reserve i,j for Nat;

theorem Th19:
  for K being Field, a being Element of K, M being Matrix of K st
  1<=i & i<=width M holds Col(a*M,i)=a*Col(M,i)
proof
  let K be Field,a be Element of K, M be Matrix of K;
  assume
A1: 1<=i & i<=width M;
A2: Seg len (a*M)=dom (a*M) by FINSEQ_1:def 3;
A3: len (a*M)=len M by MATRIX_3:def 5;
  then
A4: dom M=dom (a*M) by FINSEQ_3:29;
A5: len (a*Col(M,i))=len (Col(M,i)) & len (Col(M,i))=len M by Th16,
MATRIX_0:def 8;
  then
A6: dom (a*Col(M,i))=Seg len (a*M) by A3,FINSEQ_1:def 3;
  for j st j in dom (a*M) holds (a*Col(M,i)).j = (a*M)*(j,i)
  proof
    let j;
    assume
A7: j in dom (a*M);
    i in Seg width M by A1,FINSEQ_1:1;
    then [j,i] in Indices M by A4,A7,ZFMISC_1:87;
    then
A8: (a*M)*(j,i)=a*(M*(j,i)) by MATRIX_3:def 5;
    (Col(M,i)).j = M*(j,i) by A4,A7,MATRIX_0:def 8;
    hence thesis by A6,A2,A7,A8,FVSUM_1:50;
  end;
  hence thesis by A5,A3,MATRIX_0:def 8;
end;
