reserve F for Field,
  x for Element of F,
  V for VectSp of F,
  v for Element of V;

theorem Th15:
  for F being add-associative right_zeroed right_complementable
non empty addLoopStr, x,y being Element of F holds (x+(-y)=0.F iff x=y) & (x-y
  =0.F iff x=y)
proof
  let F be add-associative right_zeroed right_complementable non empty
  addLoopStr, x,y be Element of F;
  x+(-y)=0.F implies x=y
  proof
    assume x+(-y)=0.F;
    then x+((-y)+y)=0.F+y by RLVECT_1:def 3;
    then x+0.F=0.F+y by RLVECT_1:5;
    then x=0.F+y by RLVECT_1:4;
    hence thesis by RLVECT_1:4;
  end;
  hence thesis by RLVECT_1:5;
end;
