theorem Th120:
  H + a = H + b iff b + -a in H
proof
  thus H + a = H + b implies b + -a in H
  proof
    assume
A1: H + a = H + b;
    carr(H) = H + 0_G by Th37
      .= H + (a + -a) by Def5
      .= H + b + -a by A1,ThB34
      .= H + (b + -a) by ThB34;
    hence thesis by Th119;
  end;
  assume b + -a in H;
  hence H + a = H + (b + -a) + a by Th119
    .= H + (b + -a + a) by ThB34
    .= H + (b + (-a + a)) by RLVECT_1:def 3
    .= H + (b + (0_G)) by Def5
    .= H + b by Def4;
end;
