reserve a, b, c, d, x, y, z for Complex;
reserve r for Real;

theorem
  a <> 0 & b <> 0 implies (Im(a.|.b) = |.a.|*|.b.| or Im(a.|.b) = -|.a.|
  *|.b .| iff angle(a,0,b) = PI/2 or angle(a,0,b) = 3/2*PI)
proof
  assume
A1: a<>0 & b<>0;
  hereby
    assume Im(a.|.b)=|.a.|*|.b.| or Im(a.|.b)=-(|.a.|*|.b.|);
    then Re (a.|.b)=0 by Th48;
    hence angle(a,0,b)=PI/2 or angle(a,0,b)=3/2*PI by A1,Th73;
  end;
  hereby
    assume angle(a,0,b)=PI/2 or angle(a,0,b)=3/2*PI;
    then Re (a.|.b)=0 by A1,Th73;
    hence Im(a.|.b)=|.a.|*|.b.| or Im(a.|.b)=-(|.a.|*|.b.|) by Th48;
  end;
end;
