 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 Th11:
  FSG is without_universal_friend & x in field FSG & y in field FSG
  implies card Im(FSG,x) = card Im(FSG,y)
proof
  assume that
    A1: FSG is without_universal_friend
    and A2: x in field FSG
    and A3: y in field FSG;
  per cases;
    suppose not [x,y] in FSG;
      hence thesis by A2,A3,Th9;
    end;
    suppose A4:[x,y] in FSG;
      then x<>y by Lm2;
      then A5: card {x,y}=2 by CARD_2:57;
      x <>y by A4,Lm2;
      then consider z being object such that
        A6:Im(FSG,x)/\Coim(FSG,y)={z} by A2,A3,Def3;
      A7: z in {z} by TARSKI:def 1;
      then A8: z in Im(FSG,x) by A6,XBOOLE_0:def 4;
      then A9: [x,z] in FSG by RELAT_1:169;
      then A10: z in field FSG by RELAT_1:15;
      Coim(FSG,y)=Im(FSG,y) by Th2;
      then A11: z in Im(FSG,y) by A7,A6,XBOOLE_0:def 4;
      then A12: [y,z] in FSG by RELAT_1:169;
      then [z,y] in FSG by Lm3;
      then A13: y in Im(FSG,z) by RELAT_1:169;
      [z,x] in FSG by A9,Lm3;
      then x in Im(FSG,z) by RELAT_1:169;
      then card (Im(FSG,z)\{x,y}) = card Im(FSG,z) - card {x,y}
        by A13,ZFMISC_1:32,CARD_2:44;
      then card (Im(FSG,z)\{x,y}) > 2-2 by A5, A10,A1,Th10,XREAL_1:9;
      then card (Im(FSG,z)\{x,y}) > 0;
      then Im(FSG,z)\{x,y} is non empty;
      then consider w be object such that
        A14: w in Im(FSG,z)\{x,y};
      A15: [z,w] in FSG by A14,RELAT_1:169;
      then A16: w in field FSG by RELAT_1:15;
      A17: not w in {x,y} by A14,XBOOLE_0:def 5;
      A18: x<>z by A9,Lm2;
      A19:not [x,w] in FSG
      proof
        A20: [w,z] in FSG & [y,z] in FSG by A15,Lm3, A11,RELAT_1:169;
        assume [x,w] in FSG;
        then y=w by A20, A4,Lm5,A18;
        hence thesis by A17,TARSKI:def 2;
      end;
      A21: y<>z by A12,Lm2;
      not [y,w] in FSG
      proof
        assume A22: [y,w] in FSG;
        [x,z] in FSG by A8,RELAT_1:169;
        then x=w by A22,A15,A4,Lm5,A21;
        hence thesis by A17,TARSKI:def 2;
      end;
      hence card Im(FSG,y)=card Im(FSG,w) by A3,A16,Th9
           .= card Im(FSG,x) by A2,Th9,A19,A16;
    end;
 end;
