 reserve Omega for non empty set;
 reserve r for Real;
 reserve Sigma for SigmaField of Omega;
 reserve P for Probability of Sigma;

theorem Th3:
  (Omega --> 1) = chi(Omega, Omega)
  proof
    set E0 = (Omega --> 1);
    A1: dom (chi(Omega,Omega)) = Omega by FUNCT_3:def 3;
    A2: dom E0 = Omega by FUNCT_2:def 1;
    now let x be object;
      assume A3:x in dom (chi(Omega,Omega));
      per cases;
      suppose x in Omega;
        hence (chi(Omega,Omega)).x = 1 by FUNCT_3:def 3
        .=E0.x by A3,FUNCOP_1:7;
      end;
      suppose not x in Omega;
        hence (chi(Omega,Omega)).x=E0.x by A3;
      end;
    end;
    hence thesis by A1,A2,FUNCT_1:2;
  end;
