
theorem Th41:
  for L be non empty ZeroStr for z0 be Element of L st z0 <> 0.L
  holds len <%z0,0.L%> = 1
proof
  let L be non empty ZeroStr;
  let z0 be Element of L;
A1: 1 is_at_least_length_of <%z0,0.L%>
  proof
    let n be Nat;
    assume
A2: n >= 1;
    per cases by A2,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;
  assume z0 <> 0.L;
  then <%z0,0.L%>.0 <> 0.L by Th38;
  then for n be Nat st n is_at_least_length_of <%z0,0.L%> holds 0+1 <= n
  by NAT_1:13;
  hence thesis by A1,ALGSEQ_1:def 3;
end;
