let g, h be continuous PartFunc of REAL, the carrier of X; :: thesis: ( g = (Fredholm (G,a,b,y0)) . u & h = (Fredholm (G,a,b,y0)) . v implies for t being Real st t in ['a,b'] holds
||.((g /. t) - (h /. t)).|| <= (r * (t - a)) * ||.(u - v).|| )

assume A4: ( g = (Fredholm (G,a,b,y0)) . u & h = (Fredholm (G,a,b,y0)) . v ) ; :: thesis: for t being Real st t in ['a,b'] holds
||.((g /. t) - (h /. t)).|| <= (r * (t - a)) * ||.(u - v).||

set F = Fredholm (G,a,b,y0);
consider f1, g1, Gf1 being continuous PartFunc of REAL, the carrier of X such that
A5: ( 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
A6: ( 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 A8: ( dom Gf1 = ['a,b'] & dom Gf2 = ['a,b'] ) by ;
reconsider Gf12 = Gf1 - Gf2 as continuous PartFunc of REAL, the carrier of X ;
A10: ['a,b'] = [.a,b.] by ;
then A18: a in ['a,b'] by A1;
let t be Real; :: thesis: ( t in ['a,b'] implies ||.((g /. t) - (h /. t)).|| <= (r * (t - a)) * ||.(u - v).|| )
assume A11: t in ['a,b'] ; :: thesis: ||.((g /. t) - (h /. t)).|| <= (r * (t - a)) * ||.(u - v).||
then A12: ex g being Real st
( t = g & a <= g & g <= b ) by A10;
then A20: ['a,t'] c= ['a,b'] by INTEGR19:1;
X1: ['a,t'] = [.a,t.] by ;
A13: dom Gf12 = (dom Gf1) /\ (dom Gf2) by VFUNCT_1:def 2;
for x being Real st x in ['a,t'] holds
||.(Gf12 /. x).|| <= r * ||.(u - v).||
proof
let x be Real; :: thesis: ( x in ['a,t'] implies ||.(Gf12 /. x).|| <= r * ||.(u - v).|| )
assume A19: x in ['a,t'] ; :: thesis: ||.(Gf12 /. x).|| <= r * ||.(u - v).||
then A21: Gf12 /. x = (Gf1 /. x) - (Gf2 /. x) by ;
A24: r * ||.((f1 /. x) - (f2 /. x)).|| <= r * ||.(u - v).|| by ;
A22: 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 ||.((Gf1 /. x) - (Gf2 /. x)).|| <= r * ||.((f1 /. x) - (f2 /. x)).|| by ;
hence ||.(Gf12 /. x).|| <= r * ||.(u - v).|| by ; :: thesis: verum
end;
then A25: ||.(integral (Gf12,a,t)).|| <= (r * ||.(u - v).||) * (t - a) by Lm14a, X1, A8, A13, A18, A10, A11, A12;
( g /. t = y0 + (integral (Gf1,a,t)) & h /. t = y0 + (integral (Gf2,a,t)) ) by A4, A5, A6, A11;
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)) * ||.(u - v).|| by A25; :: thesis: verum