reserve i,j for Nat;

theorem Th60:
  for K being Ring,j being Nat,A,B being Matrix of K st len A=len
  B & j in Seg(width A) holds Col(A+B,j)=Col(A,j)+Col(B,j)
proof
  let K be Ring,j be Nat, A,B be Matrix of K;
A1: len (A+B)=len A by MATRIX_3:def 3;
  assume that
A2: len A=len B and
A3: j in Seg width A;
  reconsider a = Col(A,j),b=Col(B,j) as Element of (len A)-tuples_on (the
  carrier of K) by A2;
A4: width (A+B)=width A by MATRIX_3:def 3;
  then
A5: Indices (A+B)=Indices A by A1,Th55;
A6: for k being Nat st 1<=k & k<=len (Col(A+B,j)) holds (Col(A+B,j)).k=(Col(
  A,j)+Col(B,j)).k
  proof
    let k be Nat;
    assume
A7: 1<=k & k<=len (Col(A+B,j));
A8: len (Col(A+B,j)) = len A by A1,MATRIX_0:def 8;
    then k in Seg len A by A7,FINSEQ_1:1;
    then
A9: k in dom (A+B) by A1,FINSEQ_1:def 3;
    len (Col(B,j))=len(B) by MATRIX_0:def 8;
    then k in Seg len (Col(B,j)) by A2,A7,A8,FINSEQ_1:1;
    then k in dom Col(B,j) by FINSEQ_1:def 3;
    then reconsider e=Col(B,j).k as Element of K by FINSEQ_2:11;
A10: dom A=Seg len A by FINSEQ_1:def 3
      .=dom B by A2,FINSEQ_1:def 3;
A11: len (Col(A,j))=len(A) by MATRIX_0:def 8;
    then
A12: k in Seg len (Col(A,j)) by A7,A8,FINSEQ_1:1;
    then k in dom Col(A,j) by FINSEQ_1:def 3;
    then reconsider d = Col(A,j).k as Element of K by FINSEQ_2:11;
    len (Col(A,j)+Col(B,j)) = len (a+b) .= len A by CARD_1:def 7
      .= len (Col(A,j)) by CARD_1:def 7;
    then k in dom (Col(A,j)+Col(B,j)) by A12,FINSEQ_1:def 3;
    then
A13: (Col(A,j)+Col(B,j)).k=d+e by FVSUM_1:17;
A14: [k,j] in Indices (A+B) by A3,A4,A9,ZFMISC_1:87;
A15: (Col(A+B,j)).k= (A+B)*(k,j) by A9,MATRIX_0:def 8
      .= A*(k,j)+B*(k,j) by A5,A14,MATRIX_3:def 3;
A16: k in dom A by A11,A12,FINSEQ_1:def 3;
    then Col(A,j).k=A*(k,j) by MATRIX_0:def 8;
    hence thesis by A15,A13,A10,A16,MATRIX_0:def 8;
  end;
A17: len (Col(A,j)+Col(B,j)) = len (a+b) .= len A by CARD_1:def 7;
  len (Col(A+B,j)) = len A by A1,MATRIX_0:def 8;
  hence thesis by A17,A6,FINSEQ_1:14;
end;
