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;

theorem
  J is linear implies J.(0.K) = 0.L & J.(-x) = -J.x & J.(x-y) = J.x-J.y
  by Lm6,Lm7,Lm8;
