
theorem lemP:
for R being preordered non degenerated Ring,
    P being Preordering of R
holds P + P^+ c= P^+ & P^+ + P c= P^+
proof
let R be preordered non degenerated Ring; let P be Preordering of R;
set S = P \ {0.R};
H: P + P c= P by REALALG1:def 14;
now let o be object;
   assume o in P + S; then
   consider a,b being Element of R such that
A: o = a + b & a in P & b in S;
B: b in P & not b in {0.R} by A,XBOOLE_0:def 5; then
C: a + b in P + P by A;
   now assume a + b in {0.R};
     then a + b = 0.R by TARSKI:def 1;
     then a = -b by RLVECT_1:6;
     then --b in -P by A;
     then b in P /\ -P by B;
     hence contradiction by B,REALALG1:def 7;
     end;
   hence o in S by C,A,H,XBOOLE_0:def 5;
  end;
hence P + P^+ c= P^+;
now let o be object;
   assume o in S + P; then
   consider a,b being Element of R such that
A: o = a + b & a in S & b in P;
B: a in P & not a in {0.R} by A,XBOOLE_0:def 5; then
C: a + b in P + P by A;
   now assume a + b in {0.R};
     then a + b = 0.R by TARSKI:def 1;
     then b = -a by RLVECT_1:6;
     then --a in -P by A;
     then a in P /\ -P by B;
     hence contradiction by B,REALALG1:def 7;
     end;
   hence o in S by C,A,H,XBOOLE_0:def 5;
  end;
hence P^+ + P c= P^+;
end;
