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 Th06:
  M = <* <* a,b,c *>,
         <* d,e,f *>,
         <* g,h,i *> *> &
  M is symmetric implies b = d & c = g & h = f
  proof
    assume that 
A1: M = <* <* a,b,c *>,
         <* d,e,f *>,
         <* g,h,i *> *> and
A2: M is symmetric;
A3: M = M@ by A2,MATRIX_6:def 5;
    M@ = <* <* a, d, g *>,
            <* b, e, h *>,
            <* c, f, i *> *> by A1,Th05;
    hence thesis by A1,A3,Th02;
  end;
