reserve x for set,
  D for non empty set,
  k,n,m,i,j,l for Nat,
  K for Field;

theorem Th26:
  for A,B being Matrix of REAL st len A=len B & width A=width B
  holds for i being Nat st 1<=i & i<=width A holds Col(A-B,i)=Col(A,i)-Col(B,i)
proof
  let A,B be Matrix of REAL;
  assume that
A1: len A=len B and
A2: width A=width B;
A3: len (A-B)=len A by A1,A2,Th6;
  let i be Nat;
A4: len Col(A,i)=len A by MATRIX_0:def 8;
  assume 1<=i & i<=width A;
  then
A5: i in Seg width A;
A6: len Col(B,i)=len B by MATRIX_0:def 8;
A7: dom A=dom B by A1,FINSEQ_3:29;
A8: for j being Nat st j in dom (A-B) holds (Col(A,i)-Col(B,i)).j = (A-B)*(
  j,i)
  proof
    let j be Nat;
    assume j in dom (A-B);
    then j in Seg len (A-B) by FINSEQ_1:def 3;
    then
A9: j in dom A by A3,FINSEQ_1:def 3;
    then
A10: [j,i] in Indices A by A5,ZFMISC_1:87;
    reconsider j as Nat;
    Col(A,i).j=A*(j,i) & Col(B,i).j=B*(j,i) by A7,A9,MATRIX_0:def 8;
    then Col(A,i).j-Col(B,i).j=(A-B)*(j,i) by A1,A2,A10,Th6;
    hence thesis by A1,A4,A6,Lm1;
  end;
  len (Col(A,i)-Col(B,i))=len (Col(A,i)) by A1,A4,A6,RVSUM_1:116;
  hence thesis by A4,A3,A8,MATRIX_0:def 8;
end;
