reserve x for Real;

theorem Th35:
  for x be Real st x >= 0 holds Arg x = 0
proof
  let x be Real;
A1: 0 <= Arg (x+0*<i>) & Arg (x+0*<i>) < 2*PI by Th34;
  assume
A2: x >= 0;
  per cases by A2;
  suppose
A3: x > -0;
    then
A4: (x+0*<i>) = |. (x+0*<i>) .|*cos Arg (x+0*<i>)+ |. (x+0*<i>) .|*sin Arg
    (x+0*<i>)*<i> by Def1;
    |. (x+0*<i>) .| <> 0 by A3,COMPLEX1:45;
    then sin Arg (x+0*<i>) = 0 by A4,COMPLEX1:77;
    then Arg (x+0*<i>) = 0 or |. (x+0*<i>) .|*-1 = x by A1,A4,Th17,SIN_COS:77;
    then Arg (x+0*<i>) = 0 or |. (x+0*<i>) .| < 0 by A3;
    hence thesis by COMPLEX1:46;
  end;
  suppose
    x+0*<i> = 0;
    hence thesis by Def1;
  end;
end;
