
theorem c3:
for R being preordered Ring,
    P being Preordering of R,
    a,b,c being Element of R st a <=P, b & b <=P, c holds a <=P, c
proof
let R be preordered Ring, P be Preordering of R;
let a,b,c be Element of R;
assume a <=P, b & b <=P, c;
then a <=_(OrdRel P), b & b <=_(OrdRel P), c;
hence thesis by lemOP,REALALG1:3;
end;
