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;
