theorem
  for f be linear-transformation of V1,V2 for W1,W2 be Subspace of V1,U1
,U2 be Subspace of V2 st ( dim W1 =0 implies dim U1 = 0 )& ( dim W2 =0 implies
dim U2 = 0 )& V2 is_the_direct_sum_of U1,U2 for fW1 be linear-transformation of
W1,U1, fW2 be linear-transformation of W2,U2 st fW1 = f | W1 & fW2 = f | W2 for
  w1 be OrdBasis of W1, w2 be OrdBasis of W2, u1 be OrdBasis of U1, u2 be
OrdBasis of U2 st w1^w2 = b1 & u1^u2 = b2 holds AutMt(f,b1,b2) = block_diagonal
  (<*AutMt(fW1,w1,u1),AutMt(fW2,w2,u2)*>,0.K)
proof
  let f be linear-transformation of V1,V2;
  let W1,W2 be Subspace of V1,U1,U2 be Subspace of V2 such that
A1: dim W1 =0 implies dim U1 = 0 and
A2: dim W2 =0 implies dim U2 = 0 and
A3: V2 is_the_direct_sum_of U1,U2;
A4: U1/\U2=(0).V2 by A3,VECTSP_5:def 4;
  let fW1 be linear-transformation of W1,U1;
  let fW2 be linear-transformation of W2,U2 such that
A5: fW1=f|W1 and
A6: fW2=f|W2;
  let w1 be OrdBasis of W1,w2 be OrdBasis of W2, u1 be OrdBasis of U1,u2 be
  OrdBasis of U2 such that
A7: w1^w2=b1 and
A8: u1^u2=b2;
A9: len b1=len w1+len w2 by A7,FINSEQ_1:22;
A10: U1 + U2=(Omega).V2 by A3,VECTSP_5:def 4;
  set A=AutMt(f,b1,b2);
A11: len b1=len A by MATRLIN:def 8;
  set A2=AutMt(fW2,w2,u2);
A12: len w2=dim W2 & len u2=dim U2 by Th21;
  then
A13: len w2=len A2 by A2,MATRIX13:1;
  set A1=AutMt(fW1,w1,u1);
A14: len w1=dim W1 & len u1=dim U1 by Th21;
  then
A15: len w1=len A1 by A1,MATRIX13:1;
A16: len u2=width A2 by A2,A12,MATRIX13:1;
A17: len u1=width A1 by A1,A14,MATRIX13:1;
A18: now
    reconsider uu=u1^u2 as OrdBasis of U1+U2 by A4,Th26;
    let i;
A19: dom A=Seg len A by FINSEQ_1:def 3;
    reconsider fb=f.(b1/.i),fbi=f.(b1/.(i+len A1)) as Vector of U1+U2 by A10;
A20: dom A=dom b1 by A11,FINSEQ_3:29;
A21: dom A1=dom w1 by A15,FINSEQ_3:29;
A22: dom fW1=the carrier of W1 by FUNCT_2:def 1;
    thus i in dom A1 implies Line(A,i)=Line(A1,i)^(width A2 |-> 0.K)
    proof
      assume
A23:  i in dom A1;
A24:  dom A1=Seg len A1 by FINSEQ_1:def 3;
      then
A25:  Line(A1,i) = A1.i by A15,A23,MATRIX_0:52
        .= A1/.i by A23,PARTFUN1:def 6
        .= fW1.(w1/.i) |-- u1 by A21,A23,MATRLIN:def 8;
      len A1 <=len A by A9,A15,A11,NAT_1:11;
      then
A26:  Seg len A1 c= Seg len A by FINSEQ_1:5;
      then b1/.i = b1.i by A19,A20,A23,A24,PARTFUN1:def 6
        .= w1.i by A7,A21,A23,FINSEQ_1:def 7
        .= w1/.i by A21,A23,PARTFUN1:def 6;
      then
A27:  fb = fW1.(w1/.i) by A5,A22,FUNCT_1:47;
      thus Line(A,i) = A.i by A11,A23,A24,A26,MATRIX_0:52
        .= A/.i by A19,A23,A24,A26,PARTFUN1:def 6
        .= f.(b1/.i) |--b2 by A19,A20,A23,A24,A26,MATRLIN:def 8
        .= fb|--uu by A10,A8,Th25
        .= (fb+0.(U1+U2)) |-- uu by RLVECT_1:def 4
        .= (fW1.(w1/.i) |-- u1)^(0.U2|--u2) by A4,A27,Th24,VECTSP_4:12
        .= Line(A1,i)^(width A2|->0.K) by A16,A25,Th20;
    end;
A28: dom A2=dom w2 by A13,FINSEQ_3:29;
A29: dom fW2 = the carrier of W2 by FUNCT_2:def 1;
    thus i in dom A2 implies Line(A,i+len A1)=(width A1|->0.K)^Line(A2,i)
    proof
      assume
A30:  i in dom A2;
A31:  dom A2=Seg len A2 by FINSEQ_1:def 3;
      then
A32:  i+len A1 in dom A by A9,A15,A13,A11,A19,A30,FINSEQ_1:60;
      b1/.(i+len A1) = b1.(i+len A1) by A9,A15,A13,A11,A19,A20,A30,A31,
FINSEQ_1:60,PARTFUN1:def 6
        .= w2.i by A7,A15,A28,A30,FINSEQ_1:def 7
        .= w2/.i by A28,A30,PARTFUN1:def 6;
      then
A33:  fbi = fW2.(w2/.i) by A6,A29,FUNCT_1:47;
A34:  Line(A2,i) = A2.i by A13,A30,A31,MATRIX_0:52
        .= A2/.i by A30,PARTFUN1:def 6
        .= fW2.(w2/.i) |-- u2 by A28,A30,MATRLIN:def 8;
      thus Line(A,i+len A1) = A.(i+len A1) by A9,A15,A13,A11,A19,A30,A31,
FINSEQ_1:60,MATRIX_0:52
        .= A/.(i+len A1) by A9,A15,A13,A11,A19,A30,A31,FINSEQ_1:60
,PARTFUN1:def 6
        .= f.(b1/.(i+len A1)) |--b2 by A20,A32,MATRLIN:def 8
        .= fbi|--uu by A10,A8,Th25
        .= (0.(U1+U2)+fbi) |-- uu by RLVECT_1:def 4
        .= (0.U1|--u1)^(fW2.(w2/.i) |-- u2) by A4,A33,Th24,VECTSP_4:12
        .= (width A1|->0.K)^Line(A2,i) by A17,A34,Th20;
    end;
  end;
  set A12=<*A1,A2*>;
A35: Sum Len A12=len A1+len A2 & Sum Width A12=width A1+width A2 by MATRIXJ1:16
,20;
  len b2=len u1+len u2 by A8,FINSEQ_1:22;
  hence thesis by A9,A15,A13,A17,A16,A35,A18,MATRIXJ1:23;
end;
