reserve x,y,y1,y2 for object;
reserve GF for add-associative right_zeroed right_complementable Abelian
  associative well-unital distributive non empty doubleLoopStr,
  V,X,Y for Abelian add-associative right_zeroed right_complementable
  vector-distributive scalar-distributive scalar-associative scalar-unital non
  empty ModuleStr over GF;
reserve a for Element of GF;
reserve u,u1,u2,v,v1,v2 for Element of V;
reserve W,W1,W2 for Subspace of V;
reserve V1 for Subset of V;
reserve w,w1,w2 for Element of W;
reserve B,C for Coset of W;

theorem
  for GF be add-associative right_zeroed right_complementable Abelian
  commutative associative well-unital distributive non empty doubleLoopStr, V
  be Abelian add-associative right_zeroed right_complementable
  vector-distributive scalar-distributive scalar-associative scalar-unital non
empty ModuleStr over GF, a,b being Element of GF, v being Element of V holds (
  a - b) * v = a * v - b * v by Lm1;
