reserve i,j for Nat;

theorem Th55:
  for A,B being Matrix of REAL st 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
A1: width A=width B;
  let i be Nat;
A2: len Line(A,i)=width A by MATRIX_0:def 7;
  assume 1<=i & i<=len A;
  then
A3: i in dom A by FINSEQ_3:25;
A4: width (A+B)=width A by Th25;
  len Line(B,i)=width B by MATRIX_0:def 7;
  then
A5: len (Line(A,i)+Line(B,i))=len Line(A,i) by A1,A2,RVSUM_1:115;
  then
A6: dom (Line(A,i)+Line(B,i))=Seg width A by A2,FINSEQ_1:def 3;
  for j st j in Seg width (A+B) holds (Line(A,i)+Line(B,i)).j = (A+B)*(i,j )
  proof
    let j;
    assume
A7: j in Seg width (A+B);
    then
A8: j in Seg width A by Th25;
    then
A9: [i,j] in Indices A & Line(A,i).j=A*(i,j) by A3,MATRIX_0:def 7,ZFMISC_1:87;
    reconsider j as Element of NAT by ORDINAL1:def 12;
    Line(B,i).j=B*(i,j) by A1,A8,MATRIX_0:def 7;
    then Line(A,i).j+Line(B,i).j=(A+B)*(i,j) by A9,Th25;
    hence thesis by A4,A6,A7,VALUED_1:def 1;
  end;
  hence thesis by A2,A4,A5,MATRIX_0:def 7;
end;
