reserve A,B,C for Ordinal,
  K,L,M,N for Cardinal,
  x,y,y1,y2,z,u for object,X,Y,Z,Z1,Z2 for set,
  n for Nat,
  f,f1,g,h for Function,
  Q,R for Relation;
reserve ff for Cardinal-Function;

theorem
  pi({g},x) = {g.x}
proof
  thus pi({g},x) c= {g.x}
  proof
    let y be object;
    assume y in pi({g},x);
    then consider f such that
A1: f in {g} and
A2: y = f.x by Def6;
    f = g by A1,TARSKI:def 1;
    hence thesis by A2,TARSKI:def 1;
  end;
  let y be object;
  assume
A3: y in {g.x};
A4: g in {g} by TARSKI:def 1;
  y = g.x by A3,TARSKI:def 1;
  hence thesis by A4,Def6;
end;
