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

theorem Th25:
  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<=len A holds Line(A-B,i)=Line(A,i)-Line(B,i)
proof
  let A,B be Matrix of REAL;
  assume that
A1: len A=len B and
A2: width A=width B;
A3: width (A-B)=width A by A1,A2,Th6;
  let i be Nat;
A4: len Line(A,i)=width A by MATRIX_0:def 7;
A5: len Line(B,i)=width B by MATRIX_0:def 7;
  assume 1<=i & i<=len A;
  then
A6: i in dom A by FINSEQ_3:25;
A7: for j being Nat st j in Seg width (A-B) holds (Line(A,i)-Line(B,i)).j =
  (A-B)*(i,j)
  proof
    reconsider i2=i as Nat;
    let j be Nat;
    reconsider j2=j as Nat;
A8: (Line(A,i2)-Line(B,i2)).j=Line(A,i2).j2-Line(B,i2).j2 by A2,A4,A5,Lm1;
    assume
A9: j in Seg width (A-B);
    then [i,j] in Indices A by A6,A3,ZFMISC_1:87;
    then
A10: (A-B)*(i2,j2) = A*(i2,j2) - B*(i2,j2) by A1,A2,Th6;
A11: j in Seg width A by A1,A2,A9,Th6;
    then Line(A,i).j=A*(i,j) by MATRIX_0:def 7;
    hence thesis by A2,A11,A10,A8,MATRIX_0:def 7;
  end;
  len (Line(A,i)-Line(B,i))=len Line(A,i) by A2,A4,A5,RVSUM_1:116;
  hence thesis by A4,A3,A7,MATRIX_0:def 7;
end;
