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 Th28:
  for L being right_zeroed non empty addLoopStr
  for z0,z1,z2,z3 being Element of L holds
  <%z0,z1%> + <%z2,z3%> = <%z0+z2,z1+z3%>
  proof
    let L be right_zeroed non empty addLoopStr;
    let z0,z1,z2,z3 be Element of L;
    set p = <%z0,z1%>;
    set q = <%z2,z3%>;
    set r = <%z0+z2,z1+z3%>;
    let n be Element of NAT;
    (n = 0 or ... or n = 1) or n > 1;
    then per cases;
    suppose n = 0;
      then p.n = z0 & q.n = z2 & r.n = z0+z2 by POLYNOM5:38;
      hence thesis by NORMSP_1:def 2;
    end;
    suppose n = 1;
      then p.n = z1 & q.n = z3 & r.n = z1+z3 by POLYNOM5:38;
      hence thesis by NORMSP_1:def 2;
    end;
    suppose n > 1;
      then n >= 1+1 by NAT_1:13;
      then
A1:   p.n = 0.L & q.n = 0.L & r.n = 0.L by POLYNOM5:38;
      0.L + 0.L = 0.L by RLVECT_1:def 4;
      hence thesis by A1,NORMSP_1:def 2;
    end;
  end;
