reserve a,b,c,h for Integer;
reserve k,m,n for Nat;
reserve i,j,z for Integer;
reserve p for Prime;

theorem Th107:
  for m being Complex holds
  for x,y,z being non zero Complex holds
  x/y + y/z + z/x = m iff x^2*z + y^2*x + z^2*y = m*x*y*z
  proof
    let m be Complex;
    let x,y,z be non zero Complex;
    x/y + y/z = (x*z+y*y)/(y*z) by XCMPLX_1:116;
    then
A1: x/y + y/z + z/x = ((x*z+y*y)*x+z*y*z)/(y*z*x) by XCMPLX_1:116;
    thus x/y + y/z + z/x = m implies x^2*z + y^2*x + z^2*y = m*x*y*z
    proof
      assume x/y + y/z + z/x = m;
      then ((x*z+y*y)*x+z*y*z)/(x*y*z)*(x*y*z) = m*(x*y*z) by A1;
      hence thesis by XCMPLX_1:87;
    end;
    assume x^2*z + y^2*x + z^2*y = m*x*y*z;
    then (x^2*z + y^2*x + z^2*y)/(x*y*z) = m*(x*y*z)/(x*y*z);
    hence thesis by A1,XCMPLX_1:89;
  end;
