
theorem
  for L be non empty ZeroStr for z0 be Element of L holds <%z0,0.L%> = <%z0%>
proof
  let L be non empty ZeroStr;
  let z0 be Element of L;
  per cases;
  suppose
A1: z0 = 0.L;
    hence <%z0,0.L%> = 0_.(L) by Th42
      .= <%z0%> by A1,Th34;
  end;
  suppose
A2: z0 <> 0.L;
    then
A3: len <%z0%> = 0+1 by Th33;
A4: now
      let n be Nat;
      assume n < len <%z0%>;
      then
A5:   n = 0 by A3,NAT_1:13;
      hence <%z0,0.L%>.n = z0 by Th38
        .= <%z0%>.n by A5,ALGSEQ_1:def 5;
    end;
    len <%z0,0.L%> = 1 by A2,Th41;
    hence thesis by A2,A4,Th33,ALGSEQ_1:12;
  end;
end;
