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;
reserve C,C1,C2 for Element of CSp(X);
reserve g for Element of FuncsC(X);
reserve l,l1,l2,l3 for Element of MapsC(X);

theorem
  x in Toler_on_subsets(X) iff ex A st A c= X & x is Tolerance of A
proof
  set f = the set of all Toler(Y) where Y is Subset of X;
  thus x in Toler_on_subsets(X) implies ex A st A c= X & x is Tolerance of A
  proof
    assume x in Toler_on_subsets(X);
    then consider a such that
A1: x in a and
A2: a in f by TARSKI:def 4;
    consider Y be Subset of X such that
A3: a = Toler(Y) by A2;
    take Y;
    thus thesis by A1,A3,Def15;
  end;
  given A such that
A4: A c= X and
A5: x is Tolerance of A;
  reconsider A as Subset of X by A4;
A6: Toler(A) in f;
  x in Toler(A) by A5,Def15;
  hence thesis by A6,TARSKI:def 4;
end;
