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 Th65:
  v + W = u + W implies ex v1 st v1 in W & v - v1 = u
proof
  assume v + W = u + W;
  then u in v + W by Th44;
  then consider u1 such that
A1: u = v + u1 and
A2: u1 in W;
  take v1 = v - u;
  0.V = (v + u1) - u by A1,VECTSP_1:19
    .= u1 + (v - u) by RLVECT_1:def 3;
  then v1 = - u1 by VECTSP_1:16;
  hence v1 in W by A2,Th22;
  thus v - v1 = (v - v) + u by RLVECT_1:29
    .= 0.V + u by VECTSP_1:19
    .= u by RLVECT_1:4;
end;
