reserve i,j,m,n,k for Nat,
  x,y for set,
  K for Field,
  a for Element of K;
reserve V for non trivial VectSp of K,
  V1,V2 for VectSp of K,
  f for linear-transformation of V1,V1,
  v,w for Vector of V,
  v1 for Vector of V1,
  L for Scalar of K;
reserve S for 1-sorted,
  F for Function of S,S;

theorem Th37:
  for I be Linear_Compl of UnionKers (f+(-L)*id V1), fI be
  linear-transformation of I,I st fI = f|I for v be Vector of I st fI.v = L * v
  holds v = 0.V1
proof
  set V=V1;
  set fi=f+(-L)*id V;
  let I be Linear_Compl of UnionKers fi, fI be linear-transformation of I,I
  such that
A1: fI = f|I;
  let v be Vector of I such that
A2: fI.v = L * v;
  reconsider v1=v as Vector of V by VECTSP_4:10;
A3: f.v = fI.v by A1,FUNCT_1:49
    .= L*v1 by A2,VECTSP_4:14;
  (fi|I).v1 = fi.v1 by FUNCT_1:49
    .= f.v1+((-L)*id V).v1 by MATRLIN:def 3
    .= f.v1 +(-L)*(id V.v1) by MATRLIN:def 4
    .= L*v1 +(-L)*v1 by A3
    .= (L+(-L))*v1 by VECTSP_1:def 15
    .= 0.K*v1 by VECTSP_1:19
    .= 0.V by VECTSP_1:14;
  then
A4: v1 in ker (fi|I) by RANKNULL:10;
  fi|I is one-to-one by Th36;
  then ker (fi|I)=(0).I by MATRLIN2:43;
  hence v = 0.I by A4,VECTSP_4:35
    .= 0.V by VECTSP_4:11;
end;
