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 Th17:
  for D be non empty set, A be Matrix of n,m,D, B be Matrix of n,k
  ,D for i st i in Seg width B holds Col(A^^B,width A+i) = Col(B,i)
proof
  let D be non empty set, A be Matrix of n,m,D, B be Matrix of n,k,D;
  let i such that
A1: i in Seg width B;
  set AB=A^^B;
A2: len AB=n by MATRIX_0:def 2;
A3: len B=n by MATRIX_0:def 2;
  now
A4: dom B=Seg n by A3,FINSEQ_1:def 3;
    let j such that
A5: j in Seg n;
    n<>0 by A5;
    then width AB=width A+width B by MATRIX_0:23;
    then
A6: width A+i in Seg width AB by A1,FINSEQ_1:60;
A7: dom Line(B,j)=Seg width B & len Line(A,j)=width A by CARD_1:def 7
,FINSEQ_2:124;
    dom AB=Seg n by A2,FINSEQ_1:def 3;
    hence Col(AB,width A+i).j = AB*(j,width A+i) by A5,MATRIX_0:def 8
      .= Line(AB,j).(width A+i) by A6,MATRIX_0:def 7
      .= (Line(A,j)^Line(B,j)).(width A+i) by A5,Th15
      .= Line(B,j).i by A1,A7,FINSEQ_1:def 7
      .= B*(j,i) by A1,MATRIX_0:def 7
      .= Col(B,i).j by A5,A4,MATRIX_0:def 8;
  end;
  hence thesis by A3,A2,FINSEQ_2:119;
end;
