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 Th5:
  T = Web(C) iff for x,y being object holds [x,y] in T iff {x,y} in C
proof
  thus T = Web(C) implies
for x,y being object holds [x,y] in T iff {x,y} in C
  proof
    assume
A1: T = Web(C);
    let x,y be object;
    thus [x,y] in T implies {x,y} in C
    proof
      assume [x,y] in T;
      then consider X such that
A2:   X in C and
A3:   x in X & y in X by A1,Def2;
      {x,y} c= X by A3,ZFMISC_1:32;
      hence thesis by A2,CLASSES1:def 1;
    end;
A4: x in {x,y} & y in {x,y} by TARSKI:def 2;
    assume {x,y} in C;
    hence thesis by A1,A4,Def2;
  end;
  assume
A5: for x,y being object holds [x,y] in T iff {x,y} in C;
  for x,y being object holds [x,y] in T iff ex X st X in C & x in X & y in X
  proof
    let x,y be object;
    thus [x,y] in T implies ex X st X in C & x in X & y in X
    proof
      assume
A6:   [x,y] in T;
      take {x,y};
      thus thesis by A5,A6,TARSKI:def 2;
    end;
    given X such that
A7: X in C and
A8: x in X & y in X;
    {x,y} c= X by A8,ZFMISC_1:32;
    then {x,y} in C by A7,CLASSES1:def 1;
    hence thesis by A5;
  end;
  hence thesis by Def2;
end;
