 reserve R for Ring;
 reserve x, y, y1 for set;
 reserve a, b for Element of R;
 reserve V for LeftMod of R;
 reserve v, w for Vector of V;
 reserve u,v,w for Vector of V;
 reserve F,G,H,I for FinSequence of V;
 reserve j,k,n for Nat;
 reserve f,f9,g for sequence of V;
 reserve R for Ring;
 reserve V, X, Y for LeftMod of R;
 reserve u, u1, u2, v, v1, v2 for Vector of V;
 reserve a for Element of R;
 reserve V1, V2, V3 for Subset of V;
 reserve x for set;
 reserve W, W1, W2 for Submodule of V;
 reserve w, w1, w2 for Vector of W;
 reserve D for non empty set;
 reserve d1 for Element of D;
 reserve A for BinOp of D;
 reserve M for Function of [:the carrier of R,D:],D;
reserve B,C for Coset of W;
 reserve V for LeftMod of R;
 reserve W, W1, W2, W3 for Submodule of V;
 reserve u, u1, u2, v, v1, v2 for Vector of V;
 reserve a, a1, a2 for Element of R;
 reserve X, Y, y, y1, y2 for set;

theorem
  for V being Z_Module, W being with_Linear_Compl Submodule of V,
  L being Linear_Compl of W holds W is Linear_Compl of L
  proof
    let V be Z_Module, W be with_Linear_Compl Submodule of V,
    L be Linear_Compl of W;
    V is_the_direct_sum_of L,W by Def19; then
A1: V is_the_direct_sum_of W,L by VECTSP_5:41;
    then L is with_Linear_Compl;
    hence thesis by Def19,A1;
  end;
