
theorem Th11:
  for L being Field
  for a being Element of L
  for b being non zero Element of L holds
  multiplicity(<%a,b%>,-a/b) = 1
  proof
    let L be Field;
    let a be Element of L;
    let b be non zero Element of L;
    set p = <%a,b%>;
    set x = -a/b;
    set r = <%-x,1.L%>;
    set j = multiplicity(p,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
    p = (r`^k) *' q} and
A2: j = max F by UPROOTS:def 8;
    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 p = (r`^k) *' q by A1;
    consider q being Polynomial of L such that
A5: p = (r`^k) *' q by A4;
    b <> 0.L;
    then
A6: len p = 2 by POLYNOM5:40;
A7: now
      assume len q = 0;
      then q = 0_. L by POLYNOM4:5;
      then p = 0_. L by A5,POLYNOM4:2;
      hence contradiction by A6,POLYNOM4:3;
    end;
    then
A8: q is non-zero by UPROOTS:17;
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,UPROOTS:40;
    end;
    j >= 1 by Th9,UPROOTS:52;
    then k = 1 by A3,A9,XXREAL_0:1;
    then 1 in F by A1,A5;
    then j >= 1 by A2,XXREAL_2:def 8;
    hence thesis by A3,A9,XXREAL_0:1;
  end;
