reserve a,b,c,d,m,x,n,j,k,l for Nat,
  t,u,v,z for Integer,
  f,F for FinSequence of NAT;
reserve p,q,r,s for real number;
reserve a,b,c,d,m,x,n,k,l for Nat,
  t,z for Integer,
  f,F,G for FinSequence of REAL;
reserve q,r,s for real number;
reserve D for set;

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;
