 reserve x,y,z for object, X for set,
         i,k,n,m for Nat,
         R for Relation,
         P for finite Relation,
         p,q for FinSequence;
reserve FSG for Friendship_Graph;

theorem Th10:
  FSG is without_universal_friend & x in field FSG
    implies card Im(FSG,x) > 2
proof
  assume A1: FSG is without_universal_friend;
  set I=Im(FSG,x);
  assume A2:x in field FSG;
  assume A3:card I <= 2;
  reconsider xx=x as Element of field FSG by A2;
  card I = 0 or ... or card I = 2 by A3; then
  per cases;
    suppose A4:card I=0;
      xx is universal_friend
      proof
        let y;
        assume A5: y in field FSG\{xx};
        then xx<>y by ZFMISC_1:56;
        then consider z being object such that
          A6: I/\Coim(FSG,y) ={z} by A5,Def3;
        z in {z} by TARSKI:def 1;
        then z in I by A6,XBOOLE_0:def 4;
        hence thesis by A4;
      end;
      hence thesis by A1;
    end;
    suppose card I=1;
      hence thesis by Th8,INT_2:13;
    end;
    suppose A7:card I=2;
      then consider y1,y2 be object such that
            A8: y1<>y2
        and A9: I={y1,y2} by CARD_2:60;
      consider z1 be object such that
            A10: z1 in field FSG\{x}
        and A11: not [xx,z1] in FSG by A1,Def1;
      A12: z1<>x by A10,ZFMISC_1:56;
      then consider p be object such that
        A13: I/\Coim(FSG,z1)={p} by A10, A2,Def3;
      A14: for y1,y2 be object st y1<>y2 & I={y1,y2} holds
        I/\Im(FSG,z1) <> {y1}
      proof
        let y1,y2 be object such that A15: y1<>y2
        and  A16: I={y1,y2};
        A17: y1 in {y1} by TARSKI:def 1;
        assume A18:I/\Im(FSG,z1) = {y1};
        then y1 in Im(FSG,z1) by A17,XBOOLE_0:def 4;
        then A19:[z1,y1] in FSG by RELAT_1:169;
        then y1 in field FSG & z1<>y1 by RELAT_1:15,Lm2;
        then consider t be object such that
          A20: Im(FSG,y1)/\Coim(FSG,z1)={t} by Def3,A10;
        A21: t in {t} by TARSKI:def 1;
        then t in Im(FSG,y1) by A20,XBOOLE_0:def 4;
        then A22: [y1,t] in FSG by RELAT_1:169;
        then A23: y1<>t by Lm2;
        A24: y1 in I by A16,TARSKI:def 2;
        then A25: [x,y1] in FSG by RELAT_1:169;
        then A26: x<>y1 & y1 in field FSG by Lm2,RELAT_1:15;
        then consider x1x be object such that
          A27:I/\Coim(FSG,y1) ={x1x} by A2,Def3;
        A28: x1x in {x1x} by TARSKI:def 1;
        then A29: x1x in I by A27,XBOOLE_0:def 4;
        Coim(FSG,y1)=Im(FSG,y1) by Th2;
        then x1x in Im(FSG,y1) by A28,A27,XBOOLE_0:def 4;
        then A30:[y1,x1x] in FSG by RELAT_1:169;
        then y1 <>x1x by Lm2;
        then A31:[y1,y2] in FSG by A29,A16,TARSKI:def 2,A30;
        y2 in I by A16,TARSKI:def 2;
        then A32: [xx,y2] in FSG by RELAT_1:169;
        consider z2 be object such that
            A33: z2 in field FSG\{y1}
        and A34: not [y1,z2] in FSG by A26,A1,Def1;
        A35: Coim(FSG,z2) = Im(FSG,z2) by Th2;
        z1<>z2 by A19,Lm3,A34;
        then consider w be object such that
          A36: Im(FSG,z2)/\Coim(FSG,z1)={w} by A10, A33,Def3;
        A37: Coim(FSG,z1) = Im(FSG,z1) by Th2;
        then A38: t in Im(FSG,z1) by A21,A20,XBOOLE_0:def 4;
        A39: w in {w} by TARSKI:def 1;
        then A40: w in Im(FSG,z1) by A36,A37,XBOOLE_0:def 4;
        w in Im(FSG,z2) by A39,A36,XBOOLE_0:def 4;
        then A41: [z2,w] in FSG by RELAT_1:169;
        A42: [z1,w] in FSG by A40,RELAT_1:169;
        A43: t<>w
        proof
          x <> z2 by A34, A25,Lm3;
          then consider s be object such that
            A44: I/\Coim(FSG,z2) ={s} by A2, A33,Def3;
          A45: s in {s} by TARSKI:def 1;
          then s in Im(FSG,z2) by A35,A44,XBOOLE_0:def 4;
          then [z2,s] in FSG by RELAT_1:169;
          then A46:[s,z2] in FSG by Lm3;
          assume A47:t=w;
          A48: [x,y1] in FSG & [y1,z1] in FSG by A24,RELAT_1:169, A19,Lm3;
          A49: y1<>z2 & [w,z2] in FSG by A33,ZFMISC_1:56, A41,Lm3;
          s in I by A45,A44,XBOOLE_0:def 4;
          then [y2,z2] in FSG by A46,A34,A16,TARSKI:def 2;
          then y2 = t by A49,A47,A31,A22,Lm5;
          then [y2,z1] in FSG by A42,Lm3,A47;
          hence contradiction by A48, A32,Lm5,A12,A15;
        end;
        y1<>w by A41,Lm3,A34;
        then A50:card {y1,t,w} =3 by A43,CARD_2:58,A23;
        A51: y1 in Im(FSG,z1) by A17,A18,XBOOLE_0:def 4;
        card Im(FSG,z1)=2 by A11,  A10,Th9,A7;
        hence thesis by ZFMISC_1:133, A38,A40,A51,NAT_1:43,A50;
      end;
      Coim(FSG,z1)=Im(FSG,z1) by Th2;
      then A52: p<>y1 & p<>y2 by A14,A13,A8,A9;
      p in {p} by TARSKI:def 1;
      then p in I by XBOOLE_0:def 4,A13;
      hence thesis by A52,A9,TARSKI:def 2;
    end;
 end;
