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({f,g},x) = {f.x,g.x}
proof
  thus pi({f,g},x) c= {f.x,g.x}
  proof
    let y be object;
    assume y in pi({f,g},x);
    then consider f1 such that
A1: f1 in {f,g} and
A2: y = f1.x by Def6;
    f1 = f or f1 = g by A1,TARSKI:def 2;
    hence thesis by A2,TARSKI:def 2;
  end;
  let y be object;
  assume
A3: y in {f.x,g.x};
A4: f in {f,g} by TARSKI:def 2;
A5: g in {f,g} by TARSKI:def 2;
  y = g.x or y = f.x by A3,TARSKI:def 2;
  hence thesis by A4,A5,Def6;
end;
