reserve i,j for Nat;

theorem Th59:
  for K being Ring,j being Nat,A,B being Matrix of K
  st width A=width B & (i in dom A) holds
  Line(A+B,i)=Line(A,i)+Line(B,i)
proof
  let K be Ring,j be Nat,A,B be Matrix of K;
  assume that
A1: width A=width B and
A2: i in dom A;
  reconsider wA=width A as Element of NAT by ORDINAL1:def 12;
  reconsider a= Line(A,i),b=Line(B,i) as Element of wA-tuples_on (the
  carrier of K) by A1;
A3: width (A+B)=width A by MATRIX_3:def 3;
A4: i in Seg len A by A2,FINSEQ_1:def 3;
A5: len (A+B)=len A by MATRIX_3:def 3;
  then
A6: Indices (A+B)=Indices A by A3,Th55;
A7: for k being Nat st 1<=k & k<=len (Line(A+B,i)) holds (Line(A+B,i)).k=(
  Line(A,i)+Line(B,i)).k
  proof
    let k be Nat;
    assume
A8: 1<=k & k<=len Line(A+B,i);
A9: len Line(A+B,i) = width A by A3,MATRIX_0:def 7;
    then
A10: k in Seg width (A+B) by A3,A8,FINSEQ_1:1;
    len Line(B,i)=width(B) by MATRIX_0:def 7;
    then k in Seg len (Line(B,i)) by A1,A8,A9,FINSEQ_1:1;
    then k in dom Line(B,i) by FINSEQ_1:def 3;
    then reconsider e=Line(B,i).k as Element of K by FINSEQ_2:11;
    i in dom (A+B) by A5,A4,FINSEQ_1:def 3;
    then
A11: [i,k] in Indices (A+B) by A10,ZFMISC_1:87;
A12: (Line(A+B,i)).k= (A+B)*(i,k) by A10,MATRIX_0:def 7
      .= A*(i,k)+B*(i,k) by A6,A11,MATRIX_3:def 3;
A13: len Line(A,i)=width(A) by MATRIX_0:def 7;
    then
A14: k in Seg len (Line(A,i)) by A8,A9,FINSEQ_1:1;
    then k in dom Line(A,i) by FINSEQ_1:def 3;
    then reconsider d=Line(A,i).k as Element of K by FINSEQ_2:11;
    len (Line(A,i)+Line(B,i))= len (a+b) .= width A by CARD_1:def 7
      .= len (Line(A,i)) by CARD_1:def 7;
    then k in dom (Line(A,i)+Line(B,i)) by A14,FINSEQ_1:def 3;
    then
A15: (Line(A,i)+Line(B,i)).k=d+e by FVSUM_1:17;
    Line(A,i).k=A*(i,k) by A13,A14,MATRIX_0:def 7;
    hence thesis by A1,A13,A12,A14,A15,MATRIX_0:def 7;
  end;
A16: len (Line(A,i)+Line(B,i))=len (a+b) .= width A by CARD_1:def 7;
  len (Line(A+B,i)) = width A by A3,MATRIX_0:def 7;
  hence thesis by A16,A7,FINSEQ_1:14;
end;
