 reserve R for domRing;
 reserve p for odd prime Nat, m for positive Nat;
 reserve g for non zero Polynomial of INT.Ring;
 reserve f for Element of the carrier of Polynom-Ring INT.Ring;
 reserve z0 for non zero Element of F_Real;

theorem LMM:
  number_e is algebraic implies
  ex g be INT -valued Polynomial of F_Rat st @g is irreducible &
    Ext_eval(g,In(number_e,F_Real)) = 0 & deg g >=2 & g.0 <> 0.F_Rat
    proof
      set PRat = Polynom-Ring F_Rat;
      assume number_e is algebraic; then
      consider x being Element of F_Complex such that
A2:   x = number_e &
      x is_integral_over F_Rat by ALGNUM_1:def 4;
      reconsider e1 = number_e as Element of F_Real by XREAL_0:def 1;
      consider f0 be Element of Polynom-Ring F_Rat such that
A3:   f0 <> 0_.F_Rat & {f0}-Ideal = Ann_Poly(x,F_Rat) & f0 = NormPolynomial(f0)
        by ALGNUM_1:3,A2,ALGNUM_1:33;
      f0 in Ann_Poly(x,F_Rat) by A3,IDEAL_1:66; then
      f0 in {p where p is Polynomial of F_Rat: Ext_eval(p,x) = 0.F_Complex}
      by ALGNUM_1:def 8; then
      consider f be Polynomial of F_Rat such that
A4:   f0 = f & Ext_eval(f,x) = 0.F_Complex;
      f0 <> 0.Polynom-Ring F_Rat by A3,POLYNOM3:def 10; then
reconsider f0 as non zero Element of Polynom-Ring F_Rat by STRUCT_0:def 12;
      {f0}-Ideal is maximal by A2,E_TRANS1:11,ALGNUM_1:3,A3; then
A7:   f0 is irreducible by RING_2:25;
      reconsider f0 as non zero Element of the carrier of Polynom-Ring F_Rat
        by A3,UPROOTS:def 5;
      reconsider m = Product denomi-seq(f0) as non zero Nat by E_TRANS1:9;
A8:   m*f0 = (In(m,F_Rat)|F_Rat)*'(~f0) by Lm8a;
      reconsider mf0 = m*f0 as Element of the carrier of Polynom-Ring F_Rat;
A14:  rng mf0 c= INT
      proof
        let y be object;
        assume y in rng mf0; then
        consider x1 be object such that
A10:    x1 in dom mf0 & y = mf0.x1 by FUNCT_1:def 3;
        reconsider i = x1 as Element of NAT by A10;
        reconsider f0i = f0.i as Rational;
A11:    mf0.i = (In(m,F_Rat))*f0.i by A8,RINGDER1:27;
        consider z be Integer such that
A12:    z*(denominator (f0.i)) = m by E_TRANS1:10;
        m*f0i = z*((denominator (f0i))*f0i) by A12
        .= z*numerator(f0i) by RAT_1:def 4;
        hence thesis by A11,A10,INT_1:def 2;
       end;
reconsider g = (In(m,F_Rat)|F_Rat)*'(~f0) as INT -valued Polynomial of F_Rat
         by A14,RELAT_1:def 19, A8;
A16:   Ext_eval(g,x) = Ext_eval((In(m,F_Rat)|F_Rat),x) *0.F_Complex
       by A4,ALGNUM_1:3,20
       .= 0 by MATRIX_5:2;
       take g;
       @g = m*f0 by Lm8a; then
A18:   @g is irreducible by A7,E_TRANS1:12;
reconsider FR = F_Real as Subring of F_Complex by LIOUVIL2:3;
        g is Polynomial of F_Real by FIELD_4:8; then
reconsider g1 = g as Element of the carrier of Polynom-Ring F_Real
          by POLYNOM3:def 10;
        g is Polynomial of F_Complex by FIELD_4:8; then
reconsider gc = g as Element of the carrier of Polynom-Ring F_Complex
          by POLYNOM3:def 10;
reconsider g0 = g as Element of the carrier of Polynom-Ring F_Rat
          by POLYNOM3:def 10;
A19:    Ext_eval(g0,x) = 0 by A16;
A20:    Ext_eval(g0,e1) = eval(g1,e1) by FIELD_4:26
        .= eval(gc,x) by A2,FIELD_4:27
        .= 0 by A19,FIELD_4:26;
        then deg g >= 2 by A18,E_TRANS1:13;
        hence thesis by A18,A20,E_TRANS1:14;
      end;
