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
  for L1,L2 be Scalar of K st f is with_eigenvalues & L1 <> L2 & L2 is
  eigenvalue of f for I be Linear_Compl of UnionKers (f+(-L1)*id V1), fI be
linear-transformation of I,I st fI = f|I holds fI is with_eigenvalues & not L1
  is eigenvalue of fI & L2 is eigenvalue of fI & UnionKers (f+(-L2)*id V1) is
  Subspace of I
proof
  let L1,L2 be Scalar of K such that
A1: f is with_eigenvalues and
A2: L1 <> L2 and
A3: L2 is eigenvalue of f;
  set V=V1;
  consider v be Vector of V such that
A4: v <> 0.V and
A5: f.v = L2 * v by A1,A3,Def2;
  set f1=(f+(-L1)*id V);
  set U=UnionKers f1;
  reconsider fU=f|U as linear-transformation of U,U by Th31;
  set f2=(f+(-L2)*id V);
  let I be Linear_Compl of U;
  let fI be linear-transformation of I,I such that
A6: fI = f|I;
A7: now
    let v be Vector of V;
    assume v in UnionKers f2;
    then consider m such that
A8: (f2|^m).v =0.V by Th24;
    set v1=v|--(I,U);
    set i1=v1`1;
    set u1=v1`2;
A9: V is_the_direct_sum_of I,U by VECTSP_5:def 5;
    then
A10: v=i1 + u1 by VECTSP_5:def 6;
    defpred P[Nat] means (f2|^$1).u1 = 0.V;
    defpred Q[Nat] means for W be Subspace of V st W=I or W=U for w be Vector
    of V st w in W holds (f2|^$1).w in W;
    set L21=L2-L1;
    set f21=f2+L21*id V;
A11: 0.V in I & 0.V in U by VECTSP_4:17;
A12: for n st Q[n] holds Q[n+1]
    proof
      let n such that
A13:  Q[n];
      let W be Subspace of V such that
A14:  W=I or W=U;
      let w be Vector of V such that
A15:  w in W;
      set fnw=(f2|^n).w;
A16:  fnw in W by A13,A14,A15;
A17:  now
        per cases by A14;
        suppose
A18:      W=I;
          then reconsider F=fnw as Vector of I by A16;
          f.fnw = fI.F by A6,FUNCT_1:49;
          hence f.fnw in W by A18;
        end;
        suppose
A19:      W=U;
          then reconsider F=fnw as Vector of U by A16;
          f.fnw = fU.F by FUNCT_1:49;
          hence f.fnw in W by A19;
        end;
      end;
      ((-L2)*id V).fnw = (-L2)*((id V).fnw) by MATRLIN:def 4
        .= (-L2)*fnw;
      then
A20:  ((-L2)*id V).fnw in W by A16,VECTSP_4:21;
A21:  dom (f2|^n) = [#]V by FUNCT_2:def 1;
      (f2|^(n+1)).w = ((f2|^1)*(f2|^n)).w by Th20
        .= (f2|^1).fnw by A21,FUNCT_1:13
        .= (f+(-L2)*id V).fnw by Th19
        .= f.fnw+((-L2)*id V).fnw by MATRLIN:def 3;
      hence thesis by A17,A20,VECTSP_4:20;
    end;
A22: 0.V+0.V = 0.V by RLVECT_1:def 4
      .= (f2|^m).i1 +(f2|^m).u1 by A10,A8,VECTSP_1:def 20;
A23: u1 in U by A9,VECTSP_5:def 6;
    then consider n such that
A24: (f1|^n).u1 =0.V by Th24;
A25: Q[0]
    proof
      let W be Subspace of V such that
      W=I or W=U;
      let w be Vector of V such that
A26:  w in W;
      (f2|^0).w = (id V).w by Th18
        .= w;
      hence thesis by A26;
    end;
A27: for n holds Q[n] from NAT_1:sch 2(A25,A12);
    then
A28: (f2|^m).u1 in U by A23;
A29: i1 in I by A9,VECTSP_5:def 6;
    then (f2|^m).i1 in I by A27;
    then (f2|^m).u1=0.V by A9,A28,A11,A22,VECTSP_5:48;
    then
A30: ex m st P[m];
    consider MIN be Nat such that
A31: P[MIN] and
A32: for n st P[n] holds MIN <= n from NAT_1:sch 5(A30);
    assume
A33: not v is Vector of I;
A34: u1 <>0.V
    by A10,RLVECT_1:def 4,A29,A33;
    n<>0
    proof
      assume n=0;
      then 0.V = (id V).u1 by A24,Th18
        .= u1;
      hence thesis by A34;
    end;
    then consider h be linear-transformation of V,V such that
A35: f21 |^ n = f2 * h + ((L21*id V) |^ n) and
A36: for i holds (f2 |^ i) * h = h * (f2 |^ i) by Th38,NAT_1:14;
A37: dom (f21|^n)=[#]V by FUNCT_2:def 1;
    MIN <>0
    proof
      assume MIN=0;
      then 0.V = (id V).u1 by A31,Th18
        .= u1;
      hence thesis by A34;
    end;
    then reconsider M1=MIN-1 as Element of NAT by NAT_1:20;
A38: (f2|^M1)*(f2 * h) = (f2|^M1)*f2 * h by RELAT_1:36
      .= (f2|^M1)*(f2|^1)*h by Th19
      .= (f2|^(M1+1))*h by Th20
      .= h*(f2|^MIN) by A36;
    dom((L21*id V) |^ n)=[#]V by FUNCT_2:def 1;
    then
A39: ((f2|^M1) * ((L21*id V) |^ n)).u1 = (f2|^M1).(((L21*id V) |^ n).u1)
    by FUNCT_1:13
      .= (f2|^M1).(((power K).(L21,n) * id V).u1) by Lm9
      .= (f2|^M1).((power K).(L21,n)*((id V).u1)) by MATRLIN:def 4
      .= (f2|^M1).( (power K).(L21,n)*u1)
      .= (power K).(L21,n) * ((f2|^M1).u1) by MOD_2:def 2;
A40: (power K).(L21,n) <>0.K
    proof
      assume 0.K=(power K).(L21,n);
      then 0.K = Product (n|->L21) by MATRIXJ1:5;
      then
A41:  ex k be Nat st k in dom (n|->L21) & (n|->L21) .k=0.K by
FVSUM_1:82;
      dom (n|->L21)=Seg n by FINSEQ_2:124;
      then L21=0.K by A41,FINSEQ_2:57;
      hence thesis by A2,VECTSP_1:19;
    end;
    dom (f2|^MIN) =[#]V by FUNCT_2:def 1;
    then
A42: (h*(f2|^MIN)).u1 = h.((f2|^MIN).u1) by FUNCT_1:13
      .= 0.V by A31,RANKNULL:9;
    f21 = f+((-L2)*id V + (L21*id V)) by Lm8
      .= f+((-L2+L21)*id V) by Lm10
      .= f+((-L2+L2-L1)*id V) by RLVECT_1:def 3
      .= f+((0.K+-L1)*id V) by VECTSP_1:19
      .= f1 by RLVECT_1:def 4;
    then 0.V =(f2|^M1).((f21|^n).u1) by A24,RANKNULL:9
      .= ((f2|^M1)*(f2 * h + ((L21*id V) |^ n))).u1 by A35,A37,FUNCT_1:13
      .= (h*(f2|^MIN) + (f2|^M1) * ((L21*id V) |^ n)).u1 by A38,Lm7
      .= 0.V + (power K).(L21,n) * ((f2|^M1).u1) by A42,A39,MATRLIN:def 3
      .= (power K).(L21,n) * ((f2|^M1).u1) by RLVECT_1:def 4;
    then (f2|^M1).u1 = 0.V by A40,VECTSP_1:15;
    then M1+1<=M1 by A32;
    hence contradiction by NAT_1:13;
  end;
  v is eigenvector of f,L2 by A1,A3,A5,Def3;
  then v in ker(f2) by A1,A3,Th17;
  then 0.V = f2.v by RANKNULL:10
    .= (f2|^1).v by Th19;
  then v in UnionKers f2 by Th24;
  then reconsider vI=v as Vector of I by A7;
A43: 0.V = 0.I & L2*v=L2*vI by VECTSP_4:11,14;
A44: f.v=fI.vI by A6,FUNCT_1:49;
  hence
A45: fI is with_eigenvalues by A4,A5,A43;
  not L1 is eigenvalue of fI
  proof
    assume L1 is eigenvalue of fI;
    then consider w be Vector of I such that
A46: w<>0.I and
A47: fI.w = L1 * w by A45,Def2;
    w=0.V by A6,A47,Th37;
    hence thesis by A46,VECTSP_4:11;
  end;
  hence
  not L1 is eigenvalue of fI & L2 is eigenvalue of fI by A4,A5,A43,A44,A45,Def2
;
  the carrier of UnionKers f2 c= the carrier of I
  proof
    let x be object;
    assume x in the carrier of UnionKers f2;
    then
A48: x in UnionKers f2;
    then x in V by VECTSP_4:9;
    then reconsider v=x as Vector of V;
    v is Vector of I by A7,A48;
    hence thesis;
  end;
  hence thesis by VECTSP_4:27;
end;
