
theorem Th19:
  for I be non empty set, G be Group,
      i be Element of I, g be Element of G,
      a be Function of I,G
  st a = (I --> 1_G) +* (i,g)
  holds support(a) c= {i}
  proof
    let I be non empty set, G be Group,
        i be Element of I, g be Element of G,
        a be Function of I,G;
    assume
    A1: a = (I --> 1_G) +* (i,g);
    for j be object holds j in support(a) implies j in {i}
    proof
      let j be object;
      assume
      A2: j in support(a);
      j = i
      proof
        assume j <> i; then
        a.j = (I --> 1_G).j by A1,FUNCT_7:32
           .= 1_G by A2,FUNCOP_1:7;
        hence contradiction by A2,Def2;
      end;
      hence thesis by TARSKI:def 1;
    end;
    hence thesis;
  end;
