reserve x,y,z,a,b,c,X,A for set;
reserve C,D for Coherence_Space;
reserve T for Tolerance of union C;

theorem Th6:
  a in C iff for x,y st x in a & y in a holds {x,y} in C
proof
  defpred P[object,object] means {$1} = $2;
  thus a in C implies for x,y st x in a & y in a holds {x,y} in C
  proof
    assume
A1: a in C;
    let x,y;
    assume x in a & y in a;
    then {x,y} c= a by ZFMISC_1:32;
    hence thesis by A1,CLASSES1:def 1;
  end;
A2: for x,y,z being object st P[x,y] & P[x,z] holds y = z;
  consider X such that
A3: for x being object holds x in X iff
    ex y being object st y in a & P[y,x] from TARSKI:sch 1(A2);
  assume
A4: for x,y st x in a & y in a holds {x,y} in C;
A5: for b,c st b in X & c in X holds b \/ c in C
  proof
    let b,c;
    assume that
A6: b in X and
A7: c in X;
    consider z being object such that
A8: z in a and
A9: {z} = c by A3,A7;
    consider y being object such that
A10: y in a and
A11: {y} = b by A3,A6;
    {y,z} in C by A4,A10,A8;
    hence thesis by A11,A9,ENUMSET1:1;
  end;
A12: union X = a
  proof
    thus union X c= a
    proof
      let x be object;
      assume x in union X;
      then consider Z be set such that
A13:  x in Z and
A14:  Z in X by TARSKI:def 4;
      ex y being object st y in a & Z = {y} by A3,A14;
      hence thesis by A13,TARSKI:def 1;
    end;
    let x be object;
    assume x in a;
    then
A15: {x} in X by A3;
    x in {x} by TARSKI:def 1;
    hence thesis by A15,TARSKI:def 4;
  end;
  X c= C
  proof
    let x be object;
    assume x in X;
    then consider y being object such that
A16: y in a and
A17: {y} = x by A3;
    {y,y} in C by A4,A16;
    hence thesis by A17,ENUMSET1:29;
  end;
  hence thesis by A5,A12,Def1;
end;
