reserve Y,Z for non empty set;
reserve PA,PB for a_partition of Y;
reserve A,B for Subset of Y;
reserve i,j,k for Nat;
reserve x,y,z,x1,x2,y1,z0,X,V,a,b,d,t,SFX,SFY for set;

theorem
  for P being a_partition of Y for y being Element of Y
  ex A being Subset of Y st y in A & A in P
proof
  let P be a_partition of Y;
  let y be Element of Y;
  consider R being Equivalence_Relation of Y such that
A1: P = Class R by EQREL_1:34;
  take Class(R,y);
  thus y in Class(R,y) by EQREL_1:20;
  thus thesis by A1,EQREL_1:def 3;
end;
