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

theorem Th41:
  for F being FinSequence of COMPLEX,M being Matrix of COMPLEX st
  len F = width M & len F >= 1 holds (M*F)*'=(M*')*(F*')
proof
  let F be FinSequence of COMPLEX,M be Matrix of COMPLEX;
  assume that
A1: len F = width M and
A2: len F >= 1;
A3: len (F*') = len F by COMPLSP2:def 1;
A4: width (M*')=width M by Def1;
A5: len ((M*F)*')=len (M*F) by COMPLSP2:def 1
    .= len M by Def6;
A6: len (M*') = len M by Def1;
A7: now
    let i be Nat;
    assume that
A8: 1 <= i and
A9: i <= len ((M*F)*');
A10: i in Seg len M by A5,A8,A9,FINSEQ_1:1;
    len (Line((M*'),i)) = len (F*') by A1,A3,A4,MATRIX_0:def 7;
    then
A11: len (mlt(Line((M*'),i),(F*'))) >= 1 by A2,A3,FINSEQ_2:72;
A12: i in Seg len (M*') by A5,A6,A8,A9,FINSEQ_1:1;
    i <= len ((M*F)) by A9,COMPLSP2:def 1;
    hence ((M*F)*').i = ((M*F).i)*' by A8,COMPLSP2:def 1
      .= Sum(mlt(F,Line(M,i)))*' by A10,Def6
      .= Sum(mlt(F,(Line((M*'),i))*'))*' by A10,Th39
      .= Sum((mlt(Line((M*'),i),(F*')))*')*' by A1,Th40
      .= (Sum(mlt(Line((M*'),i),(F*'))))*'*' by A11,Th21
      .= ((M*')*(F*')).i by A12,Def6;
  end;
  len (M*'*(F*'))=len (M*') by Def6
    .= len M by Def1;
  hence thesis by A5,A7,FINSEQ_1:14;
end;
