+
+lemma bi_TC_decomp_r: ∀A,B. ∀R:bi_relation A B.
+ ∀a1,a2,b1,b2. bi_TC … R a1 b1 a2 b2 →
+ R a1 b1 a2 b2 ∨
+ ∃∃a,b. bi_TC … R a1 b1 a b & R a b a2 b2.
+#A #B #R #a1 #a2 #b1 #b2 * -a2 -b2 /2 width=1/ /3 width=4/
+qed-.
+
+lemma bi_TC_decomp_l: ∀A,B. ∀R:bi_relation A B.
+ ∀a1,a2,b1,b2. bi_TC … R a1 b1 a2 b2 →
+ R a1 b1 a2 b2 ∨
+ ∃∃a,b. R a1 b1 a b & bi_TC … R a b a2 b2.
+#A #B #R #a1 #a2 #b1 #b2 #H @(bi_TC_ind_dx … a1 b1 H) -a1 -b1
+[ /2 width=1/
+| #a1 #a #b1 #b #Hab1 #Hab2 #_ /3 width=4/
+]
+qed-.
+
+lemma s_r_trans_TC1: ∀A,B,R,S. s_r_trans A B R S → s_rs_trans A B R S.
+#A #B #R #S #HRS #L2 #T1 #T2 #H elim H -T2 [ /3 width=3/ ]
+#T #T2 #_ #HT2 #IHT1 #L1 #HL12
+lapply (HRS … HT2 … HL12) -HRS -HT2 /3 width=3/
+qed-.
+
+lemma s_r_trans_TC2: ∀A,B,R,S. s_rs_trans A B R S → s_r_trans A B R (TC … S).
+#A #B #R #S #HRS #L2 #T1 #T2 #HT12 #L1 #H @(TC_ind_dx … L1 H) -L1 /2 width=3/ /3 width=3/
+qed-.