reserve x for set;
reserve i,j for Integer;
reserve n,n1,n2,n3 for Nat;
reserve K,K1,K2,K3 for Field;
reserve SK1,SK2 for Subfield of K;
reserve ek,ek1,ek2 for Element of K;
reserve p for Prime;
reserve a,b,c for Element of GF(p);
reserve F for FinSequence of GF(p);
reserve Px,Py,Pz for Element of GF(p);

theorem Th54:
  for X be non empty finite set,
  R be Equivalence_Relation of X,
  S be Class(R)-valued Function, i be set st i in dom S holds
  S.i is finite Subset of X
  proof
    let X be non empty finite set,
    R be Equivalence_Relation of X, S be Class(R)-valued Function,
    i be set;
    assume i in dom S; then
    S.i in Class(R) by FUNCT_1:102;
    hence thesis;
  end;
