reserve a,b for Complex;
reserve z for Complex;
reserve n0 for non zero Nat;

theorem Th9:
  for r being Real st r > 0 holds n0-root(z/r) = n0-root z/ n0-root r
proof
  let r be Real;
  reconsider r9=1/r as Real;
  reconsider n=n0 as Element of NAT by ORDINAL1:def 12;
A1: n>=0+1 by NAT_1:13;
  assume
A2: r > 0;
  then
A3: |.r.|>0 by COMPLEX1:47;
  z/r = z*(1/r) & Arg(z*r9) = Arg z by A2,COMPLEX2:27,XCMPLX_1:99;
  hence
  n0-root(z/r) = (n -real-root(|.z.|/|.r.|))*(cos((Arg z)/n) + sin((Arg z
  )/n)*<i>) by COMPLEX1:67
    .= n -real-root |.z.|/(n -real-root |.r.|)*(cos((Arg z)/n) + sin((Arg z)
  /n)*<i>) by A3,A1,COMPLEX1:46,POWER:13
    .= (cos((Arg z)/n) + sin((Arg z)/n)*<i>)/ ((n -real-root |.r.|)/(n
  -real-root |.z.|)) by XCMPLX_1:79
    .= n -real-root |.z.|*(cos((Arg z)/n) + sin((Arg z)/n)*<i>)/(n
  -real-root |.r.|) by XCMPLX_1:77
    .= n0-root z/(n -real-root r) by A2,COMPLEX1:43
    .= n0-root z/n0-root r by A2,Th8;
end;
