reserve x for Real;

theorem Th36:
  for x be Real st x < 0 holds Arg x = PI
proof
  let x be Real;
A1: 0 <= Arg (x+0*<i>) & Arg (x+0*<i>) < 2*PI by Th34;
  assume
A2: x < 0;
  then
A3: (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 A2,COMPLEX1:45;
  then sin Arg (x+0*<i>) = 0 by A3,COMPLEX1:77;
  then Arg (x+0*<i>) = PI or |. (x+0*<i>) .|*1 = x by A1,A3,Th17,SIN_COS:31;
  hence thesis by A2,COMPLEX1:46;
end;
