reserve G,F for RealLinearSpace;

theorem
  for G,F be non empty addLoopStr holds
  ( for x be set holds
  (x is Point of [:G,F:]
  iff ex x1 be Point of G, x2 be Point of F st x=[x1,x2]) )
  & ( for x,y be Point of [:G,F:],
  x1,y1 be Point of G ,x2,y2 be Point of F
  st x=[x1,x2] & y=[y1,y2] holds x+y = [x1+y1,x2+y2] )
  & 0.[:G,F:] = [0.G,0.F] by Lm1,Def1;
