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 Th36:
  for I be Linear_Compl of UnionKers f holds f|I is one-to-one
proof
  let I be Linear_Compl of UnionKers f;
  set fI=f|I;
  set U=UnionKers f;
  the carrier of ker fI c= {0.I}
  proof
    let x be object;
    assume x in the carrier of ker fI;
    then
A1: x in ker fI;
    then
A2: x in I by VECTSP_4:9;
    then reconsider v=x as Vector of I;
    reconsider w=v as Vector of V1 by VECTSP_4:10;
    0.I = 0.V1 by VECTSP_4:11
      .= (f|I).v by A1,RANKNULL:10
      .= f.v by FUNCT_1:49;
    then (f|^1).w = 0.I by Th19
      .= 0.V1 by VECTSP_4:11;
    then v in U by Th24;
    then U/\I = (0).V1 & v in U/\I by A2,VECTSP_5:3,40;
    then v in the carrier of (0).V1;
    then v in {0.V1} by VECTSP_4:def 3;
    then v = 0.V1 by TARSKI:def 1
      .= 0.I by VECTSP_4:11;
    hence thesis by TARSKI:def 1;
  end;
  then the carrier of ker fI = {0.I} by ZFMISC_1:33
    .= the carrier of (0).I by VECTSP_4:def 3;
  then ker fI = (0).I by VECTSP_4:29;
  hence thesis by MATRLIN2:43;
end;
