
theorem Th3:
  for I be non empty set,
      G be Group,
      H be Subgroup of G,
      x be Function of I,G,
      y be Function of I,H
  st x = y
  holds support x = support y
  proof
    let I be non empty set,
        G be Group,
        H be Subgroup of G,
        x be Function of I,G,
        y be Function of I,H;
    assume
    A1: x = y;
    for i be object holds i in support x iff i in support y
    proof
      let i be object;
      A2: i in support x iff x.i <> 1_G & i in I by GROUP_19:def 2;
      i in support y iff y.i <> 1_H & i in I by GROUP_19:def 2;
      hence thesis by A1,A2,GROUP_2:44;
    end;
    hence thesis by TARSKI:2;
  end;
