reserve n for Nat;
reserve K for Field;
reserve a,b,c,d,e,f,g,h,i,a1,b1,c1,d1,e1,f1,g1,h1,i1 for Element of K;
reserve M,N for Matrix of 3,K;
reserve p for FinSequence of REAL;

theorem Th04:
  M = <* <* a,b,c *>,
         <* d,e,f *>,
         <* g,h,i *> *> 
  implies
  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)
  proof
    assume M = <* <* a,b,c *>,
                  <* d,e,f *>,
                 <* g,h,i *> *>;
    then
A1: M.1 = <*a,b,c*> & M.2 = <*d,e,f*> & 
      M.3 = <*g,h,i*> by FINSEQ_1:45;
A2: [1,1] in Indices M & [1,2] in Indices M & [1,3] in Indices M &
    [2,1] in Indices M & [2,2] in Indices M &
    [2,3] in Indices M &[3,1] in Indices M & [3,2] in Indices M & 
      [3,3] in Indices M by MATRIX_0:24,ANPROJ_8:1;
    then ex s be FinSequence of K st s = M.1 & M*(1,1) = s.1 
      by MATRIX_0:def 5; 
    hence M*(1,1) = a by A1,FINSEQ_1:45;
    ex s be FinSequence of K st s = M.1 & M*(1,2) = s.2 
      by A2,MATRIX_0:def 5; 
    hence M*(1,2) = b by A1,FINSEQ_1:45;
    ex s be FinSequence of K st s = M.1 & M*(1,3) = s.3 
      by A2,MATRIX_0:def 5; 
    hence M*(1,3) = c by A1,FINSEQ_1:45;
    ex s be FinSequence of K st s = M.2 & M*(2,1) = s.1 
      by A2,MATRIX_0:def 5; 
    hence M*(2,1) = d by A1,FINSEQ_1:45;
    ex s be FinSequence of K st s = M.2 & M*(2,2) = s.2 
      by A2,MATRIX_0:def 5; 
    hence M*(2,2) = e by A1,FINSEQ_1:45;
    ex s be FinSequence of K st s = M.2 & M*(2,3) = s.3 
      by A2,MATRIX_0:def 5; 
    hence M*(2,3) = f by A1,FINSEQ_1:45;
    ex s be FinSequence of K st s = M.3 & M*(3,1) = s.1 
      by A2,MATRIX_0:def 5; 
    hence M*(3,1) = g by A1,FINSEQ_1:45;
    ex s be FinSequence of K st s = M.3 & M*(3,2) = s.2 
      by A2,MATRIX_0:def 5; 
    hence M*(3,2) = h by A1,FINSEQ_1:45;
    ex s be FinSequence of K st s = M.3 & M*(3,3) = s.3 
      by A2,MATRIX_0:def 5; 
    hence M*(3,3) = i by A1,FINSEQ_1:45;        
  end;
