let L be non empty non degenerated right_complementable almost_left_invertible well-unital distributive add-associative right_zeroed associative commutative doubleLoopStr ; :: thesis: for m being Element of NAT
for x being Element of L st x is_primitive_root_of_degree m holds
for i, j being Nat st 1 <= i & i <= m & 1 <= j & j <= m & i <> j holds
pow (x,(i - j)) <> 1. L
let m be Element of NAT ; :: thesis: for x being Element of L st x is_primitive_root_of_degree m holds
for i, j being Nat st 1 <= i & i <= m & 1 <= j & j <= m & i <> j holds
pow (x,(i - j)) <> 1. L
let x be Element of L; :: thesis: ( x is_primitive_root_of_degree m implies for i, j being Nat st 1 <= i & i <= m & 1 <= j & j <= m & i <> j holds
pow (x,(i - j)) <> 1. L )
assume A1: x is_primitive_root_of_degree m ; :: thesis: for i, j being Nat st 1 <= i & i <= m & 1 <= j & j <= m & i <> j holds
pow (x,(i - j)) <> 1. L
then A2: x <> 0. L by Th30;
let i, j be Nat; :: thesis: ( 1 <= i & i <= m & 1 <= j & j <= m & i <> j implies pow (x,(i - j)) <> 1. L )
assume that
A3: 1 <= i and
A4: i <= m and
A5: 1 <= j and
A6: j <= m and
A7: i <> j ; :: thesis: pow (x,(i - j)) <> 1. L

per cases ( i > j or i <= j ) ;

suppose A8: i > j ; :: thesis: pow (x,(i - j)) <> 1. L

then reconsider k = i - j as Element of NAT by INT_1:5;
k <= i - 1 by A5, XREAL_1:13;
then k + 1 <= (i - 1) + 1 by XREAL_1:6;
then k < i by NAT_1:13;
then A9: k < m by A4, XXREAL_0:2;
i - j > j - j by A8, XREAL_1:14;
then x |^ k <> 1. L by A1, A9;
hence pow (x,(i - j)) <> 1. L by Def2; :: thesis: verum

end;

suppose i <= j ; :: thesis: pow (x,(i - j)) <> 1. L

then A10: i < j by A7, XXREAL_0:1;
then A11: j - i > i - i by XREAL_1:14;
A12: i - j < j - j by A10, XREAL_1:14;
then A13: |.(i - j).| = - (i - j) by ABSVALUE:def 1;
then reconsider k = - (i - j) as Element of NAT ;
j - i <= j - 1 by A3, XREAL_1:13;
then k + 1 <= (j - 1) + 1 by XREAL_1:6;
then k < j by NAT_1:13;
then A14: k < m by A6, XXREAL_0:2;
A15: x |^ k <> 0. L by A2, Th1;

now :: thesis: not (x |^ k) " = 1. L

assume (x |^ k) " = 1. L ; :: thesis: contradiction
then 1. L = (x |^ k) * (1. L) by A15, VECTSP_1:def 10
.= x |^ k ;
hence contradiction by A1, A11, A14; :: thesis: verum

end;

hence pow (x,(i - j)) <> 1. L by A12, A13, Def2; :: thesis: verum

end;

end;