reserve i,j,k,n,m,l,s,t for Nat;
reserve a,b for Real;
reserve F for real-valued FinSequence;
reserve z for Complex;
reserve x,y for Complex;
reserve r,s,t for natural Number;

theorem Th11:
  s >= 1 implies 0|^s = 0
proof
A0: s is Nat by TARSKI:1;
  defpred P[Nat] means 0|^$1 = 0;
A1: now
    let n be Nat;
    assume n>=1 & P[n];
    0|^(n+1) = 0|^n * 0 by Th6
      .= 0;
    hence P[n+1];
  end;
A2: P[1];
  for n be Nat st n >=1 holds P[n] from NAT_1:sch 8(A2,A1);
  hence thesis by A0;
end;
