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;
reserve e,u,v for object, E,X,Y,X1 for set;
reserve X,Y,Z for non empty set;

theorem
  for D being non empty a_partition of X, p being Element of D
  holds p = proj D " {p}
proof
  let D be non empty a_partition of X, p be Element of D;
  thus p c= proj D " {p}
  proof
    let e be object;
    assume
A1: e in p;
    then (proj D).e = p by Th65;
    then (proj D).e in {p} by TARSKI:def 1;
    hence thesis by A1,FUNCT_2:38;
  end;
  let e be object;
  assume
A2: e in proj D " {p};
  then (proj D).e in {p} by FUNCT_1:def 7;
  then (proj D).e = p by TARSKI:def 1;
  hence e in p by A2,Def9;
end;
