+implied rec lemma pc3_left_ind (c: C) (P: (T \to (T \to Prop))) (f: (\forall
+(t: T).(P t t))) (f0: (\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to
+(\forall (t3: T).((pc3_left c t2 t3) \to ((P t2 t3) \to (P t1 t3)))))))) (f1:
+(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (t3:
+T).((pc3_left c t1 t3) \to ((P t1 t3) \to (P t2 t3)))))))) (t: T) (t0: T) (p:
+pc3_left c t t0) on p: P t t0 \def match p with [(pc3_left_r t1) \Rightarrow
+(f t1) | (pc3_left_ur t1 t2 p0 t3 p1) \Rightarrow (f0 t1 t2 p0 t3 p1
+((pc3_left_ind c P f f0 f1) t2 t3 p1)) | (pc3_left_ux t1 t2 p0 t3 p1)
+\Rightarrow (f1 t1 t2 p0 t3 p1 ((pc3_left_ind c P f f0 f1) t1 t3 p1))].
+
+fact pc3_ind_left__pc3_left_pr3: