:: Mazur-Ulam Theorem
:: by Artur Korni{\l}owicz
::
:: Received December 21, 2010
:: Copyright (c) 2010-2019 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 ;
end;

reconsider D = DYADIC as Subset of I[01] by ;

Lm1:
proof end;

theorem :: MAZURULM:1
proof end;

theorem Th2: :: MAZURULM:2
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;
coherence
r ** A is bounded_above
proof end;
end;

registration
let A be real-membered bounded_above set ;
let r be non positive Real;
coherence
r ** A is bounded_below
proof end;
end;

registration
let A be real-membered bounded_below set ;
let r be non negative Real;
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;
coherence
r ** A is bounded_above
proof end;
end;

theorem Th5: :: MAZURULM:5
for t being Real
for f being Real_Sequence holds f + () = t + f
proof end;

theorem Th6: :: MAZURULM:6
for r being Element of REAL holds lim () = 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 (#) () = f * a
proof end;

theorem Th9: :: MAZURULM:9
for E being RealNormSpace
for a being Point of E holds lim () = 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 ;
coherence
id E is isometric
;
end;

registration
let E be non empty RLSStruct ;
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 ;
then A2: (f /") /" = f by ;
A3: f /" = f " by ;
A4: f /" is bijective by ;
f = (f /") " by ;
hence f . ((f /") . a) = a by ; :: 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;
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 ; :: 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;
coherence 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 () * () = id E
proof end;

registration
let E be RealNormSpace;
let a be Point of E;
coherence
proof end;
end;

theorem Th18: :: MAZURULM:18
for E being RealNormSpace
for a being Point of E holds
( () . a = a & ( for b being Point of E st () . b = b holds
a = b ) )
proof end;

theorem Th19: :: MAZURULM:19
for E being RealNormSpace
for a, b being Point of E holds (() . b) - a = a - b
proof end;

theorem Th20: :: MAZURULM:20
for E being RealNormSpace
for a, b being Point of E holds ||.((() . b) - a).|| = ||.(b - a).||
proof end;

theorem Th21: :: MAZURULM:21
for E being RealNormSpace
for a, b being Point of E holds (() . b) - b = 2 * (a - b)
proof end;

theorem Th22: :: MAZURULM:22
for E being RealNormSpace
for a, b being Point of E holds ||.((() . b) - b).|| = 2 * ||.(b - a).||
proof end;

theorem Th23: :: MAZURULM:23
for E being RealNormSpace
for a being Point of E holds () /" = a -reflection
proof end;

registration
let E be RealNormSpace;
let a be Point of E;
coherence
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 ; then A8: g /" is bijective by ; 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 ; A12: dom g = [#] E by FUNCT_2:def 1; A13: (g /") . b = b by ; A14: (g /") . a = a by ; 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 .= ((((1 / 2) * (a + b)) -reflection) * (g /")) . ((((1 / 2) * (a + b)) -reflection) . a) by FUNCT_2:15 .= ((((1 / 2) * (a + b)) -reflection) * (g /")) . b by .= (((1 / 2) * (a + b)) -reflection) . ((g /") . b) by FUNCT_2:15 .= (2 * ((1 / 2) * (a + b))) - b by .= a by ; ((((((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 .= ((((1 / 2) * (a + b)) -reflection) * (g /")) . ((((1 / 2) * (a + b)) -reflection) . b) by FUNCT_2:15 .= ((((1 / 2) * (a + b)) -reflection) * (g /")) . a by .= (((1 / 2) * (a + b)) -reflection) . ((g /") . a) by FUNCT_2:15 .= (2 * ((1 / 2) * (a + b))) - a by .= b by ; 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 ; (((((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 ; 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 ; 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 ; then reconsider A = l as non empty bounded_above Subset of REAL by ; 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 ; then sup (2 ** A) <= sup A by XXREAL_2:def 3; then (2 * (sup A)) - (sup A) <= (sup A) - (sup A) by ; then ||.((g . ((1 / 2) * (a + b))) - ((1 / 2) * (a + b))).|| = 0 by ; 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 .= (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 ; 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 ; 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 ; 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 ; 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 ; 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 .= ((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 ; 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 ; 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 ;
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
.= t * (f . b) by
.= (0. F) + (t * (f . b))
.= ((1 - t) * (f . a)) + (t * (f . b)) by ;
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
.= ((1 - t) * (f . a)) + (0. F)
.= ((1 - t) * (f . a)) + (t * (f . b)) by ;
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 ;
A16: x / (2 |^ n) = ((2 |^ n) / (2 |^ n)) - (t / (2 |^ n)) by A14
.= 1 - (t / (2 |^ n)) by ;
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
.= ((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 ; :: 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 ;
set w = t - (2 |^ n);
A20: t - (2 |^ n) <= (2 * (2 |^ n)) - (2 |^ n) by ;
A21: (2 |^ n) / (2 |^ n) = 1 by ;
A22: (2 |^ (n + 1)) / (2 |^ n) = (2 * (2 |^ n)) / (1 * (2 |^ n)) by NEWTON:6
.= 2 / 1 by ;
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 ;
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
.= ((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 ; :: 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;