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;
reserve X for non empty set,
  x for Element of X;
reserve F for Part-Family of X;

theorem
  for A being set,X being non empty set,S being a_partition of X holds A
  in S implies ex x being Element of X st A = EqClass(x,S)
proof
  let A be set,X be non empty set,S be a_partition of X;
  assume
A1: A in S;
  then A is non empty by Def4;
  then consider x being object such that
A2: x in A;
  take x;
  thus thesis by A1,A2,Def6;
end;
