reserve k,m,n for Nat;
reserve R for commutative Ring,
        p,q for Polynomial of R,
        z0,z1 for Element of R;

theorem Th4:
  for L being left_zeroed right_zeroed non empty addLoopStr
    for z0,z1 being Element of L holds
      <% z0,z1 %> = <%z0%> + <%0.L,z1%>
proof
  let L be left_zeroed right_zeroed non empty addLoopStr;
  let z0,z1 be Element of L;
  let n be Element of NAT;
  A1: (<%z0%> + <%0.L,z1%>).n = (<%z0%>.n) + (<%0.L,z1%>.n) by NORMSP_1:def 2;
  per cases by NAT_1:23;
  suppose A2: n = 0;
    then <%z0%>.n = z0 & <%0.L,z1%>.n = 0.L by POLYNOM5:32,38;
    then (<%z0%> + <%0.L,z1%>).n = z0 by A1,RLVECT_1:def 4;
    hence thesis by A2,POLYNOM5:38;
  end;
  suppose A3:n =1;
    then <%z0%>.n = 0.L & <%0.L,z1%>.n = z1 by POLYNOM5:32,38;
    then (<%z0%> + <%0.L,z1%>).n = z1 by A1, ALGSTR_1:def 2;
    hence thesis by A3,POLYNOM5:38;
  end;
  suppose A4:n>=2;
    then <%z0%>.n = 0.L & <%0.L,z1%>.n = 0.L by XXREAL_0:2,POLYNOM5:32,38;
    then (<%z0%> + <%0.L,z1%>).n = 0.L by A1, ALGSTR_1:def 2;
    hence thesis by A4,POLYNOM5:38;
  end;
end;
