reserve V for RealLinearSpace;
reserve W,W1,W2,W3 for Subspace of V;
reserve u,u1,u2,v,v1,v2 for VECTOR of V;
reserve a,a1,a2 for Real;
reserve X,Y,x,y,y1,y2 for set;

theorem
  for V being RealLinearSpace, W being Subspace of V, L being
  Linear_Compl of W holds W is Linear_Compl of L
proof
  let V be RealLinearSpace, W be Subspace of V, L be Linear_Compl of W;
  V is_the_direct_sum_of L,W by Def5;
  then V is_the_direct_sum_of W,L by Lm16;
  hence thesis by Def5;
end;
