reserve i,j for Nat;

theorem Th54:
  for A,B being Matrix of REAL st len A=len B holds for i being
  Nat st 1<=i & i<=width A holds Col(A+B,i)=Col(A,i)+Col(B,i)
proof
  let A,B be Matrix of REAL;
  assume
A1: len A=len B;
  then
A2: dom A=dom B by FINSEQ_3:29;
  let i be Nat;
A3: len Col(A,i)=len A by MATRIX_0:def 8;
  len Col(B,i)=len B by MATRIX_0:def 8;
  then
A4: len (Col(A,i)+Col(B,i))=len (Col(A,i)) by A1,A3,RVSUM_1:115;
  assume 1<=i & i<=width A;
  then
A5: i in Seg width A by FINSEQ_1:1;
A6: len (A+B)=len A by Th25;
  Seg len (A+B)=dom (A+B) by FINSEQ_1:def 3;
  then
A7: dom (Col(A,i)+Col(B,i))=dom (A+B) by A3,A6,A4,FINSEQ_1:def 3;
  for j st j in dom (A+B) holds (Col(A,i)+Col(B,i)).j = (A+B)*(j,i)
  proof
    let j;
    assume
A8: j in dom (A+B);
    then j in Seg len (A+B) by FINSEQ_1:def 3;
    then
A9: j in dom A by A6,FINSEQ_1:def 3;
    then
A10: [j,i] in Indices A by A5,ZFMISC_1:87;
    reconsider j as Element of NAT by ORDINAL1:def 12;
    Col(A,i).j=A*(j,i) & Col(B,i).j=B*(j,i) by A2,A9,MATRIX_0:def 8;
    then Col(A,i).j+Col(B,i).j=(A+B)*(j,i) by A10,Th25;
    hence thesis by A7,A8,VALUED_1:def 1;
  end;
  hence thesis by A3,A6,A4,MATRIX_0:def 8;
end;
