reserve X,Y,Z for set, x,y,z for object;
reserve i,j for Nat;
reserve A,B,C for Subset of X;
reserve R,R1,R2 for Relation of X;
reserve AX for Subset of [:X,X:];
reserve SFXX for Subset-Family of [:X,X:];
reserve EqR,EqR1,EqR2,EqR3 for Equivalence_Relation of X;

theorem
  for R being transitive Tolerance of X holds y in Class(R,x) & z in
  Class(R,x) implies [y,z] in R
proof
  let R be transitive Tolerance of X;
  assume that
A1: y in Class(R,x) and
A2: z in Class(R,x);
  [z,x] in R by A2,Th19;
  then
A3: [x,z] in R by Th6;
  [y,x] in R by A1,Th19;
  hence thesis by A3,Th7;
end;
