theorem Th20:
  for M be Matrix of len b1,len B2,K, F be FinSequence_of_Matrix
  of K st M = block_diagonal(F,0.K) for i,m st i in dom b1 & m = min(Len F,i)
holds Mx2Tran(M,b1,B2).(b1/.i) = Sum lmlt(Line(F.m,i-'Sum (Len F| (m-'1))),
(B2|Sum(Width F|m))/^Sum (Width F| (m-'1))) & len ((B2|Sum(Width F|m))/^
Sum (Width F| (m-'1))) = width (F.m)
proof
  set ONE=1.(K,len b1);
  let M be Matrix of len b1,len B2,K, F be FinSequence_of_Matrix of K such
  that
A1: M = block_diagonal(F,0.K);
A2: width M=Sum Width F by A1,MATRIXJ1:def 5;
  len ONE=len b1 by MATRIX_0:def 2;
  then
A3: dom ONE=dom b1 by FINSEQ_3:29;
  set L=Len F;
  set W=Width F;
  let i,m such that
A4: i in dom b1 and
A5: m = min(Len F,i);
A6: 1<=i & i<=len b1 by A4,FINSEQ_3:25;
  then
A7: len M=len b1 by MATRIX_0:23;
  set Fm=F.m;
  set Wm1=Sum (W| (m-'1));
A8: len (Sum (W| (m-'1)) |->0.K)=Sum (W| (m-'1)) & len Line(Fm,i-'Sum (Len F| (
  m-'1) )) =width Fm by CARD_1:def 7;
  set Wm=Sum (W|m);
  W = (W|m)^(W/^m) by RFINSEQ:8;
  then
A9: Sum W=Wm+Sum (W/^m) by RVSUM_1:75;
  then
A10: Sum W-'Wm = Sum W-Wm by NAT_1:11,XREAL_1:233
    .= Sum (W/^m) by A9;
  then
A11: len ((Sum W-'Sum (W|m)) |->0.K)=Sum (W/^m) by CARD_1:def 7;
A12: dom b1=Seg len b1 & len M=Sum Len F by A1,FINSEQ_1:def 3,MATRIXJ1:def 5;
  then
A13: m in dom Len F by A4,A5,A7,MATRIXJ1:def 1;
  then
A14: 1<=m by FINSEQ_3:25;
  reconsider wF= width Fm as Element of NAT by ORDINAL1:def 12;
  dom L=dom F & dom W=dom F by MATRIXJ1:def 3,def 4;
  then m<=len W & W.m=width Fm by A13,FINSEQ_3:25,MATRIXJ1:def 4;
  then W = (W| (m-'1))^<*width Fm*>^(W/^m) by A5,A14,POLYNOM4:1;
  then
A15: Sum W = Sum ((W| (m-'1))^<*wF*>)+Sum (W/^m) by RVSUM_1:75
    .= Wm1 + width Fm+Sum (W/^m) by RVSUM_1:74;
  then
A16: len ((Sum (W| (m-'1)) |->0.K)^Line(Fm,i-'Sum (Len F| (m-'1))))=Wm by A9,
CARD_1:def 7;
  set B2W=B2|Wm;
A17: B2W/^Wm1 = (B2|Sum(Width F|m))/^Wm1 by MATRIXJ1:21
    .= (B2|Sum(Width F|m))/^Sum (Width F| (m-'1)) by MATRIXJ1:21;
A18: width M=len B2 by A6,MATRIX_0:23;
  then
A19: len B2W =Wm by A2,A9,FINSEQ_1:59,NAT_1:11;
  Sum W=Wm1+(width Fm+Sum (W/^m)) by A15;
  then
A20: len (B2|Wm1) =Wm1 by A2,A18,FINSEQ_1:59,NAT_1:11;
  Wm>=Wm1 by A9,A15,NAT_1:11;
  then Seg Wm1 c= Seg Wm by FINSEQ_1:5;
  then B2W|Wm1=B2|Wm1 by RELAT_1:74;
  then
A21: B2W=(B2|Wm1)^(B2W/^Wm1) by RFINSEQ:8;
  then
A22: len B2W=len (B2|Wm1)+len (B2W/^Wm1) by FINSEQ_1:22;
A23: B2=B2W^(B2/^Wm) by RFINSEQ:8;
  then
A24: len B2=len B2W+ len (B2/^Wm) by FINSEQ_1:22;
  Mx2Tran(M,b1,B2).(b1/.i) = Sum lmlt (Line(LineVec2Mx(b1/.i|--b1) * M,1)
  ,B2) by MATRLIN2:def 3
    .= Sum lmlt (Line(LineVec2Mx(Line(ONE,i)) * M,1),B2) by A4,MATRLIN2:19
    .= Sum lmlt (Line(LineVec2Mx(Line(ONE*M,i)),1),B2) by A4,A6,A3,A7,
MATRIX_0:23,MATRLIN2:35
    .= Sum lmlt (Line(LineVec2Mx(Line(M,i)),1),B2) by A7,MATRIXR2:68
    .= Sum lmlt (Line(M,i),B2) by MATRIX15:25
    .= Sum lmlt((Sum (Width F| (m-'1)) |->0.K)^Line(Fm,i-'Sum (Len F| (m-'1)))^
  ((Sum W-'Sum (Width F|m)) |->0.K),B2) by A1,A4,A5,A7,A12,MATRIXJ1:37
    .= Sum lmlt((Sum (W| (m-'1)) |->0.K) ^ Line(Fm,i-'Sum (Len F| (m-'1)))^ ((
  Sum W-'Sum (Width F|m)) |->0.K),B2) by MATRIXJ1:21
    .= Sum lmlt((Sum (W| (m-'1)) |->0.K) ^ Line(Fm,i-'Sum (Len F| (m-'1)))^ ((
  Sum W-'Sum (W|m)) |->0.K),B2) by MATRIXJ1:21
    .= Sum (lmlt((Sum (W| (m-'1)) |->0.K) ^ Line(Fm,i-'Sum (Len F| (m-'1))),B2W
 )^ lmlt(((Sum W-'Sum (W|m)) |->0.K),B2/^Wm)) by A2,A18,A9,A23,A24,A19,A11,A16,
MATRLIN2:9
    .= Sum lmlt((Sum (W| (m-'1)) |->0.K)^Line(Fm,i-'Sum (Len F| (m-'1))),B2W)+
  Sum lmlt(((Sum W-'Sum (W|m)) |->0.K),B2/^Wm) by RLVECT_1:41
    .= Sum lmlt((Sum (W| (m-'1)) |->0.K) ^ Line(Fm,i-'Sum (Len F| (m-'1))),B2W)
  + 0.K* Sum (B2/^Wm) by A2,A18,A9,A10,A24,A19,MATRLIN2:11
    .= Sum lmlt((Sum (W| (m-'1)) |->0.K)^ Line(Fm,i-'Sum (Len F| (m-'1))),B2W)+
  0.V2 by VECTSP_1:14
    .= Sum lmlt((Sum (W| (m-'1)) |->0.K) ^ Line(Fm,i-'Sum (Len F| (m-'1))),B2W)
  by RLVECT_1:def 4
    .= Sum (lmlt(Sum (W| (m-'1)) |->0.K,B2|Wm1) ^ lmlt(Line(Fm,i-'Sum (Len F| (
  m-'1))),B2W/^Wm1)) by A9,A15,A19,A21,A22,A20,A8,MATRLIN2:9
    .= Sum lmlt(Sum (W| (m-'1)) |->0.K,B2|Wm1) + Sum lmlt(Line(Fm,i-'Sum (Len
  F| (m-'1))),B2W/^Wm1) by RLVECT_1:41
    .= 0.K*Sum (B2|Wm1) + Sum lmlt(Line(Fm,i-'Sum (Len F| (m-'1))),B2W/^Wm1)
  by A20,MATRLIN2:11
    .= 0.V2+Sum lmlt(Line(Fm,i-'Sum (Len F| (m-'1))),B2W/^Wm1) by VECTSP_1:14
    .= Sum lmlt(Line(Fm,i-'Sum (Len F| (m-'1))),B2W/^Wm1) by RLVECT_1:def 4;
  hence thesis by A9,A15,A19,A22,A20,A17;
end;
