reserve i,j,k for Element of NAT;
reserve a,b for Element of REAL;
reserve r,s,t for Element of RAT+;
reserve i,j,k for Element of omega;

theorem Th2:
  RAT c< REAL
proof
  reconsider two = 2 as ordinal Element of RAT+ by Lm1;
  reconsider a9 = 1 as Element of RAT+ by Lm1;
  defpred P[Element of RAT+] means $1 *' $1 < two;
  set X = { s : P[s] };
  reconsider X as Subset of RAT+ from DOMAIN_1:sch 7;
A1: 2 *^ 2 = two *' two & 1*^2 = 2 by ARYTM_3:59,ORDINAL2:39;
  2 = succ 1 .= 1 \/ {1};
  then
A2: a9 <=' two by Lm10,XBOOLE_1:7;
  then
A3: a9 < two by ARYTM_3:68;
A4: a9 *' a9 = a9 by ARYTM_3:53;
  then
A5: 1 in X by A3;
A6: for r,t st r in X & t <=' r holds t in X
  proof
    let r,t;
    assume r in X;
    then
A7: ex s st r = s & s *' s < two;
    assume t <=' r;
    then t *' t <=' t *' r & t *' r <=' r *' r by ARYTM_3:82;
    then t *' t <=' r *' r by ARYTM_3:67;
    then t *' t < two by A7,ARYTM_3:69;
    hence thesis;
  end;
  then
A8: 0 in X by A5,ARYTM_3:64;
  now
    assume X = [0,0];
    then X = {{0}, {0}} by ENUMSET1:29
      .= {{0}} by ENUMSET1:29;
    hence contradiction by A8,TARSKI:def 1;
  end;
  then
A9: not X in {[0,0]} by TARSKI:def 1;
  reconsider O9 = 0 as Element of RAT+ by Lm1;
  set DD = { A where A is Subset of RAT+: r in A implies (for s st s <=' r
  holds s in A) & ex s st s in A & r < s };
  consider half being Element of RAT+ such that
A10: a9 = two*'half by ARYTM_3:55,Lm11;
A11: one <=' two by Lm13;
  then
A12: one < two by ARYTM_3:68;
A13: now
    assume X in {{ s: s < t}: t <> 0};
    then consider t0 being Element of RAT+ such that
A14: X = { s: s < t0} and
A15: t0 <> 0;
    set n = numerator t0, d = denominator t0;
    now
      assume
A16:  t0 *' t0 <> two;
      per cases by A16,ARYTM_3:66;
      suppose
        t0 *' t0 < two;
        then t0 in X;
        then ex s st s = t0 & s < t0 by A14;
        hence contradiction;
      end;
      suppose
A17:    two < t0 *' t0;
        consider es being Element of RAT+ such that
A18:    two + es = t0 *' t0 or t0 *' t0 + es = two by ARYTM_3:92;
A19:    now
          assume O9 = es;
          then two + es = two by ARYTM_3:50;
          hence contradiction by A17,A18;
        end;
        O9 <=' es by ARYTM_3:64;
        then O9 < es by A19,ARYTM_3:68;
        then consider s such that
A20:    O9 < s and
A21:    s < es by ARYTM_3:93;
        now
          per cases;
          suppose
A22:        s < one;
A23:        s <> 0 by A20;
            then s *' s < s *' one by A22,ARYTM_3:80;
            then
A24:        s *' s < s by ARYTM_3:53;
A25:        now
              assume
A26:          t0 <=' one;
              then t0 *' t0 <=' t0 *' one by ARYTM_3:82;
              then t0 *' t0 <=' t0 by ARYTM_3:53;
              then t0 *' t0 <=' one by A26,ARYTM_3:67;
              hence contradiction by A11,A17,ARYTM_3:69;
            end;
            then
A27:        one *' one < one *' t0 by ARYTM_3:80;
            one *' t0 < two *' t0 by A12,A15,ARYTM_3:80;
            then
A28:        one *' one < two *' t0 by A27,ARYTM_3:70;
            consider t02s2 being Element of RAT+ such that
A29:        s *' s + t02s2 = t0 *' t0 or t0 *' t0 + t02s2 = s *' s by
ARYTM_3:92;
            s < t0 by A22,A25,ARYTM_3:70;
            then
A30:        s *' s < t0 by A24,ARYTM_3:70;
            consider T2t9 being Element of RAT+ such that
A31:        (two *' t0) *' T2t9 = one by A15,ARYTM_3:55,78;
            set x = s *' s *' T2t9;
            consider t0x being Element of RAT+ such that
A32:        x + t0x = t0 or t0 + t0x = x by ARYTM_3:92;
            x *' (two *' t0) = s *' s *' one by A31,ARYTM_3:52;
            then x <=' s *' s or two *' t0 <=' one by ARYTM_3:83;
            then
A33:        x < t0 by A28,A30,ARYTM_3:53,69;
            then
A34:        t0x <=' t0 by A32;
A35:        x *' t0x + x *' t0 + x *' x = x *' t0x + x *' x + x *' t0 by
ARYTM_3:51
              .= x *' t0 + x *' t0 by A32,A33,ARYTM_3:57
              .= x *' t0 *' one + x *' t0 by ARYTM_3:53
              .= x *' t0 *' one + x *' t0 *' one by ARYTM_3:53
              .= t0 *' x *' two by Lm13,ARYTM_3:57
              .= x *' (t0 *' two) by ARYTM_3:52
              .= s *' s *' one by A31,ARYTM_3:52
              .= s *' s by ARYTM_3:53;
            es <=' t0 *' t0 by A17,A18;
            then s < t0 *' t0 by A21,ARYTM_3:69;
            then
A36:        s *' s < t0 *' t0 by A24,ARYTM_3:70;
            then t02s2 + x *' x + s *' s = (t0x + x) *' t0 + x *' x by A29,A32
,A33,ARYTM_3:51
              .= t0x *' (t0x + x) + x *' t0 + x *' x by A32,A33,ARYTM_3:57

              .= t0x *' t0x + x *' t0x + x *' t0 + x *' x by ARYTM_3:57
              .= t0x *' t0x + x *' t0x + (x *' t0 + x *' x) by ARYTM_3:51
              .= t0x *' t0x + (x *' t0x + (x *' t0 + x *' x)) by ARYTM_3:51
              .= t0x *' t0x + s *' s by A35,ARYTM_3:51;
            then t0x *' t0x = t02s2 + x *' x by ARYTM_3:62;
            then
A37:        t02s2 <=' t0x *' t0x;
            now
              assume
A38:          x = 0;
              per cases by A38,ARYTM_3:78;
              suppose
                s *' s = 0;
                hence contradiction by A23,ARYTM_3:78;
              end;
              suppose
                T2t9 = 0;
                hence contradiction by A31,ARYTM_3:48;
              end;
            end;
            then t0x <> t0 by A32,A33,ARYTM_3:84;
            then t0x < t0 by A34,ARYTM_3:68;
            then t0x in X by A14;
            then
A39:        ex s st s = t0x & s *' s < two;
            s *' s < es by A21,A24,ARYTM_3:70;
            then two + s *' s < two + es by ARYTM_3:76;
            then two < t02s2 by A17,A18,A29,A36,ARYTM_3:76;
            hence contradiction by A37,A39,ARYTM_3:69;
          end;
          suppose
A40:        one <=' s;
            half *' two = one *' one by A10,ARYTM_3:53;
            then
A41:        half <=' one by A12,ARYTM_3:83;
            half <> one by A10,ARYTM_3:53;
            then
A42:        half < one by A41,ARYTM_3:68;
            then half < s by A40,ARYTM_3:69;
            then
A43:        half < es by A21,ARYTM_3:70;
            one <=' two by Lm13;
            then one < two by ARYTM_3:68;
            then
A44:        one *' t0 < two *' t0 by A15,ARYTM_3:80;
A45:        now
              assume
A46:          t0 <=' one;
              then t0 *' t0 <=' t0 *' one by ARYTM_3:82;
              then t0 *' t0 <=' t0 by ARYTM_3:53;
              then t0 *' t0 <=' one by A46,ARYTM_3:67;
              hence contradiction by A11,A17,ARYTM_3:69;
            end;
            then one *' one < one *' t0 by ARYTM_3:80;
            then
A47:        one *' one < two *' t0 by A44,ARYTM_3:70;
            set s = half;
            consider t02s2 being Element of RAT+ such that
A48:        s *' s + t02s2 = t0 *' t0 or t0 *' t0 + t02s2 = s *' s by
ARYTM_3:92;
A49:        half <> 0 by A10,ARYTM_3:48;
            then half *' half < half *' one by A42,ARYTM_3:80;
            then
A50:        half *' half < half by ARYTM_3:53;
            s < t0 by A42,A45,ARYTM_3:70;
            then
A51:        s *' s < t0 by A50,ARYTM_3:70;
            consider T2t9 being Element of RAT+ such that
A52:        (two *' t0) *' T2t9 = one by A15,ARYTM_3:55,78;
            set x = s *' s *' T2t9;
            consider t0x being Element of RAT+ such that
A53:        x + t0x = t0 or t0 + t0x = x by ARYTM_3:92;
            x *' (two *' t0) = s *' s *' one by A52,ARYTM_3:52;
            then x <=' s *' s or two *' t0 <=' one by ARYTM_3:83;
            then
A54:        x < t0 by A47,A51,ARYTM_3:53,69;
            then
A55:        t0x <=' t0 by A53;
A56:        x *' t0x + x *' t0 + x *' x = x *' t0x + x *' x + x *' t0 by
ARYTM_3:51
              .= x *' t0 + x *' t0 by A53,A54,ARYTM_3:57
              .= x *' t0 *' one + x *' t0 by ARYTM_3:53
              .= x *' t0 *' one + x *' t0 *' one by ARYTM_3:53
              .= t0 *' x *' two by Lm13,ARYTM_3:57
              .= x *' (t0 *' two) by ARYTM_3:52
              .= s *' s *' one by A52,ARYTM_3:52
              .= s *' s by ARYTM_3:53;
            es <=' t0 *' t0 by A17,A18;
            then s < t0 *' t0 by A43,ARYTM_3:69;
            then
A57:        s *' s < t0 *' t0 by A50,ARYTM_3:70;
            then t02s2 + x *' x + s *' s = t0 *' t0 + x *' x by A48,ARYTM_3:51

              .= t0x *' (t0x + x) + x *' t0 + x *' x by A53,A54,ARYTM_3:57

              .= t0x *' t0x + x *' t0x + x *' t0 + x *' x by ARYTM_3:57
              .= t0x *' t0x + x *' t0x + (x *' t0 + x *' x) by ARYTM_3:51
              .= t0x *' t0x + (x *' t0x + (x *' t0 + x *' x)) by ARYTM_3:51
              .= t0x *' t0x + s *' s by A56,ARYTM_3:51;
            then t0x *' t0x = t02s2 + x *' x by ARYTM_3:62;
            then
A58:        t02s2 <=' t0x *' t0x;
            now
              assume
A59:          x = 0;
              per cases by A59,ARYTM_3:78;
              suppose
                s *' s = 0;
                hence contradiction by A49,ARYTM_3:78;
              end;
              suppose
                T2t9 = 0;
                hence contradiction by A52,ARYTM_3:48;
              end;
            end;
            then t0x <> t0 by A53,A54,ARYTM_3:84;
            then t0x < t0 by A55,ARYTM_3:68;
            then t0x in X by A14;
            then
A60:        ex s st s = t0x & s *' s < two;
            s *' s < es by A50,A43,ARYTM_3:70;
            then two + s *' s < two + es by ARYTM_3:76;
            then two < t02s2 by A17,A18,A48,A57,ARYTM_3:76;
            hence contradiction by A58,A60,ARYTM_3:69;
          end;
        end;
        hence contradiction;
      end;
    end;
    then
A61: two/1 = (n *^ n)/(d *^ d) by ARYTM_3:40;
    d <> 0 by ARYTM_3:35;
    then d *^ d <> {} by ORDINAL3:31;
    then
A62: two*^(d *^ d) = 1*^(n *^ n) by A61,ARYTM_3:45,Lm11
      .= n *^ n by ORDINAL2:39;
    then consider k being natural Ordinal such that
A63: n = 2 *^ k by Lm12;
    two*^(d *^ d) = 2 *^ (k *^ (2 *^ k)) by A62,A63,ORDINAL3:50;
    then d *^ d = k *^ (2 *^ k) by ORDINAL3:33,Lm11
      .= 2 *^ (k *^ k)by ORDINAL3:50;
    then
A64: ex p being natural Ordinal st d = 2 *^ p by Lm12;
    n, d are_coprime by ARYTM_3:34;
    hence contradiction by A63,A64;
  end;
  2 = succ 1;
  then 1 in 2 by ORDINAL1:6;
  then
A65: 1 *^ 2 in 2 *^ 2 by ORDINAL3:19;
A66: O9 <=' a9 by ARYTM_3:64;
  now
    let r;
    assume
A67: r in X;
    then
A68: ex s st r = s & s *' s < two;
    thus for t st t <=' r holds t in X by A6,A67;
    per cases;
    suppose
A69:  r = 0;
      take a9;
      thus a9 in X by A4,A3;
      thus r < a9 by A66,A69,ARYTM_3:68;
    end;
    suppose
A70:  r <> 0;
      then consider T3r9 being Element of RAT+ such that
A71:  (r + r + r) *' T3r9 = one by ARYTM_3:54,63;
      consider rr being Element of RAT+ such that
A72:  r *' r + rr = two or two + rr = r *' r by ARYTM_3:92;
      set eps = rr *' T3r9;
A73:  now
        assume
A74:    eps = 0;
        per cases by A74,ARYTM_3:78;
        suppose
          rr = O9;
          then r *' r = two by A72,ARYTM_3:50;
          hence contradiction by A68;
        end;
        suppose
          T3r9 = O9;
          hence contradiction by A71,ARYTM_3:48;
        end;
      end;
      now
        per cases;
        suppose
          eps < r;
          then eps *' eps < r *' eps by A73,ARYTM_3:80;
          then
A75:      r *' eps + eps *' r + eps *' eps < r *' eps + eps *' r + r *'
          eps by ARYTM_3:76;
          take t = r + eps;
A76:      t *' t = r *' t + eps *' t by ARYTM_3:57
            .= r *' r + r *' eps + eps *' t by ARYTM_3:57
            .= r *' r + r *' eps + (eps *' r + eps *' eps) by ARYTM_3:57
            .= r *' r + (r *' eps + (eps *' r + eps *' eps)) by ARYTM_3:51
            .= r *' r + (r *' eps + eps *' r + eps *' eps) by ARYTM_3:51;
          r *' eps + eps *' r + r *' eps = eps *' (r + r) + r *' eps by
ARYTM_3:57
            .= eps *' (r + r + r) by ARYTM_3:57
            .= rr *' one by A71,ARYTM_3:52
            .= rr by ARYTM_3:53;
          then t *' t < two by A68,A72,A75,A76,ARYTM_3:76;
          hence t in X;
          O9 <=' eps by ARYTM_3:64;
          then O9 < eps by A73,ARYTM_3:68;
          then r + O9 < r + eps by ARYTM_3:76;
          hence r < t by ARYTM_3:50;
        end;
        suppose
A77:      r <=' eps;
          eps *' (r + r + r) = rr *' one by A71,ARYTM_3:52
            .= rr by ARYTM_3:53;
          then
A78:      r *' (r + r + r) <=' rr by A77,ARYTM_3:82;
          take t = (a9 + half) *' r;
          a9 < two *' one by A3,ARYTM_3:53;
          then half < one by A10,ARYTM_3:82;
          then one + half < two by Lm13,ARYTM_3:76;
          then
A79:      t < two *' r by A70,ARYTM_3:80;
          then
A80:      two *' r *' t < two *' r *' (two *' r) by A70,ARYTM_3:78,80;
          a9 + half <> 0 by ARYTM_3:63;
          then t *' t < two *' r *' t by A70,A79,ARYTM_3:78,80;
          then
A81:      t *' t < two *' r *' (two *' r) by A80,ARYTM_3:70;
          r *' (r + r + r) + r *' r = r *' (r + r) + r *' r + r *' r by
ARYTM_3:57
            .= r *' (r + r) + (r *' r + r *' r) by ARYTM_3:51
            .= r *' (two *' r) + (r *' r + r *' r) by Lm14
            .= r *' (two *' r) + two *' (r *' r) by Lm14
            .= two *' (r *' r) + two *' (r *' r) by ARYTM_3:52
            .= two *' (two *' (r *' r)) by Lm14
            .= two *' (two *' r *' r) by ARYTM_3:52
            .= (two *' r) *' (two *' r) by ARYTM_3:52;
          then two *' r *' (two *' r) <=' two by A68,A72,A78,ARYTM_3:76;
          then t *' t < two by A81,ARYTM_3:69;
          hence t in X;
          O9 <> half & O9 <=' half by A10,ARYTM_3:48,64;
          then O9 < half by ARYTM_3:68;
          then one + O9 < one + half by ARYTM_3:76;
          then one < one + half by ARYTM_3:50;
          then one *' r < t by A70,ARYTM_3:80;
          hence r < t by ARYTM_3:53;
        end;
      end;
      hence ex t st t in X & r < t;
    end;
  end;
  then
A82: X in DD;
  a9 *' half = half by ARYTM_3:53;
  then
A83: half in X by A10,A6,A2,A5,ARYTM_3:82;
A84: now
    assume
A85: X in RAT;
    per cases by A85,XBOOLE_0:def 3;
    suppose
A86:  X in RAT+;
      now
        per cases by A86,XBOOLE_0:def 3;
        suppose
          X in {[i,j]: i,j are_coprime & j <> {}} \ the set of all
[k,one];
          then X in {[i,j]: i,j are_coprime & j <> {}};
          then ex i,j st X = [i,j] & i,j are_coprime & j <> {};
          hence contradiction by A8,TARSKI:def 2;
        end;
        suppose
A87:      X in omega;
          2 c= X by A5,A8,Lm11,ZFMISC_1:32;
          then
A88:      not X in 2 by ORDINAL1:5;
          now
            per cases by A87,A88,ORDINAL1:14;
            suppose
              X = two;
              then half = 0 or half = 1 by A83,Lm11,TARSKI:def 2;
              hence contradiction by A10,ARYTM_3:48,53;
            end;
            suppose
              two in X;
              then ex s st s = two & s *' s < two;
              hence contradiction by A1,A65,Lm9;
            end;
          end;
          hence contradiction;
        end;
      end;
      hence contradiction;
    end;
    suppose
      X in [:{0},RAT+:];
      then ex x,y being object st X = [x,y] by RELAT_1:def 1;
      hence contradiction by A8,TARSKI:def 2;
    end;
  end;
  now
    assume two in X;
    then ex s st two = s & s *' s < two;
    hence contradiction by A1,A65,Lm9;
  end;
  then X <> RAT+;
  then not X in {RAT+} by TARSKI:def 1;
  then X in DEDEKIND_CUTS by A82,ARYTM_2:def 1,XBOOLE_0:def 5;
  then X in RAT+ \/ DEDEKIND_CUTS by XBOOLE_0:def 3;
  then X in REAL+ by A13,ARYTM_2:def 2,XBOOLE_0:def 5;
  then X in REAL+ \/ [:{0},REAL+:] by XBOOLE_0:def 3;
  then X in REAL by A9,XBOOLE_0:def 5;
  hence thesis by A84,Lm8;
end;
