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 Th24:
  for x,y st y in X holds
  Class(EqR,x) = Class(EqR,y) or Class(EqR,x) misses Class(EqR,y)
proof
  let x,y;
A1: not [x,y] in EqR implies Class(EqR,x) misses Class(EqR,y)
  proof
    assume
A2: not [x,y] in EqR;
    assume Class(EqR,x) meets Class(EqR,y);
    then consider z being object such that
A3: z in Class(EqR,x) and
A4: z in Class(EqR,y) by XBOOLE_0:3;
    [z,x] in EqR by A3,Th19;
    then
A5: [x,z] in EqR by Th6;
    [z,y] in EqR by A4,Th19;
    hence contradiction by A2,A5,Th7;
  end;
  assume
A6: y in X;
  [x,y] in EqR implies Class(EqR,x) = Class(EqR,y)
  proof
    assume [x,y] in EqR;
    then x in Class(EqR,y) by Th19;
    hence thesis by A6,Th23;
  end;
  hence thesis by A1;
end;
