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

theorem
  Web(CohSp(E)) = E
proof
  now
    let x,y be object;
    thus [x,y] in Web(CohSp(E)) implies [x,y] in E
    proof
      assume [x,y] in Web(CohSp(E));
      then
A1:   {x,y} in CohSp(E) by Th5;
      x in {x,y} & y in {x,y} by TARSKI:def 2;
      hence thesis by A1,Def3;
    end;
    assume
A2: [x,y] in E;
    then
A3: x in X & y in X by ZFMISC_1:87;
    for u,v be set st u in {x,y} & v in {x,y} holds [u,v] in E
    proof
      let u,v be set;
      assume that
A4:   u in {x,y} and
A5:   v in {x,y};
A6:   v = x or v = y by A5,TARSKI:def 2;
      u = x or u = y by A4,TARSKI:def 2;
      hence thesis by A2,A3,A6,EQREL_1:6,TOLER_1:7;
    end;
    then {x,y} in CohSp(E) by Def3;
    hence [x,y] in Web(CohSp(E)) by Th5;
  end;
  hence thesis;
end;
