let L be non degenerated comRing; :: thesis: for x being Element of L holds multiplicity <%(- x),(1. L)%>,x = 1
let x be Element of L; :: thesis: multiplicity <%(- x),(1. L)%>,x = 1
set r = <%(- x),(1. L)%>;
set j = multiplicity <%(- x),(1. L)%>,x;
consider F being non empty finite Subset of NAT such that
A1: F = { k where k is Element of NAT : ex q being Polynomial of L st <%(- x),(1. L)%> = (<%(- x),(1. L)%> `^ k) *' q } and
A2: multiplicity <%(- x),(1. L)%>,x = max F by Def8;
multiplicity <%(- x),(1. L)%>,x in F by A2, XXREAL_2:def 8;
then consider k being Element of NAT such that
A3: k = multiplicity <%(- x),(1. L)%>,x and
A4: ex q being Polynomial of L st <%(- x),(1. L)%> = (<%(- x),(1. L)%> `^ k) *' q by A1;
consider q being Polynomial of L such that
A5: <%(- x),(1. L)%> = (<%(- x),(1. L)%> `^ k) *' q by A4;
A6: len <%(- x),(1. L)%> = 2 by POLYNOM5:41;

A7: now

assume len q = 0 ; :: thesis: contradiction
then q = 0_. L by POLYNOM4:8;
then <%(- x),(1. L)%> = 0_. L by A5, POLYNOM4:5;
hence contradiction by A6, POLYNOM4:6; :: thesis: verum

end;

then A8: q is non-zero by Th19;

A9: now

assume k > 1 ; :: thesis: contradiction
then k >= 1 + 1 by NAT_1:13;
then k + (len q) > 2 + 0 by A7, XREAL_1:10;
hence contradiction by A6, A5, A8, Th42; :: thesis: verum

end;

<%(- x),(1. L)%> = <%(- x),(1. L)%> `^ 1 by POLYNOM5:17;
then <%(- x),(1. L)%> = (<%(- x),(1. L)%> `^ 1) *' (1_. L) by POLYNOM3:36;
then 1 in F by A1;
then multiplicity <%(- x),(1. L)%>,x >= 1 by A2, XXREAL_2:def 8;
hence multiplicity <%(- x),(1. L)%>,x = 1 by A3, A9, XXREAL_0:1; :: thesis: verum