theorem Th1:
  a + b + (-b) = a & a + (-b) + b = a & (-b) + b + a = a & b + (-b) + a = a
  & a + (b + (-b)) = a & a + ((-b) + b) = a & (-b) + (b + a) = a &
  b + ((-b) + a) = a
proof
  thus
A1: a + b + (-b) = a + (b + (-b)) by RLVECT_1:def 3
    .= a + 0_G by Def5
    .= a by Def4;
  thus
A2: a + (-b) + b = a + ((-b) + b) by RLVECT_1:def 3
    .= a + 0_G by Def5
    .= a by Def4;
  thus
A3: (-b) + b + a = 0_G + a by Def5
    .= a by Def4;
  thus b + (-b) + a = 0_G + a by Def5
    .= a by Def4;
  hence thesis by A1,A2,A3,RLVECT_1:def 3;
end;
