reserve

  k,n for Nat,
  x,y,X,Y,Z for set;

theorem Th7:
  for a,b,c being set st card a = n - 1 & card b = n - 1 & card c =
n - 1 & card(a /\ b) = n - 2 & card(a /\ c) = n - 2 & card(b /\ c) = n - 2 & 2
<= n holds (3 <= n implies card(a /\ b /\ c) = n - 2 & card(a \/ b \/ c) = n +
1 or card(a /\ b /\ c) = n - 3 & card(a \/ b \/ c) = n ) & (n = 2 implies card(
  a /\ b /\ c) = n - 2 & card(a \/ b \/ c) = n + 1 )
proof
  let a,b,c be set;
  assume that
A1: card a = n - 1 and
A2: card b = n - 1 and
A3: card c = n - 1 and
A4: card(a /\ b) = n - 2 and
A5: card(a /\ c) = n - 2 and
A6: card(b /\ c) = n - 2 and
A7: 2 <= n;
  2 <= n + 1 by A7,NAT_1:13;
  then
A8: 2 - 1 <= (n + 1) - 1 by XREAL_1:13;
  then
 a is finite by A1,NAT_1:21;
  then reconsider a as finite set;
A9: card(a \ (a /\ b)) = (n - 1) - (n - 2) by A1,A4,CARD_2:44,XBOOLE_1:17;
  then consider x1 being object such that
A10: {x1} = a \ (a /\ b) by CARD_2:42;
 b is finite by A2,A8,NAT_1:21;
  then reconsider b as finite set;
  card(b \ (a /\ b)) = (n - 1) - (n - 2) by A2,A4,CARD_2:44,XBOOLE_1:17;
  then consider x2 being object such that
A11: {x2} = b \ (a /\ b) by CARD_2:42;
  c is finite by A3,A8,NAT_1:21;
  then card(c \ (a /\ c)) = (n - 1) - (n - 2) by A3,A5,CARD_2:44,XBOOLE_1:17;
  then consider x3 being object such that
A12: {x3} = c \ (a /\ c) by CARD_2:42;
A13: a = (a /\ b) \/ {x1} by A10,XBOOLE_1:17,45;
A14: a /\ b /\ c = b /\ c /\ a by XBOOLE_1:16;
A15: a /\ c c= a by XBOOLE_1:17;
A16: a /\ b /\ c = a /\ c /\ b by XBOOLE_1:16;
A17: b = (a /\ b) \/ {x2} by A11,XBOOLE_1:17,45;
  x3 in {x3} by TARSKI:def 1;
  then
A18: not x3 in a /\ c by A12,XBOOLE_0:def 5;
A19: c = (a /\ c) \/ {x3} by A12,XBOOLE_1:17,45;
A20: x2 in {x2} by TARSKI:def 1;
  then
A21: not x2 in a /\ b by A11,XBOOLE_0:def 5;
A22: x1 in {x1} by TARSKI:def 1;
  then
A23: not x1 in a /\ b by A10,XBOOLE_0:def 5;
  then
A24: x1 <> x2 by A10,A11,A20,XBOOLE_0:def 4;
A25: a /\ b c= b by XBOOLE_1:17;
A26: 3 <= n implies card(a /\ b /\ c) = n - 2 & card(a \/ b \/ c) = n + 1 or
  card(a /\ b /\ c) = n - 3 & card(a \/ b \/ c) = n
  proof
    assume 3 <= n;
A27: x1 in c implies card(a /\ b /\ c) = n - 3 & card(a \/ b \/ c) = n
    proof
      a /\ b /\ c misses {x1}
      proof
        assume a /\ b /\ c meets {x1};
        then not a /\ b /\ c /\ {x1} = {} by XBOOLE_0:def 7;
        then consider x being object such that
A28:    x in a /\ b /\ c /\ {x1} by XBOOLE_0:def 1;
        x in {x1} by A28,XBOOLE_0:def 4;
        then
A29:    x = x1 by TARSKI:def 1;
        x in a /\ b /\ c by A28,XBOOLE_0:def 4;
        hence contradiction by A23,A29,XBOOLE_0:def 4;
      end;
      then
A30:  a /\ b /\ c c= (a /\ c) \ {x1} by A16,XBOOLE_1:17,86;
      (a /\ c) \ {x1} c= b
      proof
        let z be object;
        assume
A31:    z in (a /\ c) \ {x1};
        then not z in {x1} by XBOOLE_0:def 5;
        then z in a & not z in a \ (a /\ b) & not z in a or z in (a /\ b) by
A10,A31,XBOOLE_0:def 4,def 5;
        hence thesis by XBOOLE_0:def 4;
      end;
      then (a /\ c) \ {x1} c= a /\ c /\ b by XBOOLE_1:19;
      then
A32:  (a /\ c) \ {x1} c= a /\ b /\ c by XBOOLE_1:16;
A33:  a /\ b misses {x1,x2}
      proof
        assume a /\ b meets {x1,x2};
        then (a /\ b) /\ {x1,x2} <> {} by XBOOLE_0:def 7;
        then consider z1 being object such that
A34:    z1 in (a /\ b) /\ {x1,x2} by XBOOLE_0:def 1;
        z1 in a /\ b & z1 in {x1,x2} by A34,XBOOLE_0:def 4;
        hence contradiction by A23,A21,TARSKI:def 2;
      end;
      assume x1 in c;
      then x1 in a /\ c by A10,A22,XBOOLE_0:def 4;
      then
A35:  {x1} c= a /\ c by ZFMISC_1:31;
      a \/ b = (a /\ b) \/ ({x1} \/ {x2}) by A13,A17,XBOOLE_1:5;
      then
A36:  a \/ b = (a /\ b) \/ {x1,x2} by ENUMSET1:1;
      card {x1} = 1 by CARD_1:30;
      then
A37:  card((a /\ c) \ {x1}) = (n - 2) - 1 by A5,A35,CARD_2:44;
      then
A38:  card(a /\ b /\ c) = n - 3 by A30,A32,XBOOLE_0:def 10;
      x3 = x2
      proof
        assume
A39:    x2 <> x3;
        b /\ c c= a /\ b /\ c
        proof
          let z1 be object;
          assume
A40:      z1 in b /\ c;
          then z1 in b by XBOOLE_0:def 4;
          then z1 in a /\ b or z1 in {x2} by A17,XBOOLE_0:def 3;
          then
A41:      z1 in a /\ b or z1 = x2 by TARSKI:def 1;
          z1 in c by A40,XBOOLE_0:def 4;
          then z1 in a /\ c or z1 in {x3} by A19,XBOOLE_0:def 3;
          then
          (z1 in a /\ b or z1 in {x2}) & z1 in a /\ c or z1 in a /\ b & (
          z1 in a /\ c or z1 in {x3}) by A39,A41,TARSKI:def 1;
          hence thesis by A25,A11,A12,A16,XBOOLE_0:def 4;
        end;
        then Segm card(b /\ c) c= Segm card(a /\ b /\ c) by CARD_1:11;
        then - 2 + n <= - 3 + n by A6,A38,NAT_1:39;
        hence contradiction by XREAL_1:6;
      end;
      then
A42:  c c= a \/ b by A15,A11,A19,XBOOLE_1:13;
      card{x1, x2} = 2 by A24,CARD_2:57;
      then card(a \/ b) = (n - 2) + 2 by A4,A36,A33,CARD_2:40;
      hence thesis by A37,A30,A32,A42,XBOOLE_0:def 10,XBOOLE_1:12;
    end;
    not x1 in c implies card(a /\ b /\ c) = n - 2 & card(a \/ b \/ c) = n + 1
    proof
A43:  x1 <> x3 by A10,A12,A22,A18,XBOOLE_0:def 4;
A44:  card(a \ {x1}) = (n - 1) - 1 by A1,A9,A10,CARD_2:44;
      assume
A45:  not x1 in c;
A46:  a /\ c misses {x1} & a /\ b misses {x1}
      proof
        assume not a /\ c misses {x1} or not a /\ b misses {x1};
        then (a /\ c) /\ {x1} <> {} or (a /\ b) /\ {x1} <> {} by XBOOLE_0:def 7
;
        then consider z2 being object such that
A47:    z2 in (a /\ c) /\ {x1} or z2 in (a /\ b) /\ {x1} by XBOOLE_0:def 1;
        z2 in a /\ c & z2 in {x1} or z2 in a /\ b & z2 in {x1} by A47,
XBOOLE_0:def 4;
        then z2 in a & z2 in c & z2 = x1 or z2 in a /\ b & z2 = x1 by
TARSKI:def 1,XBOOLE_0:def 4;
        hence contradiction by A10,A22,A45,XBOOLE_0:def 5;
      end;
      then a /\ c c= a \ {x1} by XBOOLE_1:17,86;
      then
A48:  a /\ c = a \ {x1} by A5,A44,CARD_2:102;
      a /\ b c= a \ {x1} by A46,XBOOLE_1:17,86;
      then
A49:  a /\ b = a \ {x1} by A4,A44,CARD_2:102;
A50:  a /\ b misses {x1,x2,x3}
      proof
        assume not a /\ b misses {x1,x2,x3};
        then (a /\ b) /\ {x1,x2,x3} <> {} by XBOOLE_0:def 7;
        then consider z3 being object such that
A51:    z3 in (a /\ b) /\ {x1,x2,x3} by XBOOLE_0:def 1;
        z3 in a /\ b & z3 in {x1,x2,x3} by A51,XBOOLE_0:def 4;
        hence contradiction by A23,A21,A18,A48,A49,ENUMSET1:def 1;
      end;
      a \/ b = (a /\ b) \/ ({x1} \/ {x2}) by A13,A17,XBOOLE_1:5;
      then a \/ b = (a /\ b) \/ {x1,x2} by ENUMSET1:1;
      then a \/ b \/ c = (a /\ b) \/ ({x1,x2} \/ {x3}) by A19,A48,A49,
XBOOLE_1:5;
      then
A52:  a \/ b \/ c = (a /\ b) \/ {x1,x2,x3} by ENUMSET1:3;
      (a /\ b) /\ (a /\ c) = a /\ b by A48,A49;
      then b /\ a /\ a /\ c = a /\ b by XBOOLE_1:16;
      then
A53:  b /\ (a /\ a) /\ c = a /\ b by XBOOLE_1:16;
      then a /\ b /\ c = b /\ c by A4,A6,A14,CARD_2:102,XBOOLE_1:17;
      then x2 <> x3 by A11,A12,A20,A21,A53,XBOOLE_0:def 4;
      then card{x1,x2,x3} = 3 by A24,A43,CARD_2:58;
      then card(a \/ b \/ c) = (n - 2) + 3 by A4,A52,A50,CARD_2:40;
      hence thesis by A4,A53;
    end;
    hence thesis by A27;
  end;
A54: x1 <> x3 by A10,A12,A22,A18,XBOOLE_0:def 4;
  n = 2 implies card(a /\ b /\ c) = n - 2 & card(a \/ b \/ c) = n + 1
  proof
    assume
A55: n = 2;
    then
A56: a /\ b = {} by A4;
    then a /\ b /\ c = a /\ c by A4,A5;
    then a \/ b \/ c = (a /\ b /\ c) \/ ({x1,x2} \/ {x3}) by A10,A11,A12,A56,
ENUMSET1:1;
    then
A57: a \/ b \/ c = (a /\ b /\ c) \/ {x1,x2,x3} by ENUMSET1:3;
    a /\ b /\ c = b /\ c by A4,A6,A56;
    then x2 <> x3 by A11,A12,A20,A56,XBOOLE_0:def 4;
    hence thesis by A24,A54,A55,A56,A57,CARD_2:58;
  end;
  hence thesis by A26;
end;
