-theorem dfr_lift_bi (f) (p) (q) (t1) (t2):
- t1 โก๐๐[p,q] t2 โ โ[f]t1 โก๐๐[โ[f]p,โ[โ[pโ๐โ๐]f]q] โ[f]t2.
-#f #p #q #t1 #t2
-* #n * #H1n #Ht1 #Ht2
-@(ex_intro โฆ ((โ[pโ๐โ๐โq]f)๏ผ โงฃโจnโฉ)) @and3_intro
-[ -Ht1 -Ht2
- <lift_rmap_L_dx >lift_path_L_sn
- >list_append_rcons_sn in H1n; <reverse_append #H1n
- <(lift_path_head โฆ H1n) -H1n //
-| lapply (in_comp_lift_path_term f โฆ Ht1) -Ht2 -Ht1 -H1n
+theorem dfr_lift_bi (f) (t1) (t2) (r):
+ t1 โก๐๐[r] t2 โ ๐ ก[f]t1 โก๐๐[๐ ก[f]r] ๐ ก[f]t2.
+#f #t1 #t2 #r
+* #p #q #n #Hr #Hn #Ht1 #Ht2 destruct
+@(ex4_3_intro โฆ (๐ ก[f]p) (๐ ก[๐ ข[f](pโ๐โ๐)]q) (๐ ข[f](pโ๐โ๐โq)๏ผ ยงโจnโฉ))
+[ -Hn -Ht1 -Ht2 //
+| -Ht1 -Ht2
+ /2 width=1 by lift_path_rmap_closed_L/
+| lapply (in_comp_lift_path_term f โฆ Ht1) -Ht2 -Ht1 -Hn