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 Th7:
  a in C iff for x,y st x in a & y in a holds [x,y] in Web(C)
proof
  thus a in C implies for x,y st x in a & y in a holds [x,y] in Web(C)
  proof
    assume
A1: a in C;
    let x,y;
    assume x in a & y in a;
    then {x,y} in C by A1,Th6;
    hence thesis by Th5;
  end;
  assume
A2: for x,y st x in a & y in a holds [x,y] in Web(C);
  now
    let x,y;
    assume x in a & y in a;
    then [x,y] in Web(C) by A2;
    hence {x,y} in C by Th5;
  end;
  hence thesis by Th6;
end;
