reserve i,j for Nat;
reserve A,B for Ring;

theorem Th27:
  for A being non degenerated Ring
   for a be Element of A st A is Subring of B holds
   In(a,B) is_integral_over A
   proof
     let A be non degenerated Ring;
     let a be Element of A;
     assume
A0:  A is Subring of B;
     set p = <% -a, 1.A %>;
     p.(len p -' 1) = p.(2-'1) by POLYNOM5:40 .= p.(2-1) by XREAL_1:233
     .= p.1; then
A2:  LC p = 1.A by POLYNOM5:38;
A3:  eval(p,a) = -a + a by POLYNOM5:47
     .= a - a .= 0.A by RLVECT_1:15;
     Ext_eval(p,In(a,B)) = In(eval(p,a),B) by A0,Th16
    .= 0.B by A0,Lm5,A3;
     hence thesis by A2;
   end;
