reserve r,t for Real;
reserve i for Integer;
reserve k,n for Nat;
reserve p for Polynomial of F_Real;
reserve e for Element of F_Real;
reserve L for non empty ZeroStr;
reserve z,z0,z1,z2 for Element of L;

theorem Th20:
  rng <%z%> = {z,0.L}
  proof
    set p = <%z%>;
A1: p.0 = z by ALGSEQ_1:def 5;
A2: dom p = NAT by FUNCT_2:def 1;
    thus rng p c= {z,0.L}
    proof
      let y be object;
      assume y in rng p;
      then consider x being object such that
A3:   x in dom p and
A4:   p.x = y by FUNCT_1:def 3;
      reconsider x as Element of NAT by A3;
      per cases;
      suppose x = 0;
        hence thesis by A4,A1,TARSKI:def 2;
      end;
      suppose x <> 0;
        then p.x = 0.L by POLYNOM5:32,NAT_1:14;
        hence thesis by A4,TARSKI:def 2;
      end;
    end;
    let y be object;
    assume y in {z,0.L};
    then per cases by TARSKI:def 2;
    suppose y = z;
      hence thesis by A1,A2,FUNCT_1:def 3;
    end;
    suppose
A5:   y = 0.L;
      p.1 = 0.L by POLYNOM5:32;
      hence thesis by A2,A5,FUNCT_1:def 3;
    end;
  end;
