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 Th29:
  for L being right_zeroed non empty addLoopStr
  for z0,z1,z2,z3,z4,z5 being Element of L holds
  <%z0,z1,z2%> + <%z3,z4,z5%> = <%z0+z3,z1+z4,z2+z5%>
  proof
    let L be right_zeroed non empty addLoopStr;
    let z0,z1,z2,z3,z4,z5 be Element of L;
    set p = <%z0,z1,z2%>;
    set q = <%z3,z4,z5%>;
    set r = <%z0+z3,z1+z4,z2+z5%>;
    let n be Element of NAT;
    (n = 0 or ... or n = 2) or n > 2;
    then per cases;
    suppose n = 0;
      then p.n = z0 & q.n = z3 & r.n = z0+z3 by Th21;
      hence thesis by NORMSP_1:def 2;
    end;
    suppose n = 1;
      then p.n = z1 & q.n = z4 & r.n = z1+z4 by Th22;
      hence thesis by NORMSP_1:def 2;
    end;
    suppose n = 2;
      then p.n = z2 & q.n = z5 & r.n = z2+z5 by Th23;
      hence thesis by NORMSP_1:def 2;
    end;
    suppose n > 2;
      then n >= 2+1 by NAT_1:13;
      then
A1:   p.n = 0.L & q.n = 0.L & r.n = 0.L by Th24;
      0.L + 0.L = 0.L by RLVECT_1:def 4;
      hence thesis by A1,NORMSP_1:def 2;
    end;
  end;
