 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 Th07:
  for a11,a12,a13,a21,a22,a23,a31,a32,a33,
      b11,b12,b13,b21,b22,b23,b31,b32,b33 being Element of F_Real
  for A,B being Matrix of 3,F_Real st
  A = <* <* a11,a12,a13 *>,
         <* a21,a22,a23 *>,
         <* a31,a32,a33 *> *> &
  B = <* <* b11,b12,b13 *>,
         <* b21,b22,b23 *>,
         <* b31,b32,b33 *> *> holds
  A * B = <* <* a11 * b11 + a12 * b21 + a13 * b31, a11 * b12 +
    a12 * b22 + a13 * b32, a11 * b13 + a12 * b23 + a13 * b33 *>,
             <* a21 * b11 + a22 * b21 + a23 * b31, a21 * b12 +
    a22 * b22 + a23 * b32, a21 * b13 + a22 * b23 + a23 * b33 *>,
             <* a31 * b11 + a32 * b21 + a33 * b31, a31 * b12 +
    a32 * b22 + a33 * b32, a31 * b13 + a32 * b23 + a33 * b33 *> *>
  proof
    let a11,a12,a13,a21,a22,a23,a31,a32,a33,
        b11,b12,b13,b21,b22,b23,b31,b32,b33 be Element of F_Real;
    let A,B be Matrix of 3,F_Real;
    assume that
A1: A = <* <* a11,a12,a13 *>, <* a21,a22,a23 *>, <* a31,a32,a33 *> *> and
A2: B = <* <* b11,b12,b13 *>, <* b21,b22,b23 *>, <* b31,b32,b33 *> *>;
A3: width A = 3 & len B = 3 by MATRIX_0:23;
A4: [1,1] in Indices (A * B) & [1,2] in Indices (A * B) &
      [1,3] in Indices (A * B) & [2,1] in Indices (A * B) &
      [2,2] in Indices (A * B) & [2,3] in Indices (A * B) &
      [3,1] in Indices (A * B) & [3,2] in Indices (A * B) &
      [3,3] in Indices (A * B) by ANPROJ_8:1,MATRIX_0:23;
A5: Line(A,1) = <* a11,a12,a13 *> & Line(A,2) = <* a21,a22,a23 *> &
      Line(A,3) = <* a31,a32,a33 *> by A1,Th05;
A6: Col(B,1) = <* b11,b21,b31 *> & Col(B,2) = <* b12,b22,b32 *> &
      Col(B,3) = <* b13,b23,b33 *> by A2,Th06;
    now
      thus (A * B)*(1,1) = Line(A,1) "*" Col(B,1) by A3,A4,MATRIX_3:def 4
                        .= a11 * b11 + a12 * b21 + a13 * b31
                        by A5,A6,ANPROJ_8:7;
      thus (A * B)*(1,2) = Line(A,1) "*" Col(B,2) by A3,A4,MATRIX_3:def 4
                        .= a11 * b12 + a12 * b22 + a13 * b32
                        by A5,A6,ANPROJ_8:7;
      thus (A * B)*(1,3) = Line(A,1) "*" Col(B,3) by A3,A4,MATRIX_3:def 4
                        .= a11 * b13 + a12 * b23 + a13 * b33
                        by A5,A6,ANPROJ_8:7;
      thus (A * B)*(2,1) = Line(A,2) "*" Col(B,1) by A3,A4,MATRIX_3:def 4
                        .= a21 * b11 + a22 * b21 + a23 * b31
                        by A5,A6,ANPROJ_8:7;
      thus (A * B)*(2,2) = Line(A,2) "*" Col(B,2) by A3,A4,MATRIX_3:def 4
                        .= a21 * b12 + a22 * b22 + a23 * b32
                        by A5,A6,ANPROJ_8:7;
      thus (A * B)*(2,3) = Line(A,2) "*" Col(B,3) by A3,A4,MATRIX_3:def 4
                        .= a21 * b13 + a22 * b23 + a23 * b33
                        by A5,A6,ANPROJ_8:7;
      thus (A * B)*(3,1) = Line(A,3) "*" Col(B,1) by A3,A4,MATRIX_3:def 4
                        .= a31 * b11 + a32 * b21 + a33 * b31
                        by A5,A6,ANPROJ_8:7;
      thus (A * B)*(3,2) = Line(A,3) "*" Col(B,2) by A3,A4,MATRIX_3:def 4
                        .= a31 * b12 + a32 * b22 + a33 * b32
                        by A5,A6,ANPROJ_8:7;
      thus (A * B)*(3,3) = Line(A,3) "*" Col(B,3) by A3,A4,MATRIX_3:def 4
                        .= a31 * b13 + a32 * b23 + a33 * b33
                        by A5,A6,ANPROJ_8:7;
    end;
    hence thesis by MATRIXR2:37;
  end;
