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;
reserve G,H for AbGroup;
reserve f for Homomorphism of G,H;
reserve x,y for Element of G;
reserve K,L for Ring;
reserve J for Function of K,L;
reserve V for LeftMod of K;
reserve W for LeftMod of L;

theorem
  ZeroMap(V,W) is Homomorphism of J,V,W
proof
  set z = ZeroMap(V,W);
  ( for x,y being Vector of V holds z.(x+y) = z.x+z.y)& for a being Scalar
  of K, x being Vector of V holds z.(a*x) = J.a*z.x by Lm16,Lm17;
  hence thesis by Def17;
end;
