reserve x for Real;

theorem Th37:
  for x be Real st x > 0 holds Arg (x*<i>) = PI/2
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>) = PI/2 or |. (0+x*<i>) .|*-1 = x by A1,A4,Th18,SIN_COS:77;
  then Arg (0+x*<i>) = PI/2 or |. (0+x*<i>) .| < 0 by A2;
  hence thesis by COMPLEX1:46;
end;
