
theorem Th50:
  for L being non degenerated comRing, x being Element of L holds
  multiplicity(<%-x, 1.L%>,x) = 1
proof
  let L be non degenerated comRing, x be Element of L;
  set r = <%-x, 1.L%>;
  set j = multiplicity(r,x);
  consider F being finite non empty Subset of NAT such that
A1: F = {k where k is Element of NAT : ex q being Polynomial of L st r =
  (r`^k) *' q} and
A2: multiplicity(r,x) = max F by Def7;
  j in F by A2,XXREAL_2:def 8;
  then consider k being Element of NAT such that
A3: k = j and
A4: ex q being Polynomial of L st r = (r`^k) *' q by A1;
  consider q being Polynomial of L such that
A5: r = (r`^k) *' q by A4;
A6: len r = 2 by POLYNOM5:40;
A7: now
    assume len q = 0;
    then q = 0_. L by POLYNOM4:5;
    then r = 0_. L by A5,POLYNOM4:2;
    hence contradiction by A6,POLYNOM4:3;
  end;
  then
A8: q is non-zero by Th14;
A9: now
    assume k > 1;
    then k >= 1+1 by NAT_1:13;
    then k+len q > 2+(0 qua Nat) by A7,XREAL_1:8;
    hence contradiction by A6,A5,A8,Th37;
  end;
  r = (r`^1) by POLYNOM5:16;
  then r = (r`^1) *' 1_. L by POLYNOM3:35;
  then 1 in F by A1;
  then multiplicity(r,x) >= 1 by A2,XXREAL_2:def 8;
  hence thesis by A3,A9,XXREAL_0:1;
end;
