
theorem Th9:
  for L being Field
  for a,x being Element of L
  for b being non zero Element of L holds
  x is_a_root_of <%a,b%> iff x = -a/b
  proof
    let L be Field;
    let a,x be Element of L;
    let b be non zero Element of L;
    set p = <%a,b%>;
    hereby
      assume
A1:   x is_a_root_of p;
      b*(/b) = (/b)*b;
      then
A2:   b*x/b = x by Th4;
      a+b*x = 0.L by A1,POLYNOM5:44;
      then b*x = -a by RLVECT_1:6;
      hence x = -a/b by A2,VECTSP_1:9;
    end;
    assume x = -a/b;
    hence eval(p,x) = 0.L by Th8;
  end;
