 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 Th09:
  for a,b,c being non zero Element of F_Real
  for M1,M2 being Matrix of 3,F_Real st
  M1 = <* <* a,0,0 *>,
          <* 0,b,0 *>,
          <* 0,0,c *> *> &
  M2 = <* <* 1/a,0,0 *>,
          <* 0,1/b,0 *>,
          <* 0,0,1/c *> *>
  holds M1 * M2 = 1.(F_Real,3) & M2 * M1 = 1.(F_Real,3)
  proof
    let a,b,c be non zero Element of F_Real;
    let M1,M2 be Matrix of 3,F_Real;
    assume that
A1: M1 = <* <* a,0,0 *>, <* 0,b,0 *>, <* 0,0,c *> *>  and
A2: M2 = <* <* (1/a),0,0 *>, <* 0,(1/b),0 *>, <* 0,0,(1/c) *> *>;
    reconsider z = 0 as Element of F_Real;
    reconsider ia = 1/a, ib = 1/b, ic = 1/c as Element of F_Real
      by XREAL_0:def 1;
A4: M2 = <* <* ia,z,z *>, <* z,ib,z *>, <* z,z,ic *> *> by A2;
    a is non zero; then
A5: a * ia = 1 & b * ib = 1 & c * ic = 1 by XCMPLX_1:106;
A6: len (M1 * M2) = 3 by MATRIX_0:23;
    (M1 * M2).1 = <* a * ia + z * z + z * z, a * z + z * ib + z * z,
    a * z + z * z + z * ic *> & (M1 * M2).2 = <* z * ia + b * z + z * z,
    z * z + b * ib + z * z, z * z + b * z + z * ic *> &
    (M1 * M2).3 = <* z * ia + z * z + c * z, z * z + z * ib + c * z,
    z * z + z * z + c * ic *> by A1,A2,Lem01;
    hence M1 * M2 = 1.(F_Real,3) by A6,A5,FINSEQ_1:45,Th01;
A7: len (M2 * M1) = 3 by MATRIX_0:23;
    (M2 * M1).1 = <* ia * a + z * z + z * z, ia * z + z * b + z * z,
    ia * z + z * z + z * c *> & (M2 * M1).2 = <* z * a + ib * z + z * z,
    z * z + ib * b + z * z, z * z + ib * z + z * c *> &
    (M2 * M1).3 = <* z * a + z * z + ic * z, z * z + z * b + ic * z,
    z * z + z * z + ic * c *> by A1,A4,Lem01;
    hence M2 * M1 = 1.(F_Real,3) by A7,A5,FINSEQ_1:45,Th01;
  end;
