 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
  for x,y be Element of field FSG st x is universal_friend & x <> y
    holds
  ex z being object st Im(FSG,y) = {x,z} & Im(FSG,z) = {x,y}
 proof
   set F= field FSG;
   let x,y be Element of F such that
         A1: x is universal_friend
     and A2: x <> y;
  A3: F is non empty
proof
  assume F is empty;
  then x={} & y={} by SUBSET_1:def 1;
  hence thesis by A2;
end;
  then A4:y in F\{x} by A2,ZFMISC_1:56;
   consider z being object such that
       A5: {z} = Im(FSG,x)/\Coim(FSG,y) by A2,A3,Def3;
   A6: z in {z} by TARSKI:def 1;
   then z in Im(FSG,x) by A5,XBOOLE_0:def 4;
   then A7: [x,z] in FSG by RELAT_1:169;
   then A8: [z,x] in FSG by Lm3;
   then A9: x in Im(FSG,z) by RELAT_1:169;
    Coim(FSG,y) = Im(FSG,y) by Th2;
   then A10: z in Im(FSG,y) by A6,A5,XBOOLE_0:def 4;
   then A11: [y,z] in FSG by RELAT_1:169;
   A12: Im(FSG,y) c= {x,z}
   proof
     let d be object;
     assume d in Im(FSG,y);
     then A13:[y,d] in FSG by RELAT_1:169;
     assume A14: not d in {x,z};
     then A15: d<>x by TARSKI:def 2;
     d in F by A13,RELAT_1:15;
     then d in F\{x} by A15,ZFMISC_1:56;
     then [x,d] in FSG by A1;
     then A16: [d,x] in FSG by Lm3;
      d<>z by A14,TARSKI:def 2;
     hence thesis by A16,A2,A13,Lm5,A11,A8;
   end;
   take z;
   [x,y] in FSG by A1,A4;
   then A17: [y,x] in FSG by Lm3;
   then x in Im(FSG,y) by RELAT_1:169;
   then {x,z} c= Im(FSG,y) by A10,ZFMISC_1:32;
   hence Im(FSG,y) = {x,z} by A12;
   A18: [z,y] in FSG by A11,Lm3;
   A19: Im(FSG,z) c= {x,y}
   proof
     let d be object;
     assume d in Im(FSG,z);
     then A20:[z,d] in FSG by RELAT_1:169;
     assume A21: not d in {x,y};
     then A22: d<>x by TARSKI:def 2;
     d in F by A20,RELAT_1:15;
     then d in F\{x} by A22,ZFMISC_1:56;
     then [x,d] in FSG by A1;
     then A23: [d,x] in FSG by Lm3;
     A24: x<>z by A7,Lm2;
     d<>y by A21,TARSKI:def 2;
     hence thesis by A23,A24,A20,Lm5,A17, A18;
   end;
   y in Im(FSG,z) by A18,RELAT_1:169;
   then {x,y} c= Im(FSG,z) by A9,ZFMISC_1:32;
   hence thesis by A19;
end;
