reserve a,b,c,d,e,f for Real,
        g           for positive Real,
        x,y         for Complex,
        S,T         for Element of REAL 2,
        u,v,w       for Element of TOP-REAL 3;
reserve a,b,c for Element of F_Real,
          M,N for Matrix of 3,F_Real;
reserve D        for non empty set;
reserve d1,d2,d3 for Element of D;
reserve A        for Matrix of 1,3,D;
reserve B        for Matrix of 3,1,D;
reserve u,v for non zero Element of TOP-REAL 3;

theorem Th39:
  for r,a,b,c,d,e,f,g,h,i being Element of F_Real
  for M being Matrix of 3,F_Real st
  M = <* <* a,b,c *>,
         <* d,e,f *>,
         <* g,h,i *> *>
  holds
  r * M = <* <* r * a,r * b,r * c *>,
             <* r * d,r * e,r * f *>,
             <* r * g,r * h,r * i *> *>
  proof
    let r,a,b,c,d,e,f,g,h,i be Element of F_Real;
    let M be Matrix of 3,F_Real;
    assume
A1: M = <* <* a,b,c *>,
           <* d,e,f *>,
           <* g,h,i *> *>;
A3: Indices M = [: Seg 3, Seg 3:] by MATRIX_0:23;
    reconsider b11 = (r * M)*(1,1), b12 = (r * M)*(1,2), b13 = (r * M)*(1,3),
               b21 = (r * M)*(2,1), b22 = (r * M)*(2,2), b23 = (r * M)*(2,3),
               b31 = (r * M)*(3,1), b32 = (r * M)*(3,2), b33 = (r * M)*(3,3)
      as Element of F_Real;
    M =  <* <* M*(1,1), M*(1,2), M*(1,3) *>,
            <* M*(2,1), M*(2,2), M*(2,3) *>,
            <* M*(3,1), M*(3,2), M*(3,3) *> *> by MATRIXR2:37;
    then
A4: a = M*(1,1) & b = M*(1,2) & c =  M*(1,3) &
      d = M*(2,1) & e = M*(2,2)& f = M*(2,3) &
      g = M*(3,1) & h = M*(3,2) & i = M*(3,3) by A1,PASCAL:2;
    b11 = r * (M*(1,1)) & b12 = r * (M*(1,2)) & b13 = r * (M*(1,3)) &
    b21 = r * (M*(2,1)) & b22 = r * (M*(2,2)) & b23 = r * (M*(2,3)) &
    b31 = r * (M*(3,1)) & b32 = r * (M*(3,2)) & b33 = r * (M*(3,3))
      by A3,ANPROJ_8:1,MATRIX_3:def 5;
    hence thesis by A4,MATRIXR2:37;
  end;
