reserve A for RelStr;
reserve X for non empty set;
reserve PX,PY,PZ,Y,a,b,c,x,y for set;
reserve S1,S2 for Subset of Y;

theorem
  for EqR being Equivalence_Relation of X, x,y,z be set holds z in
  Class(EqR,x) & z in Class(EqR,y) implies Class(EqR,x) = Class(EqR,y)
proof
  let EqR being Equivalence_Relation of X, x,y,z be set;
  assume that
A1: z in Class(EqR,x) and
A2: z in Class(EqR,y);
A3: [z,y] in EqR by A2,EQREL_1:19;
  [z,x] in EqR by A1,EQREL_1:19;
  hence Class(EqR,x) = Class(EqR,z) by A1,EQREL_1:35
    .= Class(EqR,y) by A1,A3,EQREL_1:35;
