reserve x,y,z, X,Y,Z for set,
  n for Element of NAT;
reserve A for set,
  D for non empty set,
  a,b,c,l,r for Element of D,
  o,o9 for BinOp of D,
  f,g,h for Function of A,D;
reserve G for non empty multMagma;

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;
