reserve i,j,n,k for Nat,
  a for Element of COMPLEX,
  R1,R2 for Element of i-tuples_on COMPLEX;

theorem
  for i,j being Nat,M1,M2 being Matrix of COMPLEX st width M1=width M2 &
  j in Seg len M1 holds Line(M1+M2,j)=Line(M1,j)+Line(M2,j)
proof
  let i,j be Nat,M1,M2 be Matrix of COMPLEX;
  assume that
A1: width M1=width M2 and
A2: j in Seg len M1;
  len Line(M2,j)=width M1 by A1,MATRIX_0:def 7
    .= len Line(M1,j) by MATRIX_0:def 7;
  then
A3: len (Line(M1,j)+Line(M2,j))=len (Line(M1,j)) by COMPLSP2:6
    .= width M1 by MATRIX_0:def 7;
A4: len (Line(M1+M2,j))=width (M1+M2) by MATRIX_0:def 7
    .= width M1 by Th5;
A5: width (M1+M2)=width M1 by Th5;
  now
    let i be Nat;
    assume that
A6: 1 <= i and
A7: i <= len Line(M1+M2,j);
A8: i in Seg width M1 by A4,A6,A7,FINSEQ_1:1;
    i in Seg width M1 by A4,A6,A7,FINSEQ_1:1;
    then [j,i] in [:Seg len M1,Seg width M1:] by A2,ZFMISC_1:87;
    then
A9: [j,i] in Indices M1 by FINSEQ_1:def 3;
    i in Seg len (Line(M1,j)+Line(M2,j)) by A3,A4,A6,A7,FINSEQ_1:1;
    then
A10: i in dom (Line(M1,j)+Line(M2,j)) by FINSEQ_1:def 3;
A11: i in Seg width M2 by A1,A4,A6,A7,FINSEQ_1:1;
    i in Seg width (M1+M2) by A5,A4,A6,A7,FINSEQ_1:1;
    hence (Line(M1+M2,j)).i = (M1+M2)*(j,i) by MATRIX_0:def 7
      .= M1*(j,i)+M2*(j,i) by A9,Th6
      .= M1*(j,i)+(Line(M2,j)).i by A11,MATRIX_0:def 7
      .= (Line(M1,j)).i+(Line(M2,j)).i by A8,MATRIX_0:def 7
      .= (Line(M1,j)+Line(M2,j)).i by A10,COMPLSP2:1;
  end;
  hence thesis by A3,A4,FINSEQ_1:14;
end;
