:: Mazur-Ulam Theorem
:: by Artur Korni{\l}owicz
::
:: Received December 21, 2010
:: Copyright (c) 2010-2021 Association of Mizar Users

registration

cluster I[01] -> closed for SubSpace of R^1 ;

coherence
for b₁ being SubSpace of R^1 st b₁ = I[01] holds
b₁ is closed by TOPMETR:20, TREAL_1:2;

end;

reconsider D = DYADIC as Subset of I[01] by BORSUK_1:40, URYSOHN2:28;

Lm1: Cl D = [#] I[01]

proof end;

theorem :: MAZURULM:1

DYADIC is dense Subset of I[01]

proof end;

theorem Th2: :: MAZURULM:2

Cl DYADIC = [.0,1.]

proof end;

theorem Th3: :: MAZURULM:3

for E being RealNormSpace
for a being Point of E holds a + a = 2 * a

proof end;

theorem Th4: :: MAZURULM:4

for E being RealNormSpace
for a, b being Point of E holds (a + b) - b = a

proof end;

registration

let A be real-membered bounded_above set ;
let r be non negative Real;

cluster r ** A -> bounded_above ;

coherence
r ** A is bounded_above

proof end;

end;

registration

let A be real-membered bounded_above set ;
let r be non positive Real;

cluster r ** A -> bounded_below ;

coherence
r ** A is bounded_below

proof end;

end;

registration

let A be real-membered bounded_below set ;
let r be non negative Real;

cluster r ** A -> bounded_below ;

coherence
r ** A is bounded_below

proof end;

end;

registration

let A be non empty real-membered bounded_below set ;
let r be non positive Real;

cluster r ** A -> bounded_above ;

coherence
r ** A is bounded_above

proof end;

end;

theorem Th5: :: MAZURULM:5

for t being Real
for f being Real_Sequence holds f + (NAT --> t) = t + f

proof end;

theorem Th6: :: MAZURULM:6

for r being Element of REAL holds lim (NAT --> r) = r

proof end;

theorem Th7: :: MAZURULM:7

for t being Real
for f being convergent Real_Sequence holds lim (t + f) = t + (lim f)

proof end;

registration

let f be convergent Real_Sequence;
let t be Real;

cluster t + f -> convergent for Real_Sequence;

coherence
for b₁ being Real_Sequence st b₁ = t + f holds
b₁ is convergent

proof end;

end;

theorem Th8: :: MAZURULM:8

for E being RealNormSpace
for a being Point of E
for f being Real_Sequence holds f (#) (NAT --> a) = f * a

proof end;

theorem Th9: :: MAZURULM:9

for E being RealNormSpace
for a being Point of E holds lim (NAT --> a) = a

proof end;

theorem Th10: :: MAZURULM:10

for E being RealNormSpace
for a being Point of E
for f being convergent Real_Sequence holds lim (f * a) = (lim f) * a

proof end;

registration

let f be convergent Real_Sequence;
let E be RealNormSpace;
let a be Point of E;

cluster f * a -> convergent ;

coherence
f * a is convergent

proof end;

end;

definition

let E, F be non empty NORMSTR ;
let f be Function of E,F;

attr f is isometric means :Def1: :: MAZURULM:def 1
for a, b being Point of E holds ||.((f . a) - (f . b)).|| = ||.(a - b).||;

end;

:: deftheorem Def1 defines isometric MAZURULM:def 1 :
for E, F being non empty NORMSTR
for f being Function of E,F holds
( f is isometric iff for a, b being Point of E holds ||.((f . a) - (f . b)).|| = ||.(a - b).|| );

definition

let E, F be non empty RLSStruct ;
let f be Function of E,F;

attr f is Affine means :: MAZURULM:def 2
for a, b being Point of E
for t being Real st 0 <= t & t <= 1 holds
f . (((1 - t) * a) + (t * b)) = ((1 - t) * (f . a)) + (t * (f . b));

attr f is midpoints-preserving means :: MAZURULM:def 3
for a, b being Point of E holds f . ((1 / 2) * (a + b)) = (1 / 2) * ((f . a) + (f . b));

end;

:: deftheorem defines Affine MAZURULM:def 2 :
for E, F being non empty RLSStruct
for f being Function of E,F holds
( f is Affine iff for a, b being Point of E
for t being Real st 0 <= t & t <= 1 holds
f . (((1 - t) * a) + (t * b)) = ((1 - t) * (f . a)) + (t * (f . b)) );

:: deftheorem defines midpoints-preserving MAZURULM:def 3 :
for E, F being non empty RLSStruct
for f being Function of E,F holds
( f is midpoints-preserving iff for a, b being Point of E holds f . ((1 / 2) * (a + b)) = (1 / 2) * ((f . a) + (f . b)) );

registration

let E be non empty NORMSTR ;

cluster id E -> isometric ;

coherence
id E is isometric ;

end;

registration

let E be non empty RLSStruct ;

cluster id E -> Affine midpoints-preserving ;

coherence
( id E is midpoints-preserving & id E is Affine ) ;

end;

registration

let E be non empty NORMSTR ;

cluster non empty V4() V7( the carrier of E) V8( the carrier of E) Function-like total quasi_total bijective isometric Affine midpoints-preserving for Element of K10(K11( the carrier of E, the carrier of E));

existence
ex b₁ being UnOp of E st
( b₁ is bijective & b₁ is isometric & b₁ is midpoints-preserving & b₁ is Affine )

proof end;

end;

theorem Th11: :: MAZURULM:11

for E, F, G being RealNormSpace
for f being Function of E,F
for g being Function of F,G st f is isometric & g is isometric holds
g * f is isometric

proof end;

registration

let E be RealNormSpace;
let f, g be isometric UnOp of E;

cluster g * f -> isometric for UnOp of E;

coherence
for b₁ being UnOp of E st b₁ = g * f holds
b₁ is isometric by Th11;

end;

Lm2: now :: thesis: for E, F being RealNormSpace
for f being Function of E,F st f is bijective holds
for a being Point of F holds f . ((f /") . a) = a

let E, F be RealNormSpace; :: thesis: for f being Function of E,F st f is bijective holds
for a being Point of F holds f . ((f /") . a) = a
let f be Function of E,F; :: thesis: ( f is bijective implies for a being Point of F holds f . ((f /") . a) = a )
assume A1: f is bijective ; :: thesis: for a being Point of F holds f . ((f /") . a) = a
set g = f /" ;
let a be Point of F; :: thesis: f . ((f /") . a) = a
rng f = [#] F by A1, FUNCT_2:def 3;
then A2: (f /") /" = f by A1, TOPS_2:51;
A3: f /" = f " by A1, TOPS_2:def 4;
A4: f /" is bijective by A1, PENCIL_2:12;
f = (f /") " by A2, A4, TOPS_2:def 4;
hence f . ((f /") . a) = a by A1, A3, FUNCT_2:26; :: thesis: verum

end;

theorem Th12: :: MAZURULM:12

for E, F being RealNormSpace
for f being Function of E,F st f is bijective & f is isometric holds
f /" is isometric

proof end;

registration

let E be RealNormSpace;
let f be bijective isometric UnOp of E;

cluster f /" -> isometric ;

coherence
f /" is isometric by Th12;

end;

theorem Th13: :: MAZURULM:13

for E, F, G being RealNormSpace
for f being Function of E,F
for g being Function of F,G st f is midpoints-preserving & g is midpoints-preserving holds
g * f is midpoints-preserving

proof end;

registration

let E be RealNormSpace;
let f, g be midpoints-preserving UnOp of E;

cluster g * f -> midpoints-preserving for UnOp of E;

coherence
for b₁ being UnOp of E st b₁ = g * f holds
b₁ is midpoints-preserving by Th13;

end;

Lm3: now :: thesis: for E, F being RealNormSpace
for f being Function of E,F st f is bijective holds
(f /") * f = id E

let E, F be RealNormSpace; :: thesis: for f being Function of E,F st f is bijective holds
(f /") * f = id E
let f be Function of E,F; :: thesis: ( f is bijective implies (f /") * f = id E )
assume A1: f is bijective ; :: thesis: (f /") * f = id E
then A2: rng f = [#] F by FUNCT_2:def 3;
dom f = [#] E by FUNCT_2:def 1;
hence (f /") * f = id E by A1, A2, TOPS_2:52; :: thesis: verum

end;

theorem Th14: :: MAZURULM:14

for E, F being RealNormSpace
for f being Function of E,F st f is bijective & f is midpoints-preserving holds
f /" is midpoints-preserving

proof end;

registration

let E be RealNormSpace;
let f be bijective midpoints-preserving UnOp of E;

cluster f /" -> midpoints-preserving ;

coherence
f /" is midpoints-preserving by Th14;

end;

theorem Th15: :: MAZURULM:15

for E, F, G being RealNormSpace
for f being Function of E,F
for g being Function of F,G st f is Affine & g is Affine holds
g * f is Affine

proof end;

registration

let E be RealNormSpace;
let f, g be Affine UnOp of E;

cluster g * f -> Affine for UnOp of E;

coherence
for b₁ being UnOp of E st b₁ = g * f holds
b₁ is Affine by Th15;

end;

theorem Th16: :: MAZURULM:16

for E, F being RealNormSpace
for f being Function of E,F st f is bijective & f is Affine holds
f /" is Affine

proof end;

registration

let E be RealNormSpace;
let f be bijective Affine UnOp of E;

cluster f /" -> Affine ;

coherence
f /" is Affine by Th16;

end;

definition

let E be non empty RLSStruct ;
let a be Point of E;

func a -reflection -> UnOp of E means :Def4: :: MAZURULM:def 4
for b being Point of E holds it . b = (2 * a) - b;

existence
ex b₁ being UnOp of E st
for b being Point of E holds b₁ . b = (2 * a) - b

proof end;

uniqueness
for b₁, b₂ being UnOp of E st ( for b being Point of E holds b₁ . b = (2 * a) - b ) & ( for b being Point of E holds b₂ . b = (2 * a) - b ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def4 defines -reflection MAZURULM:def 4 :
for E being non empty RLSStruct
for a being Point of E
for b₃ being UnOp of E holds
( b₃ = a -reflection iff for b being Point of E holds b₃ . b = (2 * a) - b );

theorem Th17: :: MAZURULM:17

for E being RealNormSpace
for a being Point of E holds (a -reflection) * (a -reflection) = id E

proof end;

registration

let E be RealNormSpace;
let a be Point of E;

cluster a -reflection -> bijective ;

coherence
a -reflection is bijective

proof end;

end;

theorem Th18: :: MAZURULM:18

for E being RealNormSpace
for a being Point of E holds
( (a -reflection) . a = a & ( for b being Point of E st (a -reflection) . b = b holds
a = b ) )

proof end;

theorem Th19: :: MAZURULM:19

for E being RealNormSpace
for a, b being Point of E holds ((a -reflection) . b) - a = a - b

proof end;

theorem Th20: :: MAZURULM:20

for E being RealNormSpace
for a, b being Point of E holds ||.(((a -reflection) . b) - a).|| = ||.(b - a).||

proof end;

theorem Th21: :: MAZURULM:21

for E being RealNormSpace
for a, b being Point of E holds ((a -reflection) . b) - b = 2 * (a - b)

proof end;

theorem Th22: :: MAZURULM:22

for E being RealNormSpace
for a, b being Point of E holds ||.(((a -reflection) . b) - b).|| = 2 * ||.(b - a).||

proof end;

theorem Th23: :: MAZURULM:23

for E being RealNormSpace
for a being Point of E holds (a -reflection) /" = a -reflection

proof end;

registration

let E be RealNormSpace;
let a be Point of E;

cluster a -reflection -> isometric ;

coherence
a -reflection is isometric

proof end;

end;

deffunc H₁( RealNormSpace, Point of $1, Point of $1) -> set = { g where g is UnOp of $1 : ( g is bijective & g is isometric & g . $2 = $2 & g . $3 = $3 ) } ;

deffunc H₂( RealNormSpace, Point of $1, Point of $1) -> set = { ||.((g . ((1 / 2) * ($2 + $3))) - ((1 / 2) * ($2 + $3))).|| where g is UnOp of $1 : g in H₁($1,$2,$3) } ;

Lm4: now :: thesis: for E being RealNormSpace
for a, b being Point of E
for l being real-membered set st l = H₂(E,a,b) holds
2 * ||.(a - ((1 / 2) * (a + b))).|| is UpperBound of l

let E be RealNormSpace; :: thesis: for a, b being Point of E
for l being real-membered set st l = H₂(E,a,b) holds
2 * ||.(a - ((1 / 2) * (a + b))).|| is UpperBound of l
let a, b be Point of E; :: thesis: for l being real-membered set st l = H₂(E,a,b) holds
2 * ||.(a - ((1 / 2) * (a + b))).|| is UpperBound of l
let l be real-membered set ; :: thesis: ( l = H₂(E,a,b) implies 2 * ||.(a - ((1 / 2) * (a + b))).|| is UpperBound of l )
assume A1: l = H₂(E,a,b) ; :: thesis: 2 * ||.(a - ((1 / 2) * (a + b))).|| is UpperBound of l
set z = (1 / 2) * (a + b);
thus 2 * ||.(a - ((1 / 2) * (a + b))).|| is UpperBound of l :: thesis: verum

proof

let x be ExtReal; :: according to XXREAL_2:def 1 :: thesis: ( not x in l or x <= 2 * ||.(a - ((1 / 2) * (a + b))).|| )
assume x in l ; :: thesis: x <= 2 * ||.(a - ((1 / 2) * (a + b))).||
then consider g1 being UnOp of E such that
A2: x = ||.((g1 . ((1 / 2) * (a + b))) - ((1 / 2) * (a + b))).|| and
A3: g1 in H₁(E,a,b) by A1;
consider g being UnOp of E such that
A4: g1 = g and
g is bijective and
A5: g is isometric and
A6: g . a = a and
g . b = b by A3;
A7: ||.((g . ((1 / 2) * (a + b))) - a).|| = ||.(((1 / 2) * (a + b)) - a).|| by A5, A6
.= ||.(a - ((1 / 2) * (a + b))).|| by NORMSP_1:7 ;
||.((g . ((1 / 2) * (a + b))) - ((1 / 2) * (a + b))).|| <= ||.((g . ((1 / 2) * (a + b))) - a).|| + ||.(a - ((1 / 2) * (a + b))).|| by NORMSP_1:10;
hence x <= 2 * ||.(a - ((1 / 2) * (a + b))).|| by A2, A4, A7; :: thesis: verum

end;

end;

Lm5: now :: thesis: for E being RealNormSpace
for a, b being Point of E
for h being UnOp of E st h in H₁(E,a,b) holds
(((((1 / 2) * (a + b)) -reflection) * (h /")) * (((1 / 2) * (a + b)) -reflection)) * h in H₁(E,a,b)

let E be RealNormSpace; :: thesis: for a, b being Point of E
for h being UnOp of E st h in H₁(E,a,b) holds
(((((1 / 2) * (a + b)) -reflection) * (h /")) * (((1 / 2) * (a + b)) -reflection)) * h in H₁(E,a,b)
let a, b be Point of E; :: thesis: for h being UnOp of E st h in H₁(E,a,b) holds
(((((1 / 2) * (a + b)) -reflection) * (h /")) * (((1 / 2) * (a + b)) -reflection)) * h in H₁(E,a,b)
let h be UnOp of E; :: thesis: ( h in H₁(E,a,b) implies (((((1 / 2) * (a + b)) -reflection) * (h /")) * (((1 / 2) * (a + b)) -reflection)) * h in H₁(E,a,b) )
assume A1: h in H₁(E,a,b) ; :: thesis: (((((1 / 2) * (a + b)) -reflection) * (h /")) * (((1 / 2) * (a + b)) -reflection)) * h in H₁(E,a,b)
set z = (1 / 2) * (a + b);
set R = ((1 / 2) * (a + b)) -reflection ;
set gs = (((((1 / 2) * (a + b)) -reflection) * (h /")) * (((1 / 2) * (a + b)) -reflection)) * h;
consider g being UnOp of E such that
A2: g = h and
A3: g is bijective and
A4: g is isometric and
A5: g . a = a and
A6: g . b = b by A1;
A7: g /" = g " by A3, TOPS_2:def 4;
then A8: g /" is bijective by A3, GROUP_6:63;
A9: 2 * ((1 / 2) * (a + b)) = (2 * (1 / 2)) * (a + b) by RLVECT_1:def 7
.= a + b by RLVECT_1:def 8 ;
then A10: (2 * ((1 / 2) * (a + b))) - a = b by Th4;
A11: (2 * ((1 / 2) * (a + b))) - b = a by A9, Th4;
A12: dom g = [#] E by FUNCT_2:def 1;
A13: (g /") . b = b by A7, A6, A3, A12, FUNCT_1:32;
A14: (g /") . a = a by A7, A5, A3, A12, FUNCT_1:32;
A15: ((((((1 / 2) * (a + b)) -reflection) * (h /")) * (((1 / 2) * (a + b)) -reflection)) * h) . a = (((((1 / 2) * (a + b)) -reflection) * (g /")) * (((1 / 2) * (a + b)) -reflection)) . a by A5, A2, FUNCT_2:15
.= ((((1 / 2) * (a + b)) -reflection) * (g /")) . ((((1 / 2) * (a + b)) -reflection) . a) by FUNCT_2:15
.= ((((1 / 2) * (a + b)) -reflection) * (g /")) . b by A10, Def4
.= (((1 / 2) * (a + b)) -reflection) . ((g /") . b) by FUNCT_2:15
.= (2 * ((1 / 2) * (a + b))) - b by A13, Def4
.= a by A9, Th4 ;
((((((1 / 2) * (a + b)) -reflection) * (h /")) * (((1 / 2) * (a + b)) -reflection)) * h) . b = (((((1 / 2) * (a + b)) -reflection) * (g /")) * (((1 / 2) * (a + b)) -reflection)) . b by A6, A2, FUNCT_2:15
.= ((((1 / 2) * (a + b)) -reflection) * (g /")) . ((((1 / 2) * (a + b)) -reflection) . b) by FUNCT_2:15
.= ((((1 / 2) * (a + b)) -reflection) * (g /")) . a by A11, Def4
.= (((1 / 2) * (a + b)) -reflection) . ((g /") . a) by FUNCT_2:15
.= (2 * ((1 / 2) * (a + b))) - a by A14, Def4
.= b by A9, Th4 ;
hence (((((1 / 2) * (a + b)) -reflection) * (h /")) * (((1 / 2) * (a + b)) -reflection)) * h in H₁(E,a,b) by A2, A3, A4, A8, A15; :: thesis: verum

end;

Lm6: now :: thesis: for E being RealNormSpace
for a, b being Point of E
for l being non empty bounded_above Subset of REAL st l = H₂(E,a,b) holds
sup l is UpperBound of 2 ** l

let E be RealNormSpace; :: thesis: for a, b being Point of E
for l being non empty bounded_above Subset of REAL st l = H₂(E,a,b) holds
sup l is UpperBound of 2 ** l
let a, b be Point of E; :: thesis: for l being non empty bounded_above Subset of REAL st l = H₂(E,a,b) holds
sup l is UpperBound of 2 ** l
let l be non empty bounded_above Subset of REAL; :: thesis: ( l = H₂(E,a,b) implies sup l is UpperBound of 2 ** l )
assume A1: l = H₂(E,a,b) ; :: thesis: sup l is UpperBound of 2 ** l
thus sup l is UpperBound of 2 ** l :: thesis: verum

proof

set z = (1 / 2) * (a + b);
set R = ((1 / 2) * (a + b)) -reflection ;
let x be ExtReal; :: according to XXREAL_2:def 1 :: thesis: ( not x in 2 ** l or x <= sup l )
assume x in 2 ** l ; :: thesis: x <= sup l
then consider w being Element of ExtREAL such that
A2: x = 2 * w and
A3: w in l by MEMBER_1:188;
consider h being UnOp of E such that
A4: w = ||.((h . ((1 / 2) * (a + b))) - ((1 / 2) * (a + b))).|| and
A5: h in H₁(E,a,b) by A3, A1;
consider g being UnOp of E such that
A6: g = h and
A7: g is bijective and
A8: g is isometric and
( g . a = a & g . b = b ) by A5;
A9: (((1 / 2) * (a + b)) -reflection) * ((g /") * ((((1 / 2) * (a + b)) -reflection) * g)) = ((((1 / 2) * (a + b)) -reflection) * (g /")) * ((((1 / 2) * (a + b)) -reflection) * g) by RELAT_1:36
.= (((((1 / 2) * (a + b)) -reflection) * (g /")) * (((1 / 2) * (a + b)) -reflection)) * g by RELAT_1:36 ;
A10: (((1 / 2) * (a + b)) -reflection) . ((g /") . ((((1 / 2) * (a + b)) -reflection) . (g . ((1 / 2) * (a + b))))) = (((1 / 2) * (a + b)) -reflection) . ((g /") . (((((1 / 2) * (a + b)) -reflection) * g) . ((1 / 2) * (a + b)))) by FUNCT_2:15
.= (((1 / 2) * (a + b)) -reflection) . (((g /") * ((((1 / 2) * (a + b)) -reflection) * g)) . ((1 / 2) * (a + b))) by FUNCT_2:15
.= ((((1 / 2) * (a + b)) -reflection) * ((g /") * ((((1 / 2) * (a + b)) -reflection) * g))) . ((1 / 2) * (a + b)) by FUNCT_2:15 ;
A11: g /" = g " by A7, TOPS_2:def 4;
(((((1 / 2) * (a + b)) -reflection) * (g /")) * (((1 / 2) * (a + b)) -reflection)) * g in H₁(E,a,b) by A5, A6, Lm5;
then A12: ||.((((((((1 / 2) * (a + b)) -reflection) * (g /")) * (((1 / 2) * (a + b)) -reflection)) * g) . ((1 / 2) * (a + b))) - ((1 / 2) * (a + b))).|| in l by A1;
reconsider d = 2 as R_eal by XXREAL_0:def 1;
A13: (g /") . (g . ((1 / 2) * (a + b))) = (1 / 2) * (a + b) by A7, A11, FUNCT_2:26;
2 * ||.((g . ((1 / 2) * (a + b))) - ((1 / 2) * (a + b))).|| = ||.(((((1 / 2) * (a + b)) -reflection) . (g . ((1 / 2) * (a + b)))) - (g . ((1 / 2) * (a + b)))).|| by Th22
.= ||.(((g /") . ((((1 / 2) * (a + b)) -reflection) . (g . ((1 / 2) * (a + b))))) - ((g /") . (g . ((1 / 2) * (a + b))))).|| by A7, A8, Def1
.= ||.(((((1 / 2) * (a + b)) -reflection) . ((g /") . ((((1 / 2) * (a + b)) -reflection) . (g . ((1 / 2) * (a + b)))))) - ((1 / 2) * (a + b))).|| by A13, Th20 ;
hence x <= sup l by A2, A4, A9, A10, A12, A6, XXREAL_2:4; :: thesis: verum

end;

end;

Lm7: now :: thesis: for E being RealNormSpace
for a, b being Point of E
for l being real-membered set st l = H₂(E,a,b) holds
for g being UnOp of E st g in H₁(E,a,b) holds
g . ((1 / 2) * (a + b)) = (1 / 2) * (a + b)

let E be RealNormSpace; :: thesis: for a, b being Point of E
for l being real-membered set st l = H₂(E,a,b) holds
for g being UnOp of E st g in H₁(E,a,b) holds
g . ((1 / 2) * (a + b)) = (1 / 2) * (a + b)
let a, b be Point of E; :: thesis: for l being real-membered set st l = H₂(E,a,b) holds
for g being UnOp of E st g in H₁(E,a,b) holds
g . ((1 / 2) * (a + b)) = (1 / 2) * (a + b)
let l be real-membered set ; :: thesis: ( l = H₂(E,a,b) implies for g being UnOp of E st g in H₁(E,a,b) holds
g . ((1 / 2) * (a + b)) = (1 / 2) * (a + b) )
assume A1: l = H₂(E,a,b) ; :: thesis: for g being UnOp of E st g in H₁(E,a,b) holds
g . ((1 / 2) * (a + b)) = (1 / 2) * (a + b)
let g be UnOp of E; :: thesis: ( g in H₁(E,a,b) implies g . ((1 / 2) * (a + b)) = (1 / 2) * (a + b) )
assume A2: g in H₁(E,a,b) ; :: thesis: g . ((1 / 2) * (a + b)) = (1 / 2) * (a + b)
set z = (1 / 2) * (a + b);
set R = ((1 / 2) * (a + b)) -reflection ;
A3: l c= REAL by XREAL_0:def 1;
( (id E) . a = a & (id E) . b = b ) ;
then id E in H₁(E,a,b) ;
then A4: ||.(((id E) . ((1 / 2) * (a + b))) - ((1 / 2) * (a + b))).|| in l by A1;
2 * ||.(a - ((1 / 2) * (a + b))).|| is UpperBound of l by A1, Lm4;
then reconsider A = l as non empty bounded_above Subset of REAL by A3, A4, XXREAL_2:def 10;
set lambda = sup A;
reconsider d = 2 as R_eal by XXREAL_0:def 1;
A5: d * (sup A) = sup (2 ** A) by URYSOHN2:18;
A6: ||.((g . ((1 / 2) * (a + b))) - ((1 / 2) * (a + b))).|| in A by A1, A2;
sup A is UpperBound of 2 ** A by A1, Lm6;
then sup (2 ** A) <= sup A by XXREAL_2:def 3;
then (2 * (sup A)) - (sup A) <= (sup A) - (sup A) by A5, XREAL_1:9;
then ||.((g . ((1 / 2) * (a + b))) - ((1 / 2) * (a + b))).|| = 0 by A6, XXREAL_2:4;
hence g . ((1 / 2) * (a + b)) = (1 / 2) * (a + b) by NORMSP_1:6; :: thesis: verum

end;

theorem Th24: :: MAZURULM:24

for E, F being RealNormSpace
for f being Function of E,F st f is isometric holds
f is_continuous_on dom f

proof end;

Lm8: for E being RealNormSpace
for a, b being Point of E
for t being Real holds ((1 - t) * a) + (t * b) = a + (t * (b - a))

proof end;

Lm9: now :: thesis: for E, F being RealNormSpace
for f being Function of E,F
for a, b being Point of E
for n, j being Nat st f is midpoints-preserving & j <= 2 |^ n holds
ex x being Nat st
( x = (2 |^ n) - j & f . (a + ((j / (2 |^ (n + 1))) * (b - a))) = (1 / 2) * ((f . a) + (f . ((1 / (2 |^ n)) * ((x * a) + (j * b))))) )

let E, F be RealNormSpace; :: thesis: for f being Function of E,F
for a, b being Point of E
for n, j being Nat st f is midpoints-preserving & j <= 2 |^ n holds
ex x being Nat st
( x = (2 |^ n) - j & f . (a + ((j / (2 |^ (n + 1))) * (b - a))) = (1 / 2) * ((f . a) + (f . ((1 / (2 |^ n)) * ((x * a) + (j * b))))) )
let f be Function of E,F; :: thesis: for a, b being Point of E
for n, j being Nat st f is midpoints-preserving & j <= 2 |^ n holds
ex x being Nat st
( x = (2 |^ n) - j & f . (a + ((j / (2 |^ (n + 1))) * (b - a))) = (1 / 2) * ((f . a) + (f . ((1 / (2 |^ n)) * ((x * a) + (j * b))))) )
let a, b be Point of E; :: thesis: for n, j being Nat st f is midpoints-preserving & j <= 2 |^ n holds
ex x being Nat st
( x = (2 |^ n) - j & f . (a + ((j / (2 |^ (n + 1))) * (b - a))) = (1 / 2) * ((f . a) + (f . ((1 / (2 |^ n)) * ((x * a) + (j * b))))) )
let n, j be Nat; :: thesis: ( f is midpoints-preserving & j <= 2 |^ n implies ex x being Nat st
( x = (2 |^ n) - j & f . (a + ((j / (2 |^ (n + 1))) * (b - a))) = (1 / 2) * ((f . a) + (f . ((1 / (2 |^ n)) * ((x * a) + (j * b))))) ) )
set m = 2 |^ (n + 1);
set k = 2 |^ n;
set x = (2 |^ n) - j;
assume A1: f is midpoints-preserving ; :: thesis: ( j <= 2 |^ n implies ex x being Nat st
( x = (2 |^ n) - j & f . (a + ((j / (2 |^ (n + 1))) * (b - a))) = (1 / 2) * ((f . a) + (f . ((1 / (2 |^ n)) * ((x * a) + (j * b))))) ) )
assume j <= 2 |^ n ; :: thesis: ex x being Nat st
( x = (2 |^ n) - j & f . (a + ((j / (2 |^ (n + 1))) * (b - a))) = (1 / 2) * ((f . a) + (f . ((1 / (2 |^ n)) * ((x * a) + (j * b))))) )
then (2 |^ n) - j in NAT by INT_1:5;
then reconsider x = (2 |^ n) - j as Nat ;
take x = x; :: thesis: ( x = (2 |^ n) - j & f . (a + ((j / (2 |^ (n + 1))) * (b - a))) = (1 / 2) * ((f . a) + (f . ((1 / (2 |^ n)) * ((x * a) + (j * b))))) )
thus x = (2 |^ n) - j ; :: thesis: f . (a + ((j / (2 |^ (n + 1))) * (b - a))) = (1 / 2) * ((f . a) + (f . ((1 / (2 |^ n)) * ((x * a) + (j * b)))))
set z = (1 - (j / (2 |^ (n + 1)))) - (1 / 2);
A2: 2 |^ n <> 0 by NEWTON:83;
A3: 2 * (2 |^ n) = 2 |^ (n + 1) by NEWTON:6;
A4: (1 / 2) * (1 / (2 |^ n)) = 1 / (2 * (2 |^ n)) by XCMPLX_1:102;
x / (2 |^ (n + 1)) = ((1 * (2 |^ n)) / (2 |^ (n + 1))) - (j / (2 |^ (n + 1)))
.= (1 / 2) - (j / (2 |^ (n + 1))) by A2, A3, XCMPLX_1:91
.= (1 - (j / (2 |^ (n + 1)))) - (1 / 2) ;
then A5: (((1 - (j / (2 |^ (n + 1)))) - (1 / 2)) * a) + ((j / (2 |^ (n + 1))) * b) = ((1 / (2 |^ (n + 1))) * (x * a)) + (((1 / (2 |^ (n + 1))) * j) * b) by RLVECT_1:def 7
.= ((1 / (2 |^ (n + 1))) * (x * a)) + ((1 / (2 |^ (n + 1))) * (j * b)) by RLVECT_1:def 7
.= (1 / (2 |^ (n + 1))) * ((x * a) + (j * b)) by RLVECT_1:def 5
.= (1 / 2) * ((1 / (2 |^ n)) * ((x * a) + (j * b))) by A4, A3, RLVECT_1:def 7 ;
a + ((j / (2 |^ (n + 1))) * (b - a)) = a + (((j / (2 |^ (n + 1))) * b) - ((j / (2 |^ (n + 1))) * a)) by RLVECT_1:34
.= (a + ((j / (2 |^ (n + 1))) * b)) - ((j / (2 |^ (n + 1))) * a) by RLVECT_1:def 3
.= (a - ((j / (2 |^ (n + 1))) * a)) + ((j / (2 |^ (n + 1))) * b) by RLVECT_1:def 3
.= ((1 * a) - ((j / (2 |^ (n + 1))) * a)) + ((j / (2 |^ (n + 1))) * b) by RLVECT_1:def 8
.= ((1 - (j / (2 |^ (n + 1)))) * a) + ((j / (2 |^ (n + 1))) * b) by RLVECT_1:35
.= ((((1 - (j / (2 |^ (n + 1)))) * a) + (((1 - (j / (2 |^ (n + 1)))) - (1 / 2)) * a)) - (((1 - (j / (2 |^ (n + 1)))) - (1 / 2)) * a)) + ((j / (2 |^ (n + 1))) * b) by Th4
.= ((((1 - (j / (2 |^ (n + 1)))) * a) - (((1 - (j / (2 |^ (n + 1)))) - (1 / 2)) * a)) + (((1 - (j / (2 |^ (n + 1)))) - (1 / 2)) * a)) + ((j / (2 |^ (n + 1))) * b) by RLVECT_1:def 3
.= ((((1 - (j / (2 |^ (n + 1)))) - ((1 - (j / (2 |^ (n + 1)))) - (1 / 2))) * a) + (((1 - (j / (2 |^ (n + 1)))) - (1 / 2)) * a)) + ((j / (2 |^ (n + 1))) * b) by RLVECT_1:35
.= ((1 / 2) * a) + ((((1 - (j / (2 |^ (n + 1)))) - (1 / 2)) * a) + ((j / (2 |^ (n + 1))) * b)) by RLVECT_1:def 3
.= (1 / 2) * (a + ((1 / (2 |^ n)) * ((x * a) + (j * b)))) by A5, RLVECT_1:def 5 ;
hence f . (a + ((j / (2 |^ (n + 1))) * (b - a))) = (1 / 2) * ((f . a) + (f . ((1 / (2 |^ n)) * ((x * a) + (j * b))))) by A1; :: thesis: verum

end;

Lm10: now :: thesis: for E, F being RealNormSpace
for f being Function of E,F
for a, b being Point of E
for n, j being Nat st f is midpoints-preserving & j >= 2 |^ n holds
ex x being Nat st
( x = ((2 |^ n) + j) - (2 |^ (n + 1)) & f . (a + ((j / (2 |^ (n + 1))) * (b - a))) = (1 / 2) * ((f . b) + (f . ((1 / (2 |^ n)) * ((((2 |^ (n + 1)) - j) * a) + (x * b))))) )

let E, F be RealNormSpace; :: thesis: for f being Function of E,F
for a, b being Point of E
for n, j being Nat st f is midpoints-preserving & j >= 2 |^ n holds
ex x being Nat st
( x = ((2 |^ n) + j) - (2 |^ (n + 1)) & f . (a + ((j / (2 |^ (n + 1))) * (b - a))) = (1 / 2) * ((f . b) + (f . ((1 / (2 |^ n)) * ((((2 |^ (n + 1)) - j) * a) + (x * b))))) )
let f be Function of E,F; :: thesis: for a, b being Point of E
for n, j being Nat st f is midpoints-preserving & j >= 2 |^ n holds
ex x being Nat st
( x = ((2 |^ n) + j) - (2 |^ (n + 1)) & f . (a + ((j / (2 |^ (n + 1))) * (b - a))) = (1 / 2) * ((f . b) + (f . ((1 / (2 |^ n)) * ((((2 |^ (n + 1)) - j) * a) + (x * b))))) )
let a, b be Point of E; :: thesis: for n, j being Nat st f is midpoints-preserving & j >= 2 |^ n holds
ex x being Nat st
( x = ((2 |^ n) + j) - (2 |^ (n + 1)) & f . (a + ((j / (2 |^ (n + 1))) * (b - a))) = (1 / 2) * ((f . b) + (f . ((1 / (2 |^ n)) * ((((2 |^ (n + 1)) - j) * a) + (x * b))))) )
let n, j be Nat; :: thesis: ( f is midpoints-preserving & j >= 2 |^ n implies ex x being Nat st
( x = ((2 |^ n) + j) - (2 |^ (n + 1)) & f . (a + ((j / (2 |^ (n + 1))) * (b - a))) = (1 / 2) * ((f . b) + (f . ((1 / (2 |^ n)) * ((((2 |^ (n + 1)) - j) * a) + (x * b))))) ) )
set m = 2 |^ (n + 1);
set k = 2 |^ n;
set x = ((2 |^ n) + j) - (2 |^ (n + 1));
assume A1: f is midpoints-preserving ; :: thesis: ( j >= 2 |^ n implies ex x being Nat st
( x = ((2 |^ n) + j) - (2 |^ (n + 1)) & f . (a + ((j / (2 |^ (n + 1))) * (b - a))) = (1 / 2) * ((f . b) + (f . ((1 / (2 |^ n)) * ((((2 |^ (n + 1)) - j) * a) + (x * b))))) ) )
A2: 2 * (2 |^ n) = 2 |^ (n + 1) by NEWTON:6;
assume j >= 2 |^ n ; :: thesis: ex x being Nat st
( x = ((2 |^ n) + j) - (2 |^ (n + 1)) & f . (a + ((j / (2 |^ (n + 1))) * (b - a))) = (1 / 2) * ((f . b) + (f . ((1 / (2 |^ n)) * ((((2 |^ (n + 1)) - j) * a) + (x * b))))) )
then (2 |^ n) + (2 |^ n) <= (2 |^ n) + j by XREAL_1:6;
then ((2 |^ n) + j) - (2 |^ (n + 1)) in NAT by A2, INT_1:5;
then reconsider x = ((2 |^ n) + j) - (2 |^ (n + 1)) as Nat ;
take x = x; :: thesis: ( x = ((2 |^ n) + j) - (2 |^ (n + 1)) & f . (a + ((j / (2 |^ (n + 1))) * (b - a))) = (1 / 2) * ((f . b) + (f . ((1 / (2 |^ n)) * ((((2 |^ (n + 1)) - j) * a) + (x * b))))) )
thus x = ((2 |^ n) + j) - (2 |^ (n + 1)) ; :: thesis: f . (a + ((j / (2 |^ (n + 1))) * (b - a))) = (1 / 2) * ((f . b) + (f . ((1 / (2 |^ n)) * ((((2 |^ (n + 1)) - j) * a) + (x * b)))))
set z = (j / (2 |^ (n + 1))) - (1 / 2);
A3: 2 |^ n <> 0 by NEWTON:83;
A4: 2 |^ (n + 1) <> 0 by NEWTON:83;
A5: (2 |^ (n + 1)) / (2 |^ (n + 1)) = 1 by A4, XCMPLX_1:60;
then A6: 1 - (j / (2 |^ (n + 1))) = ((2 |^ (n + 1)) / (2 |^ (n + 1))) - (j / (2 |^ (n + 1)))
.= (1 / (2 |^ (n + 1))) * ((2 |^ (n + 1)) - j) ;
A7: (2 |^ n) / (2 |^ (n + 1)) = (1 * (2 |^ n)) / (2 * (2 |^ n)) by NEWTON:6
.= 1 / 2 by A3, XCMPLX_1:91 ;
A8: (1 / 2) * (1 / (2 |^ n)) = 1 / (2 * (2 |^ n)) by XCMPLX_1:102;
x / (2 |^ (n + 1)) = (((2 |^ n) + j) / (2 |^ (n + 1))) - ((2 |^ (n + 1)) / (2 |^ (n + 1)))
.= (((2 |^ n) / (2 |^ (n + 1))) + (j / (2 |^ (n + 1)))) - ((2 |^ (n + 1)) / (2 |^ (n + 1)))
.= (j / (2 |^ (n + 1))) - (1 / 2) by A5, A7 ;
then A9: (((j / (2 |^ (n + 1))) - (1 / 2)) * b) + ((1 - (j / (2 |^ (n + 1)))) * a) = ((1 / (2 |^ (n + 1))) * (((2 |^ (n + 1)) - j) * a)) + (((1 / (2 |^ (n + 1))) * x) * b) by A6, RLVECT_1:def 7
.= ((1 / (2 |^ (n + 1))) * (((2 |^ (n + 1)) - j) * a)) + ((1 / (2 |^ (n + 1))) * (x * b)) by RLVECT_1:def 7
.= (1 / (2 |^ (n + 1))) * ((((2 |^ (n + 1)) - j) * a) + (x * b)) by RLVECT_1:def 5
.= (1 / 2) * ((1 / (2 |^ n)) * ((((2 |^ (n + 1)) - j) * a) + (x * b))) by A8, A2, RLVECT_1:def 7 ;
a + ((j / (2 |^ (n + 1))) * (b - a)) = a + (((j / (2 |^ (n + 1))) * b) - ((j / (2 |^ (n + 1))) * a)) by RLVECT_1:34
.= (a + ((j / (2 |^ (n + 1))) * b)) - ((j / (2 |^ (n + 1))) * a) by RLVECT_1:def 3
.= (a - ((j / (2 |^ (n + 1))) * a)) + ((j / (2 |^ (n + 1))) * b) by RLVECT_1:def 3
.= ((1 * a) - ((j / (2 |^ (n + 1))) * a)) + ((j / (2 |^ (n + 1))) * b) by RLVECT_1:def 8
.= ((1 - (j / (2 |^ (n + 1)))) * a) + ((j / (2 |^ (n + 1))) * b) by RLVECT_1:35
.= ((((j / (2 |^ (n + 1))) * b) + (((j / (2 |^ (n + 1))) - (1 / 2)) * b)) - (((j / (2 |^ (n + 1))) - (1 / 2)) * b)) + ((1 - (j / (2 |^ (n + 1)))) * a) by Th4
.= ((((j / (2 |^ (n + 1))) * b) - (((j / (2 |^ (n + 1))) - (1 / 2)) * b)) + (((j / (2 |^ (n + 1))) - (1 / 2)) * b)) + ((1 - (j / (2 |^ (n + 1)))) * a) by RLVECT_1:def 3
.= ((((j / (2 |^ (n + 1))) - ((j / (2 |^ (n + 1))) - (1 / 2))) * b) + (((j / (2 |^ (n + 1))) - (1 / 2)) * b)) + ((1 - (j / (2 |^ (n + 1)))) * a) by RLVECT_1:35
.= ((1 / 2) * b) + ((((j / (2 |^ (n + 1))) - (1 / 2)) * b) + ((1 - (j / (2 |^ (n + 1)))) * a)) by RLVECT_1:def 3
.= (1 / 2) * (b + ((1 / (2 |^ n)) * ((((2 |^ (n + 1)) - j) * a) + (x * b)))) by A9, RLVECT_1:def 5 ;
hence f . (a + ((j / (2 |^ (n + 1))) * (b - a))) = (1 / 2) * ((f . b) + (f . ((1 / (2 |^ n)) * ((((2 |^ (n + 1)) - j) * a) + (x * b))))) by A1; :: thesis: verum

end;

Lm11: now :: thesis: for E, F being RealNormSpace
for f being Function of E,F
for a, b being Point of E
for t being Nat st f is midpoints-preserving holds
for n being Nat st t <= 2 |^ n holds
f . (((1 - (t / (2 |^ n))) * a) + ((t / (2 |^ n)) * b)) = ((1 - (t / (2 |^ n))) * (f . a)) + ((t / (2 |^ n)) * (f . b))

let E, F be RealNormSpace; :: thesis: for f being Function of E,F
for a, b being Point of E
for t being Nat st f is midpoints-preserving holds
for n being Nat st t <= 2 |^ n holds
f . (((1 - (t / (2 |^ n))) * a) + ((t / (2 |^ n)) * b)) = ((1 - (t / (2 |^ n))) * (f . a)) + ((t / (2 |^ n)) * (f . b))
let f be Function of E,F; :: thesis: for a, b being Point of E
for t being Nat st f is midpoints-preserving holds
for n being Nat st t <= 2 |^ n holds
f . (((1 - (t / (2 |^ n))) * a) + ((t / (2 |^ n)) * b)) = ((1 - (t / (2 |^ n))) * (f . a)) + ((t / (2 |^ n)) * (f . b))
let a, b be Point of E; :: thesis: for t being Nat st f is midpoints-preserving holds
for n being Nat st t <= 2 |^ n holds
f . (((1 - (t / (2 |^ n))) * a) + ((t / (2 |^ n)) * b)) = ((1 - (t / (2 |^ n))) * (f . a)) + ((t / (2 |^ n)) * (f . b))
let t be Nat; :: thesis: ( f is midpoints-preserving implies for n being Nat st t <= 2 |^ n holds
f . (((1 - (t / (2 |^ n))) * a) + ((t / (2 |^ n)) * b)) = ((1 - (t / (2 |^ n))) * (f . a)) + ((t / (2 |^ n)) * (f . b)) )
assume A1: f is midpoints-preserving ; :: thesis: for n being Nat st t <= 2 |^ n holds
f . (((1 - (t / (2 |^ n))) * a) + ((t / (2 |^ n)) * b)) = ((1 - (t / (2 |^ n))) * (f . a)) + ((t / (2 |^ n)) * (f . b))
thus for n being Nat st t <= 2 |^ n holds
f . (((1 - (t / (2 |^ n))) * a) + ((t / (2 |^ n)) * b)) = ((1 - (t / (2 |^ n))) * (f . a)) + ((t / (2 |^ n)) * (f . b)) :: thesis: verum

proof

defpred S₁[ Nat] means for w being Nat st w <= 2 |^ $1 holds
f . (((1 - (w / (2 |^ $1))) * a) + ((w / (2 |^ $1)) * b)) = ((1 - (w / (2 |^ $1))) * (f . a)) + ((w / (2 |^ $1)) * (f . b));
A2: S₁[ 0 ]

proof

let t be Nat; :: thesis: ( t <= 2 |^ 0 implies f . (((1 - (t / (2 |^ 0))) * a) + ((t / (2 |^ 0)) * b)) = ((1 - (t / (2 |^ 0))) * (f . a)) + ((t / (2 |^ 0)) * (f . b)) )
assume A3: t <= 2 |^ 0 ; :: thesis: f . (((1 - (t / (2 |^ 0))) * a) + ((t / (2 |^ 0)) * b)) = ((1 - (t / (2 |^ 0))) * (f . a)) + ((t / (2 |^ 0)) * (f . b))
A4: 2 |^ 0 = 1 by NEWTON:4;

per cases ( t = 1 or t = 0 ) by A3, A4, NAT_1:25;

suppose A5: t = 1 ; :: thesis: f . (((1 - (t / (2 |^ 0))) * a) + ((t / (2 |^ 0)) * b)) = ((1 - (t / (2 |^ 0))) * (f . a)) + ((t / (2 |^ 0)) * (f . b))

then f . (((1 - t) * a) + (t * b)) = f . ((0. E) + (t * b)) by RLVECT_1:10
.= f . (t * b)
.= f . b by A5, RLVECT_1:def 8
.= t * (f . b) by A5, RLVECT_1:def 8
.= (0. F) + (t * (f . b))
.= ((1 - t) * (f . a)) + (t * (f . b)) by A5, RLVECT_1:10 ;
hence f . (((1 - (t / (2 |^ 0))) * a) + ((t / (2 |^ 0)) * b)) = ((1 - (t / (2 |^ 0))) * (f . a)) + ((t / (2 |^ 0)) * (f . b)) by A4; :: thesis: verum

end;

suppose A6: t = 0 ; :: thesis: f . (((1 - (t / (2 |^ 0))) * a) + ((t / (2 |^ 0)) * b)) = ((1 - (t / (2 |^ 0))) * (f . a)) + ((t / (2 |^ 0)) * (f . b))

then f . (((1 - t) * a) + (t * b)) = f . ((1 * a) + (0. E)) by RLVECT_1:10
.= f . (1 * a)
.= f . a by RLVECT_1:def 8
.= (1 - t) * (f . a) by A6, RLVECT_1:def 8
.= ((1 - t) * (f . a)) + (0. F)
.= ((1 - t) * (f . a)) + (t * (f . b)) by A6, RLVECT_1:10 ;
hence f . (((1 - (t / (2 |^ 0))) * a) + ((t / (2 |^ 0)) * b)) = ((1 - (t / (2 |^ 0))) * (f . a)) + ((t / (2 |^ 0)) * (f . b)) by A4; :: thesis: verum

end;

end;

end;

A7: for n being Nat st S₁[n] holds
S₁[n + 1]

proof

let n be Nat; :: thesis: ( S₁[n] implies S₁[n + 1] )
set m = 2 |^ n;
set k = 2 |^ (n + 1);
assume A8: S₁[n] ; :: thesis: S₁[n + 1]
let t be Nat; :: thesis: ( t <= 2 |^ (n + 1) implies f . (((1 - (t / (2 |^ (n + 1)))) * a) + ((t / (2 |^ (n + 1))) * b)) = ((1 - (t / (2 |^ (n + 1)))) * (f . a)) + ((t / (2 |^ (n + 1))) * (f . b)) )
assume A9: t <= 2 |^ (n + 1) ; :: thesis: f . (((1 - (t / (2 |^ (n + 1)))) * a) + ((t / (2 |^ (n + 1))) * b)) = ((1 - (t / (2 |^ (n + 1)))) * (f . a)) + ((t / (2 |^ (n + 1))) * (f . b))
A10: 2 |^ (n + 1) = (2 |^ n) * 2 by NEWTON:6;
A11: 2 |^ n <> 0 by NEWTON:83;
A12: (1 / 2) * (t / (2 |^ n)) = (1 * t) / (2 * (2 |^ n)) by XCMPLX_1:76
.= t / (2 |^ (n + 1)) by NEWTON:6 ;

per cases ( t <= 2 |^ n or t >= 2 |^ n ) ;

suppose A13: t <= 2 |^ n ; :: thesis: f . (((1 - (t / (2 |^ (n + 1)))) * a) + ((t / (2 |^ (n + 1))) * b)) = ((1 - (t / (2 |^ (n + 1)))) * (f . a)) + ((t / (2 |^ (n + 1))) * (f . b))

then consider x being Nat such that
A14: x = (2 |^ n) - t and
A15: f . (a + ((t / (2 |^ (n + 1))) * (b - a))) = (1 / 2) * ((f . a) + (f . ((1 / (2 |^ n)) * ((x * a) + (t * b))))) by A1, Lm9;
A16: x / (2 |^ n) = ((2 |^ n) / (2 |^ n)) - (t / (2 |^ n)) by A14
.= 1 - (t / (2 |^ n)) by A11, XCMPLX_1:60 ;
A17: (1 / (2 |^ n)) * ((x * a) + (t * b)) = ((1 / (2 |^ n)) * (x * a)) + ((1 / (2 |^ n)) * (t * b)) by RLVECT_1:def 5
.= ((1 / (2 |^ n)) * (x * a)) + ((t / (2 |^ n)) * b) by RLVECT_1:def 7
.= ((x / (2 |^ n)) * a) + ((t / (2 |^ n)) * b) by RLVECT_1:def 7 ;
thus f . (((1 - (t / (2 |^ (n + 1)))) * a) + ((t / (2 |^ (n + 1))) * b)) = f . (a + ((t / (2 |^ (n + 1))) * (b - a))) by Lm8
.= ((1 / 2) * (f . a)) + ((1 / 2) * (f . ((1 / (2 |^ n)) * ((x * a) + (t * b))))) by A15, RLVECT_1:def 5
.= ((1 / 2) * (f . a)) + ((1 / 2) * (((1 - (t / (2 |^ n))) * (f . a)) + ((t / (2 |^ n)) * (f . b)))) by A16, A13, A17, A8
.= ((1 / 2) * (f . a)) + (((1 / 2) * ((1 - (t / (2 |^ n))) * (f . a))) + ((1 / 2) * ((t / (2 |^ n)) * (f . b)))) by RLVECT_1:def 5
.= (((1 / 2) * (f . a)) + ((1 / 2) * ((1 - (t / (2 |^ n))) * (f . a)))) + ((1 / 2) * ((t / (2 |^ n)) * (f . b))) by RLVECT_1:def 3
.= (((1 / 2) * (f . a)) + (((1 / 2) * (1 - (t / (2 |^ n)))) * (f . a))) + ((1 / 2) * ((t / (2 |^ n)) * (f . b))) by RLVECT_1:def 7
.= (((1 / 2) + ((1 / 2) * (1 - (t / (2 |^ n))))) * (f . a)) + ((1 / 2) * ((t / (2 |^ n)) * (f . b))) by RLVECT_1:def 6
.= ((1 - (t / (2 |^ (n + 1)))) * (f . a)) + ((t / (2 |^ (n + 1))) * (f . b)) by A12, RLVECT_1:def 7 ; :: thesis: verum

end;

suppose t >= 2 |^ n ; :: thesis: f . (((1 - (t / (2 |^ (n + 1)))) * a) + ((t / (2 |^ (n + 1))) * b)) = ((1 - (t / (2 |^ (n + 1)))) * (f . a)) + ((t / (2 |^ (n + 1))) * (f . b))

then consider x being Nat such that
A18: x = ((2 |^ n) + t) - (2 |^ (n + 1)) and
A19: f . (a + ((t / (2 |^ (n + 1))) * (b - a))) = (1 / 2) * ((f . b) + (f . ((1 / (2 |^ n)) * ((((2 |^ (n + 1)) - t) * a) + (x * b))))) by A1, Lm10;
set w = t - (2 |^ n);
A20: t - (2 |^ n) <= (2 * (2 |^ n)) - (2 |^ n) by A9, A10, XREAL_1:9;
A21: (2 |^ n) / (2 |^ n) = 1 by A11, XCMPLX_1:60;
A22: (2 |^ (n + 1)) / (2 |^ n) = (2 * (2 |^ n)) / (1 * (2 |^ n)) by NEWTON:6
.= 2 / 1 by A11, XCMPLX_1:91 ;
A23: 1 - (t / (2 |^ n)) = ((2 |^ n) / (2 |^ n)) - (t / (2 |^ n)) by A21
.= - ((t / (2 |^ n)) - ((2 |^ n) / (2 |^ n)))
.= - ((t - (2 |^ n)) / (2 |^ n)) ;
A24: (1 / (2 |^ n)) * ((2 |^ (n + 1)) - t) = ((2 |^ (n + 1)) / (2 |^ n)) - (t / (2 |^ n))
.= (1 + 1) - (t / (2 |^ n)) by A22
.= 1 - ((t - (2 |^ n)) / (2 |^ n)) by A23
.= 1 - ((t - (2 |^ n)) / (2 |^ n)) ;
(1 / (2 |^ n)) * x = (t - (2 |^ n)) / (2 |^ n) by A18, A10;
then (((1 / (2 |^ n)) * ((2 |^ (n + 1)) - t)) * a) + (((1 / (2 |^ n)) * x) * b) = ((1 - ((t - (2 |^ n)) / (2 |^ n))) * a) + (((t - (2 |^ n)) / (2 |^ n)) * b) by A24;
then A25: (1 / 2) * (f . ((((1 / (2 |^ n)) * ((2 |^ (n + 1)) - t)) * a) + (((1 / (2 |^ n)) * x) * b))) = (1 / 2) * (((1 - ((t - (2 |^ n)) / (2 |^ n))) * (f . a)) + (((t - (2 |^ n)) / (2 |^ n)) * (f . b))) by A20, A18, A10, A8;
A26: (1 / 2) * (1 - ((t - (2 |^ n)) / (2 |^ n))) = (1 / 2) * (1 - ((t / (2 |^ n)) - ((2 |^ n) / (2 |^ n))))
.= 1 - ((1 / 2) * (t / (2 |^ n))) by A21
.= 1 - ((1 * t) / (2 * (2 |^ n))) by XCMPLX_1:76
.= 1 - (t / (2 |^ (n + 1))) by NEWTON:6 ;
thus f . (((1 - (t / (2 |^ (n + 1)))) * a) + ((t / (2 |^ (n + 1))) * b)) = f . (a + ((t / (2 |^ (n + 1))) * (b - a))) by Lm8
.= ((1 / 2) * (f . b)) + ((1 / 2) * (f . ((1 / (2 |^ n)) * ((((2 |^ (n + 1)) - t) * a) + (x * b))))) by A19, RLVECT_1:def 5
.= ((1 / 2) * (f . b)) + ((1 / 2) * (f . (((1 / (2 |^ n)) * (((2 |^ (n + 1)) - t) * a)) + ((1 / (2 |^ n)) * (x * b))))) by RLVECT_1:def 5
.= ((1 / 2) * (f . b)) + ((1 / 2) * (f . ((((1 / (2 |^ n)) * ((2 |^ (n + 1)) - t)) * a) + ((1 / (2 |^ n)) * (x * b))))) by RLVECT_1:def 7
.= ((1 / 2) * (f . b)) + ((1 / 2) * (f . ((((1 / (2 |^ n)) * ((2 |^ (n + 1)) - t)) * a) + (((1 / (2 |^ n)) * x) * b)))) by RLVECT_1:def 7
.= ((1 / 2) * (f . b)) + ((1 / 2) * (((1 - ((t - (2 |^ n)) / (2 |^ n))) * (f . a)) + (((t - (2 |^ n)) / (2 |^ n)) * (f . b)))) by A25
.= ((1 / 2) * (f . b)) + (((1 / 2) * ((1 - ((t - (2 |^ n)) / (2 |^ n))) * (f . a))) + ((1 / 2) * (((t - (2 |^ n)) / (2 |^ n)) * (f . b)))) by RLVECT_1:def 5
.= ((1 / 2) * ((1 - ((t - (2 |^ n)) / (2 |^ n))) * (f . a))) + (((1 / 2) * (f . b)) + ((1 / 2) * (((t - (2 |^ n)) / (2 |^ n)) * (f . b)))) by RLVECT_1:def 3
.= ((1 / 2) * ((1 - ((t - (2 |^ n)) / (2 |^ n))) * (f . a))) + ((1 / 2) * ((f . b) + (((t - (2 |^ n)) / (2 |^ n)) * (f . b)))) by RLVECT_1:def 5
.= ((1 / 2) * ((1 - ((t - (2 |^ n)) / (2 |^ n))) * (f . a))) + ((1 / 2) * ((1 * (f . b)) + (((t - (2 |^ n)) / (2 |^ n)) * (f . b)))) by RLVECT_1:def 8
.= ((1 / 2) * ((1 - ((t - (2 |^ n)) / (2 |^ n))) * (f . a))) + ((1 / 2) * ((1 + ((t - (2 |^ n)) / (2 |^ n))) * (f . b))) by RLVECT_1:def 6
.= ((1 / 2) * ((1 - ((t - (2 |^ n)) / (2 |^ n))) * (f . a))) + (((1 / 2) * (1 + ((t - (2 |^ n)) / (2 |^ n)))) * (f . b)) by RLVECT_1:def 7
.= ((1 - (t / (2 |^ (n + 1)))) * (f . a)) + ((t / (2 |^ (n + 1))) * (f . b)) by A26, RLVECT_1:def 7 ; :: thesis: verum

end;

end;

end;

for n being Nat holds S₁[n] from NAT_1:sch 2(A2, A7);
hence for n being Nat st t <= 2 |^ n holds
f . (((1 - (t / (2 |^ n))) * a) + ((t / (2 |^ n)) * b)) = ((1 - (t / (2 |^ n))) * (f . a)) + ((t / (2 |^ n)) * (f . b)) ; :: thesis: verum

end;

end;

registration

let E, F be RealNormSpace;

cluster Function-like quasi_total bijective isometric -> midpoints-preserving for Element of K10(K11( the carrier of E, the carrier of F));

coherence
for b₁ being Function of E,F st b₁ is bijective & b₁ is isometric holds
b₁ is midpoints-preserving

proof end;

end;

registration

let E, F be RealNormSpace;

cluster Function-like quasi_total isometric midpoints-preserving -> Affine for Element of K10(K11( the carrier of E, the carrier of F));

coherence
for b₁ being Function of E,F st b₁ is isometric & b₁ is midpoints-preserving holds
b₁ is Affine

proof end;

end;