theorem
  for m be non zero Nat holds a = 0 iff ((a,b) In_Power m).1 = 0
  proof
    for m be non zero Nat holds a = 0 implies ((a,b) In_Power m).1 = 0
    proof
      let m be non zero Nat;
      assume
      A1: a = 0;
      ((a,b) In_Power m).1 = a|^m by NEWTON:28;
      hence thesis by A1;
    end;
    hence thesis by Th45;
  end;
