reserve L for non empty addLoopStr;
reserve a,b,c,x for Element of L;

theorem Th7:
  L is AddGroup iff (for a holds a + 0.L = a) & (for a ex x st a+x
  = 0.L) & for a,b,c holds (a+b)+c = a+(b+c)
proof
  thus L is AddGroup implies (for a holds a + 0.L = a) & (for a ex x st a+x =
  0.L) & for a,b,c holds (a+b)+c = a+(b+c) by Th6,RLVECT_1:def 3,def 4;
  assume that
A1: for a holds a + 0.L = a and
A2: for a ex x st a+x = 0.L and
A3: for a,b,c holds (a+b)+c = a+(b+c);
  L is right_complementable
  proof
    let a be Element of L;
    thus ex x st a+x = 0.L by A2;
  end;
  hence thesis by A1,A3,RLVECT_1:def 3,def 4;
end;
