reserve A,B,C for non empty set,
  f for Function of [:A,B:],C;
reserve K for non empty doubleLoopStr;
reserve V for non empty ModuleStr over K;
reserve W for non empty RightModStr over K;
reserve J for Function of K,K;
reserve K,L for Ring;
reserve J for Function of K,L;
reserve x,y for Scalar of K;
reserve K,L for add-associative right_zeroed right_complementable non empty
  doubleLoopStr;
reserve J for Function of K,L;
reserve K for add-associative right_zeroed right_complementable non empty
  doubleLoopStr;
reserve L for add-associative right_zeroed right_complementable well-unital
  non empty doubleLoopStr;
reserve J for Function of K,L;
reserve K for add-associative right_zeroed right_complementable well-unital
  non empty doubleLoopStr;
reserve J for Function of K,K;
reserve G,H for AddGroup;
reserve f for Homomorphism of G,H;
reserve x,y for Element of G;

theorem
  f.(0.G) = 0.H & f.(-x) = -f.x & f.(x-y) = f.x-f.y by Lm13,Lm14,Lm15;
