theorem
  b <> 0 & (for z holds Polynom(0,b,c,z) = 0) implies z = -c/b
proof
A1: 0 = 0+0*<i>;
  assume
A2: b <> 0;
  assume
A3: for z holds Polynom(0,b,c,z) = 0;
  now
    let z1;
    Polynom(0,b,c,z1) = 0 by A3;
    then b*(Re z1+(Im z1)*<i>)+c = 0 by COMPLEX1:13;
    then
A4: (b*Re z1-0+c+(b*Im z1+0)*<i>)= 0+0*<i>;
    then b*Re z1-0+c = Re 0 by COMPLEX1:12;
    then b*(Re z1)-0+c = 0 by A1,COMPLEX1:12;
    then
A5: Re z1 =(-c)/b by A2,XCMPLX_1:89;
    b*Im z1= Im 0 by A4,COMPLEX1:12;
    then Im z1 = 0 by A1,A2,COMPLEX1:12;
    then z1 = -c/b+0*<i> by A5,COMPLEX1:13;
    hence z1 = -c/b;
  end;
  hence thesis;
end;
