theorem Th31:
  for A being non empty set, a,b being Element of A holds
  (chi a).a = 1 & (b <> a implies (chi a).b = 0)
proof
  let A be non empty set, a,b be Element of A;
A1: b <> a implies not b in {a} by TARSKI:def 1;
  a in {a} by TARSKI:def 1;
  hence thesis by A1,FUNCT_3:def 3;
end;
