reserve x,y for set,
  i,j,k,l,m,n for Nat,
  K for Field,
  N for without_zero finite Subset of NAT,
  a,b for Element of K,
  A,B,B1,B2,X,X1,X2 for (Matrix of K),
  A9 for (Matrix of m,n,K),
  B9 for (Matrix of m,k,K);

theorem Th19:
  for D be non empty set, A be Matrix of n,m,D, B be Matrix of n,k
,D holds Segm(A^^B,Seg n,Seg width A) = A & Segm(A^^B,Seg n,Seg (width A+width
  B)\Seg width A) = B
proof
  let D be non empty set, A be Matrix of n,m,D, B be Matrix of n,k,D;
  set AB=A^^B;
A1: card Seg n=n by FINSEQ_1:57;
A2: len A=n by MATRIX_0:def 2;
  then reconsider A9=A as Matrix of n,width A,D by MATRIX_0:51;
  set S1=Segm(AB,Seg n,Seg width A);
A3: card Seg width A=width A by FINSEQ_1:57;
A4: len AB=n by MATRIX_0:def 2;
  now
    let i,j such that
A5: [i,j] in Indices A9;
    reconsider I=i,J=j as Element of NAT by ORDINAL1:def 12;
A6: dom A=Seg n by A2,FINSEQ_1:def 3;
    n<>0 by A5,MATRIX_0:22;
    then Indices A9=[:Seg n,Seg width A:] by MATRIX_0:23;
    then
A7: I in Seg n by A5,ZFMISC_1:87;
A8: J in Seg width A by A5,ZFMISC_1:87;
    then
A9: J = (idseq width A).J by FINSEQ_2:49
      .= Sgm (Seg (width A)).J by FINSEQ_3:48;
    dom S1 = Seg len S1 by FINSEQ_1:def 3
      .= Seg n by A1,MATRIX_0:def 2;
    hence S1*(i,j) = Col(S1,J).I by A7,MATRIX_0:def 8
      .= Col(AB,Sgm (Seg (width A)).J).I by A4,A3,A8,MATRIX13:50
      .= Col(A,J).I by A8,A9,Th16
      .= A*(i,j) by A7,A6,MATRIX_0:def 8;
  end;
  hence A=S1 by A1,A3,MATRIX_0:27;
  set w=width A+width B;
  set SS=Seg w\Seg width A;
  set S2=Segm(AB,Seg n,SS);
A10: len B=n by MATRIX_0:def 2;
  then reconsider B9=B as Matrix of n,width B,D by MATRIX_0:51;
  width A <= w by NAT_1:11;
  then card Seg (width A+width B)=width A+width B & Seg width A c= Seg w by
FINSEQ_1:5,57;
  then
A11: card SS = w-width A by A3,CARD_2:44
    .= width B;
  now
A12: dom B = Seg n by A10,FINSEQ_1:def 3;
    let i,j such that
A13: [i,j] in Indices B9;
    reconsider I=i,J=j as Element of NAT by ORDINAL1:def 12;
A14: J in Seg width B by A13,ZFMISC_1:87;
    n<>0 by A13,MATRIX_0:22;
    then Indices B9=[:Seg n,Seg width B:] by MATRIX_0:23;
    then
A15: I in Seg n by A13,ZFMISC_1:87;
    dom S2 = Seg len S2 by FINSEQ_1:def 3
      .= Seg n by A1,MATRIX_0:def 2;
    hence S2*(i,j) = Col(S2,J).I by A15,MATRIX_0:def 8
      .= Col(AB,Sgm SS.J).I by A4,A11,A14,MATRIX13:50
      .= Col(AB,width A+J).I by A14,Th8
      .= Col(B,J).I by A14,Th17
      .= B9*(i,j) by A15,A12,MATRIX_0:def 8;
  end;
  hence thesis by A1,A11,MATRIX_0:27;
end;
