theorem Th36:
  for M be Matrix of len b1,len b2,K holds AutMt(Mx2Tran(M,b1,b2), b1,b2) = M
proof
  let M be Matrix of len b1,len b2,K;
  set MX=Mx2Tran(M,b1,b2);
  set A=AutMt(MX,b1,b2);
  set ONE=1.(K,len b1);
A1: len M=len b1 by MATRIX_0:25;
A2: len A=len b1 by MATRIX_0:25;
A3: len ONE=len b1 by MATRIX_0:24;
  now
    let i such that
A4: 1<=i & i<=len M;
A5: i in Seg len b1 by A1,A4;
A6: i in dom ONE by A1,A3,A4,FINSEQ_3:25;
    reconsider Ai = A/.i as FinSequence of K by FINSEQ_1:def 11;
A7: i in dom b1 by A1,A4,FINSEQ_3:25;
    then A/.i=MX.(b1/.i) |--b2 by MATRLIN:def 8;
    then LineVec2Mx(Ai qua FinSequence of K) = LineVec2Mx(b1/.i|--b1) * M
             by A1,A4,Th32
      .= LineVec2Mx(Line(ONE,i))*M by A7,Th19
      .= LineVec2Mx(Line(ONE*M,i)) by A1,A6,Th35,MATRIX_0:24
      .= LineVec2Mx(Line(M,i)) by A1,MATRIXR2:68;
    then
A8: Ai = Line(LineVec2Mx(Line(M,i)),1) by MATRIX15:25
      .= Line(M,i) by MATRIX15:25
      .= M.i by A5,MATRIX_0:52;
    i in dom A by A1,A2,A4,FINSEQ_3:25;
    hence M.i = A.i by A8,PARTFUN1:def 6;
  end;
  hence thesis by A2,FINSEQ_1:14,MATRIX_0:25;
end;
