theorem Th29:
  i <= 0 implies i * h = -( |.i.| * h)
proof
  assume
A1: i <= 0;
  per cases by A1;
  suppose
    i < 0;
    hence thesis by Def8;
  end;
  suppose
A2: i = 0;
    hence i * h = 0_G by Lm3
      .= -(0_G) by Th8
      .= -( 0 * h) by Def7
      .= -( |.i.| * h) by A2,ABSVALUE:def 1;
  end;
end;
