reserve a,b,c,d,e,f,g,h,i for Real,
                        M for Matrix of 3,REAL;

theorem Th3:
  for r416,r415,r413,r418,r419,r412,
  r712,r746,r716,r742,r715,r743,r713,r745,
  r749,r718,r719,r748 being non zero Real st
  (- r412) * (- r713) = (- r413) * (- r712) &
  (- r415) * (- r719) = (- r419) * (- r715) &
  (- r418) * (- r716) = (- r416) * (- r718) &
  (- r745) * r416 = (- r746) * r415 &
  (- r748) * r413 = (- r743) * r418 &
  (- r742) * r419 = (- r749) * r412 &
  r712 * r746 = r716 * r742 &
  r715 * r743 = r713 * r745 holds
  r718 * r749 = r719 * r748
  proof
    let r416,r415,r413,r418,r419,r412,
    r712,r746,r716,r742,r715,r743,r713,r745,
    r749,r718,r719,r748 being non zero Real;
    assume
A1: (- r412) * (- r713) = (- r413) * (- r712) &
    (- r415) * (- r719) = (- r419) * (- r715) &
    (- r418) * (- r716) = (- r416) * (- r718) &
    (- r745) * r416 = (- r746) * r415 &
    (- r748) * r413 = (- r743) * r418 &
    (- r742) * r419 = (- r749) * r412 &
    r712 * r746 = r716 * r742 &
    r715 * r743 = r713 * r745;
    reconsider r146 = - r416 , r145 = - r415 ,
    r143 = - r413 , r148 = - r418 , r149 = - r419 ,
    r142 = - r412 , r172 = - r712 , r476 = - r746 ,
    r176 = - r716 , r472 = - r742 , r175 = - r715 ,
    r473 = - r743 , r173 = - r713 , r475 = - r745 ,
    r478 = - r748 , r479 = - r749 , r178 = - r718 ,
    r179 = - r719 as non zero Real;
A2: (r142 * r173) * (r145 * r179) *
    (r148 * r176) * (r475 * r416) *
    (r478 * r413) * (r472 * r419) *
    (r712 * r746) * (r715 * r743) =
    (r143 * r172) * (r149 * r175) *
    (r146 * r178) * (r476 * r415) *
    (r473 * r418) * (r479 * r412) *
    (r716 * r742) * (r713 * r745) by A1;
    reconsider r1 = (r142 * r173) * r145 *
    (r148 * r176) * (r475 * r416) * r413 * (r472 * r419) *
    (r712 * r746) * (r715 * r743),
    r2 = (r143 * r172) * (r149 * r175) *
    r146 * (r476 * r415) *
    (r473 * r418) * r412 *
    (r716 * r742) * (r713 * r745) as non zero Real;
A3: (r179 * r478) * r1 = (r178 * r479) * r1 by A2;
    r1 <> 0 by ORDINAL1:def 14;
    hence thesis by A3,XCMPLX_1:5;
  end;
