
theorem po3:
for F being Field, E being FieldExtension of F
for L being F-monomorphic Field,
    f being Monomorphism of F,L
for p,q,r being Element of Ext_Set(f,E) st p <= q & q <= r holds p <= r
proof
let F be Field, E being FieldExtension of F;
let L be F-monomorphic Field;
let f be Monomorphism of F,L, p,q,r be Element of Ext_Set(f,E);
assume AS: p <= q & q <= r; then
consider K1 being FieldExtension of p`1, g1 being Function of K1,L such that
B: K1 = q`1 & g1 = q`2 & g1 is (p`2)-extending;
consider K2 being FieldExtension of q`1, g2 being Function of K2,L such that
D: K2 = r`1 & g2 = r`2 & g2 is (q`2)-extending by AS;
reconsider K = p`1 as Field;
reconsider K1 as FieldExtension of K;
reconsider K2 as K1-extending FieldExtension of K by B;
reconsider L1 = L as FieldExtension of L by FIELD_4:6;
reconsider g = p`2 as Function of K,L;
reconsider g2 as Function of K2,L1;
   now let K be FieldExtension of p`1, h being Function of K,L;
   assume F: K = r`1 & h = r`2;
   g2 is g-extending by B,D,FIELD_8:41;
   hence h is (p`2)-extending by F,D;
   end;
hence thesis by AS;
end;
