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

theorem Th48:
  Re (x.|.y) = 0 iff Im (x.|.y) = |.x.|*|.y.| or Im (x.|.y) = -|.x .|*|.y.|
proof
  hereby
    assume
A1: Re (x.|.y)=0;
    (|.x.|*|.y.|)^2=(|.(x.|.y).|)^2 by Th30
      .=0^2+ (Im (x.|.y))^2 by A1,Lm2
      .= (Im (x.|.y))^2;
    hence Im (x.|.y)=|.x.|*|.y.| or Im (x.|.y)= - (|.x.|*|.y.|) by SQUARE_1:40;
  end;
  hereby
    assume Im (x.|.y)=|.x.|*|.y.| or Im (x.|.y)= - (|.x.|*|.y.|);
    then (Im (x.|.y))^2= (|.x.|*|.y.|)^2 .= (|.(x.|.y).|)^2 by Th30
      .=(Re (x.|.y))^2+(Im (x.|.y))^2 by Lm2;
    hence Re (x.|.y)=0;
  end;
end;
