theorem Th55:
  for X be non empty set,
  R be Equivalence_Relation of X,
  S be Class(R)-valued Function st S is one-to-one
  holds S is disjoint_valued
  proof
    let X be non empty set,
    R be Equivalence_Relation of X,
    S be Class(R)-valued Function;
    assume A1:S is one-to-one;
    now let x,y be object;
      assume
      A2: x <> y;
      per cases;
      suppose
        A3: x in dom S & y in dom S; then
        A4: S.x <> S.y by A1,A2,FUNCT_1:def 4;
        S.x in Class(R) & S.y in Class(R) by A3,FUNCT_1:102;
        hence S.x misses S.y by A4,EQREL_1:def 4;
      end;
      suppose
        not (x in dom S & y in dom S);
        then S.x = {} or S.y = {} by FUNCT_1:def 2;
        hence S.x misses S.y by XBOOLE_1:65;
      end;
    end;
    hence thesis by PROB_2:def 2;
  end;
