let F be commutative Skew-Field; :: thesis: for G being Subgroup of MultGroup F
for n being Nat st 0 < n holds
ex f being Polynomial of F st
( deg f = n & ( for x, xn being Element of F
for xz being Element of G st x = xz & xn = xz |^ n holds
eval (f,x) = xn - (1. F) ) )
let G be Subgroup of MultGroup F; :: thesis: for n being Nat st 0 < n holds
ex f being Polynomial of F st
( deg f = n & ( for x, xn being Element of F
for xz being Element of G st x = xz & xn = xz |^ n holds
eval (f,x) = xn - (1. F) ) )
let n be Nat; :: thesis: ( 0 < n implies ex f being Polynomial of F st
( deg f = n & ( for x, xn being Element of F
for xz being Element of G st x = xz & xn = xz |^ n holds
eval (f,x) = xn - (1. F) ) ) )
reconsider n0 = n as Element of NAT by ORDINAL1:def 12;
set f = (<%(0. F),(1_ F)%> `^ n0) - (1_. F);

A1: now :: thesis: for x, xn being Element of F
for xz being Element of G st x = xz & xn = xz |^ n holds
eval (((<%(0. F),(1_ F)%> `^ n0) - (1_. F)),x) = xn - (1_ F)

let x, xn be Element of F; :: thesis: for xz being Element of G st x = xz & xn = xz |^ n holds
eval (((<%(0. F),(1_ F)%> `^ n0) - (1_. F)),x) = xn - (1_ F)
let xz be Element of G; :: thesis: ( x = xz & xn = xz |^ n implies eval (((<%(0. F),(1_ F)%> `^ n0) - (1_. F)),x) = xn - (1_ F) )
assume that
A2: x = xz and
A3: xn = xz |^ n ; :: thesis: eval (((<%(0. F),(1_ F)%> `^ n0) - (1_. F)),x) = xn - (1_ F)
the carrier of G c= the carrier of (MultGroup F) by GROUP_2:def 5;
then reconsider xxz = xz as Element of (MultGroup F) ;
A4: xn = xxz |^ n0 by A3, GROUP_4:1
.= (power F) . (x,n0) by A2, UNIROOTS:29 ;
thus eval (((<%(0. F),(1_ F)%> `^ n0) - (1_. F)),x) = (eval ((<%(0. F),(1_ F)%> `^ n0),x)) - (eval ((1_. F),x)) by POLYNOM4:21
.= (eval ((<%(0. F),(1_ F)%> `^ n0),x)) - (1_ F) by POLYNOM4:18
.= ((power F) . ((eval (<%(0. F),(1_ F)%>,x)),n0)) - (1_ F) by POLYNOM5:22
.= xn - (1_ F) by A4, POLYNOM5:48 ; :: thesis: verum

end;

assume 0 < n ; :: thesis: ex f being Polynomial of F st
( deg f = n & ( for x, xn being Element of F
for xz being Element of G st x = xz & xn = xz |^ n holds
eval (f,x) = xn - (1. F) ) )
then A5: 0 + 1 < n + 1 by XREAL_1:8;
len (1_. F) = 1 by POLYNOM4:4;
then A6: len (- (1_. F)) = 1 by POLYNOM4:8;
len <%(0. F),(1_ F)%> = 2 by POLYNOM5:40;
then len (<%(0. F),(1_ F)%> `^ n0) = ((n * 2) - n) + 1 by POLYNOM5:23
.= n + 1 ;
then len ((<%(0. F),(1_ F)%> `^ n0) - (1_. F)) = max ((n + 1),1) by A5, A6, POLYNOM4:7
.= n + 1 by A5, XXREAL_0:def 10 ;
then deg ((<%(0. F),(1_ F)%> `^ n0) - (1_. F)) = n ;
hence ex f being Polynomial of F st
( deg f = n & ( for x, xn being Element of F
for xz being Element of G st x = xz & xn = xz |^ n holds
eval (f,x) = xn - (1. F) ) ) by A1; :: thesis: verum