reserve m,n for Nat;
reserve r for Real;
reserve c for Element of F_Complex;

theorem
  for R being Ring, S being Subring of R
  for r being Element of R, s being Element of S
  st r = s & s is_integral_over S holds r is_integral_over R
  proof
    let R be Ring;
    let S be Subring of R;
    let r be Element of R;
    let s be Element of S;
    assume
A1: r = s;
    given f being Polynomial of S such that
A2: LC f = 1.S and
A3: Ext_eval(f,s) = 0.S;
    reconsider f1 = f as Polynomial of R by Th8;
    take f1;
    LC f = LC f1 by Th9;
    hence LC f1 = 1.R by A2,C0SP1:def 3;
    r = In(r,R);
    then Ext_eval(f1,r) = Ext_eval(f,r) by Th15
    .= Ext_eval(f,s) by A1,Th16;
    hence Ext_eval(f1,r) = 0.R by A3,C0SP1:def 3;
  end;
