reserve n,m,k for Nat,
  X,Y,Z for set,
  f for Function of X,Y,
  H for Subset of X;

theorem Th16:
  m >= 2 & n >= 2 implies ex r being Nat st r <= (m + n -' 2) choose (m -' 1) &
  r >= 2 & for X being finite set,
  F being Function of the_subsets_of_card(2,X), Seg 2 st card X >= r
  ex S being Subset of X
   st card S >= m & rng(F|the_subsets_of_card(2,S)) = {1} or card S >= n &
   rng(F|the_subsets_of_card(2,S)) = {2}
proof
  defpred P[Nat,Nat] means $1 >= 2 & $2 >= 2 implies ex
r being Nat st r <= ($1 + $2 -' 2) choose ($1 -' 1) & r >= 2 & for X
being finite set, F being Function of the_subsets_of_card(2,X), Seg 2 st card X
>= r holds ex S being Subset of X st (card S >= $1 & rng(F|the_subsets_of_card(
  2,S)) = {1}) or (card S >= $2 & rng(F|the_subsets_of_card(2,S)) = {2});
A1: for n,m being Nat st P[m+1,n] & P[m,n+1] holds P[m+1,n+1]
  proof
    let n,m be Nat;
    assume
A2: P[m+1,n];
    assume
A3: P[m,n+1];
    assume that
A4: m+1 >= 2 and
A5: n+1 >= 2;
    per cases by XXREAL_0:1;
    suppose
      m+1<2 or n+1<2;
      hence thesis by A4,A5;
    end;
    suppose
A6:   m+1=2;
      set r=n+1;
      take r;
      (m+1)+(n+1) >= 2+2 by A4,A5,XREAL_1:7;
      then (m+1)+(n+1)-2 >= 4-2 by XREAL_1:9;
      then
A7:   m+n = (m+1) + (n+1) -' 2 by XREAL_0:def 2;
      (m+1) -' 1 = m & m+1-1 >= 2-1 by A4,NAT_D:34,XREAL_1:9;
      hence r <= ((m+1) + (n+1) -' 2) choose ((m+1) -' 1) by A7,Th11;
      thus r >= 2 by A5;
      let X be finite set;
      let F be Function of the_subsets_of_card(2,X), Seg 2;
      assume
A8:   card X >= r;
      F in Funcs(the_subsets_of_card(2,X), Seg 2) by FUNCT_2:8;
      then
A9:   ex f be Function st F = f & dom f = the_subsets_of_card( 2,X) & rng
      f c= Seg 2 by FUNCT_2:def 2;
      per cases;
      suppose
A10:    not 1 in rng F;
        consider S be Subset of X such that
A11:    card S = r by A8,Th10;
        card 2 <= card S by A5,A11;
        then Segm card 2 c= Segm card S by NAT_1:39;
        then the_subsets_of_card(2,S) is non empty by A11,CARD_1:27,GROUP_10:1;
        then
A12:    ex x be object st x in the_subsets_of_card(2,S) by XBOOLE_0:def 1;
        take S;
A13:    rng(F|the_subsets_of_card(2,S)) c= rng F by RELAT_1:70;
        then
A14:    rng(F|the_subsets_of_card(2,S)) c= {1,2} by A9,FINSEQ_1:2;
        the_subsets_of_card(2,S) c= the_subsets_of_card(2,X) by Lm1;
        then
A15:    F|the_subsets_of_card(2,S) <> {} by A9,A12;
        now
          let x be object;
A16:      rng(F|the_subsets_of_card(2,S)) = {} or rng(F|
the_subsets_of_card(2,S)) = {1} or rng(F|the_subsets_of_card(2,S)) = {2} or rng
          (F|the_subsets_of_card(2,S)) = {1,2} by A14,ZFMISC_1:36;
          hereby
            assume
A17:        x in rng(F|the_subsets_of_card(2,S));
            then x=1 or x=2 by A14,TARSKI:def 2;
            hence x=2 by A10,A13,A17;
          end;
          assume
A18:      x=2;
A19:      not 1 in rng(F|the_subsets_of_card(2,S)) by A10,A13;
          assume not x in rng(F|the_subsets_of_card(2,S));
          hence contradiction by A15,A18,A19,A16,TARSKI:def 1,def 2;
        end;
        hence thesis by A11,TARSKI:def 1;
      end;
      suppose
        1 in rng F;
        then consider S be object such that
A20:    S in dom F and
A21:    1 = F.S by FUNCT_1:def 3;
        S in {X9 where X9 is Subset of X: card X9 = 2} by A20;
        then
A22:    ex X9 be Subset of X st S=X9 & card X9 = 2;
        then reconsider S as Subset of X;
        the_subsets_of_card(2,S) = {S} by A22,Lm3;
        then S in the_subsets_of_card(2,S) by TARSKI:def 1;
        then
A23:    F|the_subsets_of_card(2,S).S = 1 by A21,FUNCT_1:49;
        take S;
A24:    {S} c= dom F by A20,ZFMISC_1:31;
        dom(F|the_subsets_of_card(2,S)) = dom F /\ the_subsets_of_card(2,
        S) by RELAT_1:61
          .= dom F /\ {S} by A22,Lm3
          .= {S} by A24,XBOOLE_1:28;
        hence thesis by A6,A22,A23,FUNCT_1:4;
      end;
    end;
    suppose
A25:  n+1=2;
      set r=m+1;
      take r;
A26:  n+1-1 >= 2-1 by A5,XREAL_1:9;
      (m+1)+(n+1) >= 2+2 by A4,A5,XREAL_1:7;
      then (m+1)+(n+1)-2 >= 4-2 by XREAL_1:9;
      then
A27:  m+n = (m+1) + (n+1) -' 2 by XREAL_0:def 2;
      (m+1) -' 1 = m & m+1-1 >= 2-1 by A4,NAT_D:34,XREAL_1:9;
      hence r <= ((m+1) + (n+1) -' 2) choose ((m+1) -' 1) by A27,A26,Th12;
      thus r >= 2 by A4;
      let X be finite set;
      let F be Function of the_subsets_of_card(2,X), Seg 2;
      assume
A28:  card X >= r;
      F in Funcs(the_subsets_of_card(2,X), Seg 2) by FUNCT_2:8;
      then
A29:  ex f be Function st F = f & dom f = the_subsets_of_card( 2,X) & rng
      f c= Seg 2 by FUNCT_2:def 2;
      per cases;
      suppose
A30:    not 2 in rng F;
        consider S be Subset of X such that
A31:    card S = r by A28,Th10;
        card 2 <= card S by A4,A31;
        then Segm card 2 c= Segm card S by NAT_1:39;
        then the_subsets_of_card(2,S) is non empty by A31,CARD_1:27,GROUP_10:1;
        then
A32:    ex x be object st x in the_subsets_of_card(2,S) by XBOOLE_0:def 1;
        take S;
A33:    rng(F|the_subsets_of_card(2,S)) c= rng F by RELAT_1:70;
        then
A34:    rng(F|the_subsets_of_card(2,S)) c= {1,2} by A29,FINSEQ_1:2;
        the_subsets_of_card(2,S) c= the_subsets_of_card(2,X) by Lm1;
        then
A35:    F|the_subsets_of_card(2,S) <> {} by A29,A32;
        now
          let x be object;
A36:      rng(F|the_subsets_of_card(2,S)) = {} or rng(F|
the_subsets_of_card(2,S)) = {1} or rng(F|the_subsets_of_card(2,S)) = {2} or rng
          (F|the_subsets_of_card(2,S)) = {1,2} by A34,ZFMISC_1:36;
          hereby
            assume
A37:        x in rng(F|the_subsets_of_card(2,S));
            then x=1 or x=2 by A34,TARSKI:def 2;
            hence x=1 by A30,A33,A37;
          end;
          assume
A38:      x=1;
A39:      not 2 in rng(F|the_subsets_of_card(2,S)) by A30,A33;
          assume not x in rng(F|the_subsets_of_card(2,S));
          hence contradiction by A35,A38,A39,A36,TARSKI:def 1,def 2;
        end;
        hence thesis by A31,TARSKI:def 1;
      end;
      suppose
        2 in rng F;
        then consider S be object such that
A40:    S in dom F and
A41:    2 = F.S by FUNCT_1:def 3;
        S in {X9 where X9 is Subset of X: card X9 = 2} by A40;
        then
A42:    ex X9 be Subset of X st S=X9 & card X9 = 2;
        then reconsider S as Subset of X;
        the_subsets_of_card(2,S) = {S} by A42,Lm3;
        then S in the_subsets_of_card(2,S) by TARSKI:def 1;
        then
A43:    F|the_subsets_of_card(2,S).S = 2 by A41,FUNCT_1:49;
        take S;
A44:    {S} c= dom F by A40,ZFMISC_1:31;
        dom(F|the_subsets_of_card(2,S)) = dom F /\ the_subsets_of_card(2,
        S) by RELAT_1:61
          .= dom F /\ {S} by A42,Lm3
          .= {S} by A44,XBOOLE_1:28;
        hence thesis by A25,A42,A43,FUNCT_1:4;
      end;
    end;
    suppose
A45:  m+1>2 & n+1>2;
      set t = m + n -' 1;
      set s = m -' 1;
      m + 1 - 2 >= 2 - 2 by A4,XREAL_1:9;
      then m -' 1 = m - 1 by XREAL_0:def 2;
      then
A46:  (m+1) -' 1 = s + 1 by NAT_D:34;
      (m+1) + (n+1) >= 2+2 by A4,A5,XREAL_1:7;
      then
A47:  m+n+2-3 >= 4-3 by XREAL_1:9;
      then
A48:  m + n -' 1 = m + n - 1 by XREAL_0:def 2;
A49:  m + n + 1 -' 2 = m + n + 1 - 2 by A47,XREAL_0:def 2;
      m + n >= 0;
      then
A50:  (m+1) + (n+1) -' 2 = (m+1) + (n+1) - 2 by XREAL_0:def 2
        .= t + 1 by A48;
      consider r2 be Nat such that
A51:  r2 <= (m + (n+1) -' 2) choose (m -' 1) and
      r2 >= 2 and
A52:  for X being finite set, F being Function of the_subsets_of_card
(2,X), Seg 2 st card X >= r2 holds ex S being Subset of X st card S >= m & rng(
F|the_subsets_of_card(2,S)) = {1} or card S >= n+1 & rng(F|the_subsets_of_card(
      2,S)) = {2} by A3,A45,NAT_1:13;
      consider r1 be Nat such that
A53:  r1 <= ((m+1) + n -' 2) choose ((m+1) -' 1) and
A54:  r1 >= 2 and
A55:  for X being finite set, F being Function of the_subsets_of_card
(2,X), Seg 2 st card X >= r1 holds ex S being Subset of X st card S >= m+1 &
      rng(F|the_subsets_of_card(2,S)) = {1} or card S >= n & rng(F|
      the_subsets_of_card(2,S)) = {2} by A2,A45,NAT_1:13;
      set r = r1+r2;
      take r;
      r1+r2 <= ((m+1) + n -' 2) choose ((m+1) -' 1) + (m + (n+1) -' 2)
      choose (m -' 1) by A53,A51,XREAL_1:7;
      hence r <= ((m+1) + (n+1) -' 2) choose ((m+1) -' 1) by A48,A49,A50,A46,
NEWTON:22;
      r1 + r2 >= 0 + 2 by A54,XREAL_1:7;
      hence r >= 2;
      let X be finite set;
      let F be Function of (the_subsets_of_card(2,X)), Seg 2;
      assume card X >= r;
      then consider S be Subset of X such that
A56:  card S = r by Th10;
      consider s be object such that
A57:  s in S by A54,A56,CARD_1:27,XBOOLE_0:def 1;
      set B = {s9 where s9 is Element of S: F.{s,s9} = 2 & {s,s9} in dom F};
      set A = {s9 where s9 is Element of S: F.{s,s9} = 1 & {s,s9} in dom F};
      F in Funcs(the_subsets_of_card(2,X), Seg 2) by FUNCT_2:8;
      then
A58:  ex f be Function st F = f & dom f = the_subsets_of_card( 2,X) & rng
      f c= Seg 2 by FUNCT_2:def 2;
A59:  for x being object holds x in A \/ B iff x in S \ {s}
      proof
        let x be object;
        hereby
          assume
A60:      x in A \/ B;
          per cases by A60,XBOOLE_0:def 3;
          suppose
            x in A;
            then consider s9 be Element of S such that
A61:        x = s9 and
            F.{s,s9} = 1 and
A62:        {s,s9} in dom F;
            now
              assume x in {s};
              then
A63:          x = s by TARSKI:def 1;
              {s,s9} = {s} \/ {s9} by ENUMSET1:1
                .= {s} by A61,A63;
              then {s} in the_subsets_of_card(2,X) by A62;
              then ex X9 be Subset of X st X9={s} & card X9 = 2;
              hence contradiction by CARD_1:30;
            end;
            hence x in S \ {s} by A54,A56,A61,CARD_1:27,XBOOLE_0:def 5;
          end;
          suppose
            x in B;
            then consider s9 be Element of S such that
A64:        x = s9 and
            F.{s,s9} = 2 and
A65:        {s,s9} in dom F;
            now
              assume x in {s};
              then
A66:          x = s by TARSKI:def 1;
              {s,s9} = {s} \/ {s9} by ENUMSET1:1
                .= {s} by A64,A66;
              then {s} in the_subsets_of_card(2,X) by A65;
              then ex X9 be Subset of X st X9={s} & card X9 = 2;
              hence contradiction by CARD_1:30;
            end;
            hence x in S \ {s} by A54,A56,A64,CARD_1:27,XBOOLE_0:def 5;
          end;
        end;
        assume
A67:    x in S \ {s};
        then reconsider s9=x as Element of S by XBOOLE_0:def 5;
        not s9 in {s} by A67,XBOOLE_0:def 5;
        then s<>s9 by TARSKI:def 1;
        then
A68:    card {s,s9} = 2 by CARD_2:57;
        {s,s9} c= S by A57,ZFMISC_1:32;
        then {s,s9} is Subset of X by XBOOLE_1:1;
        then
A69:    {s,s9} in dom F by A58,A68;
        then
A70:    F.{s,s9} in rng F by FUNCT_1:3;
        per cases by A58,A70,FINSEQ_1:2,TARSKI:def 2;
        suppose
          F.{s,s9} = 1;
          then x in A by A69;
          hence thesis by XBOOLE_0:def 3;
        end;
        suppose
          F.{s,s9} = 2;
          then x in B by A69;
          hence thesis by XBOOLE_0:def 3;
        end;
      end;
A71:  now
        assume A /\ B <> {};
        then consider x be object such that
A72:    x in A /\ B by XBOOLE_0:def 1;
        x in B by A72,XBOOLE_0:def 4;
        then
A73:    ex s2 be Element of S st x = s2 & F.{s,s2} = 2 & {s,s2} in dom F;
        x in A by A72,XBOOLE_0:def 4;
        then ex s1 be Element of S st x = s1 & F.{s,s1} = 1 & {s,s1} in dom F;
        hence contradiction by A73;
      end;
      S \ {s} c= S by XBOOLE_1:36;
      then
A74:  A \/ B c= S by A59;
      {s} c= S by A57,ZFMISC_1:31;
      then
A75:  card(S \ {s}) = card S - card {s} by CARD_2:44
        .= r1 + r2 - 1 by A56,CARD_1:30;
      reconsider B as finite Subset of S by A74,XBOOLE_1:11;
      reconsider A as finite Subset of S by A74,XBOOLE_1:11;
      card (A \/ B) = card A + card B - card {} by A71,CARD_2:45
        .= card A + card B;
      then
A76:  card A + card B = r1 + r2 - 1 by A59,A75,TARSKI:2;
A77:  card A >= r2 or card B >= r1
      proof
        assume card A < r2;
        then
A78:    card A + 1 <= r2 by NAT_1:13;
        assume card B < r1;
        then card A + 1 + card B < r2 + r1 by A78,XREAL_1:8;
        hence contradiction by A76;
      end;
      per cases by A77;
      suppose
A79:    card A >= r2;
        set F9 = F|the_subsets_of_card(2,A);
        A c= X by XBOOLE_1:1;
        then the_subsets_of_card(2,X) /\ the_subsets_of_card(2,A) =
        the_subsets_of_card(2,A) by Lm1,XBOOLE_1:28;
        then
A80:    dom(F|the_subsets_of_card(2,A)) = the_subsets_of_card(2,A) by A58,
RELAT_1:61;
        rng(F|the_subsets_of_card(2,A)) c= rng F by RELAT_1:70;
        then reconsider
        F9 as Function of the_subsets_of_card(2,A), Seg 2 by A58,A80,FUNCT_2:2
,XBOOLE_1:1;
        consider S9 be Subset of A such that
A81:    card S9 >= m & rng(F9|the_subsets_of_card(2,S9)) = {1} or
        card S9 >= n+1 & rng(F9|the_subsets_of_card(2,S9)) = {2} by A52,A79;
A82:    F9|the_subsets_of_card(2,S9) = F|the_subsets_of_card(2,S9) by Lm1,
RELAT_1:74;
        A c= X by XBOOLE_1:1;
        then reconsider S9 as Subset of X by XBOOLE_1:1;
        per cases by A81;
        suppose
A83:      card S9 >= n+1 & rng(F9|the_subsets_of_card(2,S9)) = {2};
          take S9;
          thus thesis by A83,Lm1,RELAT_1:74;
        end;
        suppose
A84:      card S9 >= m & rng(F9|the_subsets_of_card(2,S9)) = {1};
          set S99 = S9 \/ {s};
          {s} c= X by A57,ZFMISC_1:31;
          then reconsider S99 as Subset of X by XBOOLE_1:8;
A85:      the_subsets_of_card(2,S9) c= the_subsets_of_card(2,S99) by Lm1,
XBOOLE_1:7;
A86:      rng(F|the_subsets_of_card(2,S9)) = {1} by A84,Lm1,RELAT_1:74;
A87:      for y being object holds y in rng(F|the_subsets_of_card(2,S99))
          iff y = 1
          proof
            let y be object;
            F|the_subsets_of_card(2,S9) c= F|the_subsets_of_card(2,S99)
            by A85,RELAT_1:75;
            then
A88:        rng(F|the_subsets_of_card(2,S9)) c= rng(F|
            the_subsets_of_card(2,S99)) by RELAT_1:11;
            hereby
              assume y in rng(F|the_subsets_of_card(2,S99));
              then consider x be object such that
A89:          x in dom(F|the_subsets_of_card(2,S99)) and
A90:          y = (F|the_subsets_of_card(2,S99)).x by FUNCT_1:def 3;
A91:          x in the_subsets_of_card(2,S99) by A89,RELAT_1:57;
A92:          x in dom F by A89,RELAT_1:57;
              x in {S999 where S999 is Subset of S99: card S999 = 2} by A89,
RELAT_1:57;
              then consider S999 be Subset of S99 such that
A93:          x=S999 and
A94:          card S999 = 2;
              consider s1,s2 be object such that
A95:          s1 <> s2 and
A96:          S999 = {s1,s2} by A94,CARD_2:60;
A97:          s1 in S999 by A96,TARSKI:def 2;
A98:          s2 in S999 by A96,TARSKI:def 2;
              per cases by A97,XBOOLE_0:def 3;
              suppose
A99:            s1 in S9;
                per cases by A98,XBOOLE_0:def 3;
                suppose
                  s2 in S9;
                  then reconsider x as Subset of S9 by A93,A96,A99,ZFMISC_1:32;
                  x in the_subsets_of_card(2,S9) by A93,A94;
                  then
A100:             x in dom(F|the_subsets_of_card(2,S9)) by A92,RELAT_1:57;
                  then
A101:             x in dom(F|the_subsets_of_card(2,S99)|
                  the_subsets_of_card(2,S9)) by A85,RELAT_1:74;
                  (F|the_subsets_of_card(2,S9)).x = (F|
the_subsets_of_card(2,S99)|the_subsets_of_card(2,S9)).x by A85,RELAT_1:74
                    .= (F|the_subsets_of_card(2,S99)).x by A101,FUNCT_1:47;
                  then y in rng(F|the_subsets_of_card(2,S9)) by A90,A100,
FUNCT_1:3;
                  hence y = 1 by A82,A84,TARSKI:def 1;
                end;
                suppose
A102:             s2 in {s};
                  s1 in A by A99;
                  then
A103:             ex s99 be Element of S st s1=s99 & F.{s,s99} = 1 & {s,
                  s99} in dom F;
                  x = {s1,s} by A93,A96,A102,TARSKI:def 1;
                  hence y = 1 by A90,A91,A103,FUNCT_1:49;
                end;
              end;
              suppose
A104:           s1 in {s};
                then
A105:           s <> s2 by A95,TARSKI:def 1;
                per cases by A98,XBOOLE_0:def 3;
                suppose
                  s2 in S9;
                  then s2 in A;
                  then ex s99 be Element of S st s2=s99 & F.{s,s99} = 1 & {s,
                  s99} in dom F;
                  then F.x = 1 by A93,A96,A104,TARSKI:def 1;
                  hence y = 1 by A90,A91,FUNCT_1:49;
                end;
                suppose
                  s2 in {s};
                  hence y=1 by A105,TARSKI:def 1;
                end;
              end;
            end;
            assume y = 1;
            then y in rng(F|the_subsets_of_card(2,S9)) by A86,TARSKI:def 1;
            hence thesis by A88;
          end;
          take S99;
A106:     not s in S9
          proof
            assume s in S9;
            then s in A;
            then
A107:       ex s9 be Element of S st s=s9 & F.{s,s9} = 1 & {s,s9} in dom F;
            {s,s} = {s} \/ {s} by ENUMSET1:1
              .= {s};
            then {s} in the_subsets_of_card(2,X) by A107;
            then ex X9 be Subset of X st X9={s} & card X9 = 2;
            hence contradiction by CARD_1:30;
          end;
          card S9 + 1 >= m + 1 by A84,XREAL_1:6;
          hence thesis by A87,A106,CARD_2:41,TARSKI:def 1;
        end;
      end;
      suppose
A108:   card B >= r1;
        set F9 = F|the_subsets_of_card(2,B);
        B c= X by XBOOLE_1:1;
        then the_subsets_of_card(2,X) /\ the_subsets_of_card(2,B) =
        the_subsets_of_card(2,B) by Lm1,XBOOLE_1:28;
        then
A109:   dom(F|the_subsets_of_card(2,B)) = the_subsets_of_card(2,B) by A58,
RELAT_1:61;
        rng(F|the_subsets_of_card(2,B)) c= rng F by RELAT_1:70;
        then reconsider
        F9 as Function of the_subsets_of_card(2,B), Seg 2 by A58,A109,FUNCT_2:2
,XBOOLE_1:1;
        consider S9 be Subset of B such that
A110:   card S9 >= m+1 & rng(F9|the_subsets_of_card(2,S9)) = {1} or
        card S9 >= n & rng(F9|the_subsets_of_card(2,S9)) = {2} by A55,A108;
A111:   F9|the_subsets_of_card(2,S9) = F|the_subsets_of_card(2,S9) by Lm1,
RELAT_1:74;
        B c= X by XBOOLE_1:1;
        then reconsider S9 as Subset of X by XBOOLE_1:1;
        per cases by A110;
        suppose
A112:     card S9 >= m+1 & rng(F9|the_subsets_of_card(2,S9)) = {1};
          take S9;
          thus thesis by A112,Lm1,RELAT_1:74;
        end;
        suppose
A113:     card S9 >= n & rng(F9|the_subsets_of_card(2,S9)) = {2};
          set S99 = S9 \/ {s};
          {s} c= X by A57,ZFMISC_1:31;
          then reconsider S99 as Subset of X by XBOOLE_1:8;
A114:     the_subsets_of_card(2,S9) c= the_subsets_of_card(2,S99) by Lm1,
XBOOLE_1:7;
A115:     rng(F|the_subsets_of_card(2,S9)) = {2} by A113,Lm1,RELAT_1:74;
A116:     for y being object holds y in rng(F|the_subsets_of_card(2,S99))
          iff y = 2
          proof
            let y be object;
            F|the_subsets_of_card(2,S9) c= F|the_subsets_of_card(2,S99)
            by A114,RELAT_1:75;
            then
A117:       rng(F|the_subsets_of_card(2,S9)) c= rng(F|
            the_subsets_of_card(2,S99)) by RELAT_1:11;
            hereby
              assume y in rng(F|the_subsets_of_card(2,S99));
              then consider x be object such that
A118:         x in dom(F|the_subsets_of_card(2,S99)) and
A119:         y = (F|the_subsets_of_card(2,S99)).x by FUNCT_1:def 3;
A120:         x in the_subsets_of_card(2,S99) by A118,RELAT_1:57;
A121:         x in dom F by A118,RELAT_1:57;
              x in {S999 where S999 is Subset of S99: card S999 = 2} by A118,
RELAT_1:57;
              then consider S999 be Subset of S99 such that
A122:         x=S999 and
A123:         card S999 = 2;
              consider s1,s2 be object such that
A124:         s1 <> s2 and
A125:         S999 = {s1,s2} by A123,CARD_2:60;
A126:         s1 in S999 by A125,TARSKI:def 2;
A127:         s2 in S999 by A125,TARSKI:def 2;
              per cases by A126,XBOOLE_0:def 3;
              suppose
A128:           s1 in S9;
                per cases by A127,XBOOLE_0:def 3;
                suppose
                  s2 in S9;
                  then reconsider x as Subset of S9 by A122,A125,A128,
ZFMISC_1:32;
                  x in the_subsets_of_card(2,S9) by A122,A123;
                  then
A129:             x in dom(F|the_subsets_of_card(2,S9)) by A121,RELAT_1:57;
                  then
A130:             x in dom(F|the_subsets_of_card(2,S99)|
                  the_subsets_of_card(2,S9)) by A114,RELAT_1:74;
                  (F|the_subsets_of_card(2,S9)).x = (F|
the_subsets_of_card(2,S99)|the_subsets_of_card(2,S9)).x by A114,RELAT_1:74
                    .= (F|the_subsets_of_card(2,S99)).x by A130,FUNCT_1:47;
                  then y in rng(F|the_subsets_of_card(2,S9)) by A119,A129,
FUNCT_1:3;
                  hence y = 2 by A111,A113,TARSKI:def 1;
                end;
                suppose
A131:             s2 in {s};
                  s1 in B by A128;
                  then
A132:             ex s99 be Element of S st s1=s99 & F.{s,s99} = 2 & {s,
                  s99} in dom F;
                  x = {s1,s} by A122,A125,A131,TARSKI:def 1;
                  hence y = 2 by A119,A120,A132,FUNCT_1:49;
                end;
              end;
              suppose
A133:           s1 in {s};
                then
A134:           s <> s2 by A124,TARSKI:def 1;
                per cases by A127,XBOOLE_0:def 3;
                suppose
                  s2 in S9;
                  then s2 in B;
                  then ex s99 be Element of S st s2=s99 & F.{s,s99} = 2 & {s,
                  s99} in dom F;
                  then F.x = 2 by A122,A125,A133,TARSKI:def 1;
                  hence y = 2 by A119,A120,FUNCT_1:49;
                end;
                suppose
                  s2 in {s};
                  hence y = 2 by A134,TARSKI:def 1;
                end;
              end;
            end;
            assume y = 2;
            then y in rng(F|the_subsets_of_card(2,S9)) by A115,TARSKI:def 1;
            hence thesis by A117;
          end;
          take S99;
A135:     not s in S9
          proof
            assume s in S9;
            then s in B;
            then
A136:       ex s9 be Element of S st s=s9 & F.{s,s9} = 2 & {s,s9} in dom F;
            {s,s} = {s} \/ {s} by ENUMSET1:1
              .= {s};
            then {s} in the_subsets_of_card(2,X) by A136;
            then ex X9 be Subset of X st X9={s} & card X9 = 2;
            hence contradiction by CARD_1:30;
          end;
          card S9 + 1 >= n + 1 by A113,XREAL_1:6;
          hence thesis by A116,A135,CARD_2:41,TARSKI:def 1;
        end;
      end;
    end;
  end;
A137: for n being Nat holds P[0,n] & P[n,0];
  for n,m being Nat holds P[m,n] from BinInd2(A137,A1);
  hence thesis;
end;
