reserve x,r,a,x0,p for Real;
reserve n,i,m for Element of NAT;
reserve Z for open Subset of REAL;
reserve f,f1,f2 for PartFunc of REAL,REAL;
reserve k for Nat;

theorem Th3:
  n>=1 implies dom(( #Z n)^) = REAL\{0} & ( #Z n)"{0}={0}
proof
  assume
A1: n >= 1;
A2: ( #Z n)"{0} = {0}
  proof
    thus ( #Z n)"{0} c= {0}
    proof
      let x be object;
      assume
A3:   x in ( #Z n)"{0};
      then reconsider x as Element of REAL;
      ( #Z n).x in {0} by A3,FUNCT_1:def 7;
      then ( #Z n).x = 0 by TARSKI:def 1;
      then x #Z n = 0 by TAYLOR_1:def 1;
      then x |^ n = 0 by PREPOWER:36;
      then x = 0 by PREPOWER:5;
      hence thesis by TARSKI:def 1;
    end;
    let x be object;
A4: 0 in {0} by TARSKI:def 1;
    assume x in {0};
    then
A5: x = 0 by TARSKI:def 1;
    {In(0,REAL)} c= REAL by ZFMISC_1:31;
    then
A6: {0} c= dom( #Z n) by FUNCT_2:def 1;
    ( #Z n).0 = 0 #Z n by TAYLOR_1:def 1
      .=0|^n by PREPOWER:36
      .=0 by A1,NEWTON:11;
    hence thesis by A5,A6,A4,FUNCT_1:def 7;
  end;
  then dom(( #Z n)^) = dom ( #Z n) \ {0} by RFUNCT_1:def 2
    .= REAL \ {0} by FUNCT_2:def 1;
  hence thesis by A2;
end;
