theorem
  p has_onlyone_value_in k & i<>k implies p.i=0
proof
  assume that
A1: p has_onlyone_value_in k and
A2: i<>k;
  per cases;
  suppose
    not i in dom p;
    hence thesis by FUNCT_1:def 2;
  end;
  suppose
    i in dom p;
    hence thesis by A1,A2;
  end;
end;
