theorem Th19:
  for M be Matrix of len b1,len B2,K, M1 be Matrix of len bw1,len
Bu1,K, M2 be Matrix of len bw2,len Bu2,K st b1 = bw1 ^ bw2 & B2 = Bu1 ^ Bu2 & M
= block_diagonal(<*M1,M2*>,0.K) & width M1 = len Bu1 & width M2 = len Bu2 holds
(i in dom bw1 implies Mx2Tran(M,b1,B2).(b1/.i) = Mx2Tran(M1,bw1,Bu1).(bw1/.i))&
(i in dom bw2 implies Mx2Tran(M,b1,B2).(b1/.(i+len bw1)) = Mx2Tran(M2,bw2,Bu2).
  (bw2/.i))
proof
  let M be Matrix of len b1,len B2,K, M1 be Matrix of len bw1,len Bu1,K, M2 be
  Matrix of len bw2,len Bu2,K such that
A1: b1 = bw1 ^ bw2 and
A2: B2 = Bu1 ^ Bu2 and
A3: M = block_diagonal(<*M1,M2*>,0.K) and
A4: width M1 = len Bu1 and
A5: width M2 = len Bu2;
A6: dom bw1 c= dom b1 by A1,FINSEQ_1:26;
  rng Bu2 c= the carrier of U2 & the carrier of U2 c= the carrier of V2 by
FINSEQ_1:def 4,VECTSP_4:def 2;
  then
A7: rng Bu2 c= the carrier of V2;
  rng Bu1 c= the carrier of U1 & the carrier of U1 c= the carrier of V2 by
FINSEQ_1:def 4,VECTSP_4:def 2;
  then rng Bu1 c= the carrier of V2;
  then reconsider bu1=Bu1,bu2=Bu2 as FinSequence of V2 by A7,FINSEQ_1:def 4;
  set F1=Mx2Tran(M1,bw1,Bu1);
  set F=Mx2Tran(M,b1,B2);
A8: dom b1=Seg len b1 & len 1.(K,len b1)=len b1 by FINSEQ_1:def 3,MATRIX_0:24;
A9: dom 1.(K,len bw1)=Seg len 1.(K,len bw1) by FINSEQ_1:def 3;
A10: dom 1.(K,len b1)=Seg len 1.(K,len b1) by FINSEQ_1:def 3;
  set BI=len bw1+i;
A11: dom bw2=Seg len bw2 & len 1.(K,len bw2)=len bw2 by FINSEQ_1:def 3
,MATRIX_0:24;
  set F2=Mx2Tran(M2,bw2,Bu2);
A12: width 1.(K,len b1)=len b1 by MATRIX_0:24;
A13: len Line(M2,i)=width M2 & len (width M1|->0.K)=width M1 by CARD_1:def 7;
A14: dom bw1=Seg len bw1 & len 1.(K,len bw1)=len bw1 by FINSEQ_1:def 3
,MATRIX_0:24;
A15: len M=len b1 by MATRIX_0:def 2;
A16: len M1=len bw1 by MATRIX_0:def 2;
  then
A17: dom M1=dom bw1 by FINSEQ_3:29;
  thus i in dom bw1 implies F.(b1/.i)= F1.(bw1/.i)
  proof
A18: len Line(M1,i)=width M1 & len (width M2|->0.K)=width M2 by CARD_1:def 7;
    assume
A19: i in dom bw1;
    then
A20: Line(M,i)=Line(M1,i)^(width M2|->0.K) by A3,A17,MATRIXJ1:23;
    thus F.(b1/.i) = Sum lmlt (Line(LineVec2Mx(b1/.i|--b1) * M,1),B2) by
MATRLIN2:def 3
      .= Sum lmlt (Line(LineVec2Mx(Line(1.(K,len b1),i)) * M,1),B2) by A6,A19,
MATRLIN2:19
      .= Sum lmlt (Line(LineVec2Mx(Line(1.(K,len b1)*M,i)),1),B2) by A6,A8,A15
,A10,A19,MATRIX_0:24,MATRLIN2:35
      .= Sum lmlt (Line(LineVec2Mx(Line(M,i)),1),B2) by A15,MATRIXR2:68
      .= Sum lmlt(Line(M,i),B2) by MATRIX15:25
      .= Sum (lmlt(Line(M1,i),bu1)^lmlt(width M2|->0.K,bu2)) by A2,A4,A5,A20
,A18,MATRLIN2:9
      .= Sum lmlt(Line(M1,i),bu1) + Sum lmlt(width M2|->0.K,bu2) by RLVECT_1:41
      .= Sum lmlt(Line(M1,i),bu1) + 0.K*Sum bu2 by A5,MATRLIN2:11
      .= Sum lmlt(Line(M1,i),bu1) + 0.V2 by VECTSP_1:14
      .= Sum lmlt(Line(M1,i),bu1) by RLVECT_1:def 4
      .= Sum lmlt(Line(M1,i),Bu1) by MATRLIN2:14,15
      .= Sum lmlt (Line(LineVec2Mx(Line(M1,i)),1),Bu1) by MATRIX15:25
      .= Sum lmlt (Line(LineVec2Mx(Line(1.(K,len bw1)*M1,i)),1),Bu1) by A16,
MATRIXR2:68
      .= Sum lmlt (Line(LineVec2Mx(Line(1.(K,len bw1),i)) * M1,1),Bu1) by A14
,A9,A16,A19,MATRIX_0:24,MATRLIN2:35
      .= Sum lmlt (Line(LineVec2Mx(bw1/.i|--bw1) * M1,1),Bu1) by A19,
MATRLIN2:19
      .= F1.(bw1/.i) by MATRLIN2:def 3;
  end;
A21: dom 1.(K,len bw2)=Seg len 1.(K,len bw2) by FINSEQ_1:def 3;
  assume
A22: i in dom bw2;
  then
A23: BI in dom b1 by A1,FINSEQ_1:28;
A24: len M2=len bw2 by MATRIX_0:def 2;
  then dom M2=dom bw2 by FINSEQ_3:29;
  then
A25: Line(M,BI)=(width M1|->0.K)^Line(M2,i) by A3,A16,A22,MATRIXJ1:23;
  thus F.(b1/.(i+len bw1)) = Sum lmlt (Line(LineVec2Mx(b1/.BI|--b1) * M,1),B2)
  by MATRLIN2:def 3
    .= Sum lmlt (Line(LineVec2Mx(Line(1.(K,len b1),BI)) * M,1),B2) by A23,
MATRLIN2:19
    .= Sum lmlt (Line(LineVec2Mx(Line(1.(K,len b1)*M,BI)),1),B2) by A1,A12,A8
,A15,A10,A22,FINSEQ_1:28,MATRLIN2:35
    .= Sum lmlt (Line(LineVec2Mx(Line(M,BI)),1),B2) by A15,MATRIXR2:68
    .= Sum lmlt(Line(M,BI),B2) by MATRIX15:25
    .= Sum (lmlt(width M1|->0.K,bu1)^lmlt(Line(M2,i),bu2)) by A2,A4,A5,A25,A13,
MATRLIN2:9
    .= Sum lmlt(width M1|->0.K,bu1) + Sum lmlt(Line(M2,i),bu2) by RLVECT_1:41
    .= 0.K*Sum bu1 + Sum lmlt(Line(M2,i),bu2) by A4,MATRLIN2:11
    .= 0.V2 + Sum lmlt(Line(M2,i),bu2) by VECTSP_1:14
    .= Sum lmlt(Line(M2,i),bu2) by RLVECT_1:def 4
    .= Sum lmlt(Line(M2,i),Bu2) by MATRLIN2:14,15
    .= Sum lmlt (Line(LineVec2Mx(Line(M2,i)),1),Bu2) by MATRIX15:25
    .= Sum lmlt (Line(LineVec2Mx(Line(1.(K,len bw2)*M2,i)),1),Bu2) by A24,
MATRIXR2:68
    .= Sum lmlt (Line(LineVec2Mx(Line(1.(K,len bw2),i)) * M2,1),Bu2) by A11,A21
,A24,A22,MATRIX_0:24,MATRLIN2:35
    .= Sum lmlt (Line(LineVec2Mx(bw2/.i|--bw2) * M2,1),Bu2) by A22,MATRLIN2:19
    .= F2.(bw2/.i) by MATRLIN2:def 3;
end;
