reserve x for Real;

theorem Th38:
  for x be Real st x < 0 holds Arg (x*<i>) = 3/2*PI
proof
  let x be Real;
A1: 0 <= Arg (0+x*<i>) & Arg (0+x*<i>) < 2*PI by Th34;
  assume
A2: x < 0;
  then
A3: (0+x*<i>) <> 0;
  then
A4: (0+x*<i>) = |. (0+x*<i>) .|*cos Arg (0+x*<i>)+ |. (0+x*<i>) .|*sin Arg (
  0+x*<i>)*<i> by Def1;
  |. (0+x*<i>) .| <> 0 by A3,COMPLEX1:45;
  then cos Arg (0+x*<i>) = 0 by A4,COMPLEX1:77;
  then Arg (0+x*<i>) = 3/2*PI or |. (0+x*<i>) .|*1 = x by A1,A4,Th18,SIN_COS:77
;
  hence thesis by A2,COMPLEX1:46;
end;
