 reserve i,n for Nat;
 reserve r for Real;
 reserve ra for Element of F_Real;
 reserve a,b,c for non zero Element of F_Real;
 reserve u,v for Element of TOP-REAL 3;
 reserve p1 for FinSequence of (1-tuples_on REAL);
 reserve pf,uf for FinSequence of F_Real;
 reserve N for Matrix of 3,F_Real;
 reserve K for Field;
 reserve k for Element of K;

theorem Th10:
  for a,b,c being non zero Element of F_Real holds
  <* <* a,0,0 *>,
     <* 0,b,0 *>,
     <* 0,0,c *> *> is invertible Matrix of 3,F_Real
  proof
    let a,b,c be non zero Element of F_Real;
    reconsider ia = 1/a, ib = 1/b, ic = 1/c as Element of F_Real
      by XREAL_0:def 1;
    reconsider M = <* <* a,0,0 *>,
                      <* 0,b,0 *>,
                      <* 0,0,c *> *>,
    N = <* <* ia,0,0 *>,
           <* 0,ib,0 *>,
           <* 0,0,ic *> *> as Matrix of 3,F_Real by MATRIXR2:35;
    now
      thus N * M = 1.(F_Real,3) by Th09;
      hence N * M = M * N by Th09;
    end;
    hence thesis by MATRIX_6:def 2,MATRIX_6:def 3;
  end;
