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 Th51:
  for GF being Field, V being VectSp of GF, a being Element of GF,
  v being Element of V, W being Subspace of V st a <> 0.GF & (a * v) + W = the
  carrier of W holds v in W
proof
  let GF be Field, V be VectSp of GF, a be Element of GF, v be Element of V, W
  be Subspace of V;
  assume that
A1: a <> 0.GF and
A2: (a * v) + W = the carrier of W;
  assume not v in W;
  then not 1_GF * v in W;
  then not (a" * a) * v in W by A1,VECTSP_1:def 10;
  then not a" * (a * v) in W by VECTSP_1:def 16;
  then
A3: not a * v in W by Th21;
  0.V in W & a * v + 0.V = a * v by Th17,RLVECT_1:4;
  then a * v in {a * v + u where u is Vector of V : u in W};
  hence contradiction by A2,A3;
end;
