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

theorem Th1:
  M = <*<*a,b,c*>,
        <*d,e,f*>,
        <*g,h,i*>*> implies
  Det M = a*e*i - c*e*g - a*f*h + b*f*g - b*d*i + c*d*h
  proof
    assume
A1: M = <*<*a,b,c*>,
                 <*d,e,f*>,
                 <*g,h,i*>*>;
    reconsider ra = a,rb = b,rc = c, rd = d,re = e,rf = f,
               rg = g,rh = h, ri = i as Element of F_Real by XREAL_0:def 1;
    reconsider M1 = <*<*ra,rb,rc*>,
                      <*rd,re,rf*>,
                      <*rg,rh,ri*>*> as Matrix of 3,F_Real by MATRIXR2:35;
    Det M1 = ra*re*ri - rc*re*rg - ra*rf*rh + rb*rf*rg - rb*rd*ri + rc*rd*rh
      by MATRIX_9:46;
    hence thesis by A1,MATRIXR1:def 1;
  end;
