lemma frees_inv_lifts_SO:
∀b,f,L,U. L ⊢ 𝐅+❪U❫ ≘ f →
∀K. ⇩*[b,𝐔❨1❩] L ≘ K → ∀T. ⇧[1] T ≘ U →
- K â\8a¢ ð\9d\90\85+â\9dªTâ\9d« â\89\98 ⫱f.
+ K â\8a¢ ð\9d\90\85+â\9dªTâ\9d« â\89\98 â«°f.
#b #f #L #U #H #K #HLK #T #HTU elim(frees_inv_lifts_ex … H … HLK … HTU) -b -L -U
#f1 #Hf #Hf1 elim (coafter_inv_nxx … Hf) -Hf
/3 width=5 by frees_eq_repl_back, coafter_isid_inv_sn/
lemma frees_inv_drops_next:
∀f1,L1,T1. L1 ⊢ 𝐅+❪T1❫ ≘ f1 →
∀I2,L2,V2,i. ⇩[i] L1 ≘ L2.ⓑ[I2]V2 →
- â\88\80g1. â\86\91g1 = ⫱*[i] f1 →
+ â\88\80g1. â\86\91g1 = â«°*[i] f1 →
∃∃g2. L2 ⊢ 𝐅+❪V2❫ ≘ g2 & g2 ⊆ g1.
#f1 #L1 #T1 #H elim H -f1 -L1 -T1
[ #f1 #L1 #s #Hf1 #I2 #L2 #V2 #j #_ #g1 #H1 -I2 -L1 -s