
theorem Th42:
  for L be non empty ZeroStr holds <%0.L,0.L%> = 0_.(L)
proof
  let L be non empty ZeroStr;
  0 is_at_least_length_of <%0.L,0.L%>
  proof
    let n be Nat;
    assume n >= 0;
    per cases;
    suppose
      n = 0;
      hence thesis by Th38;
    end;
    suppose
      n > 0;
      then
A1:   n >= 0+1 by NAT_1:13;
      now
        per cases by A1,XXREAL_0:1;
        suppose
          n = 1;
          hence thesis by Th38;
        end;
        suppose
          n > 1;
          then n >= 1+1 by NAT_1:13;
          hence thesis by Th38;
        end;
      end;
      hence thesis;
    end;
  end;
  then len <%0.L,0.L%> = 0 by ALGSEQ_1:def 3;
  hence thesis by POLYNOM4:5;
end;
