ring_ker_quot_mul_rep_compat

forall (R : Sort succ u), forall (zeroR : R), forall (oneR : R), forall (addR : forall (a : R), forall (b : R), R), forall (negR : forall (a : R), R), forall (subR : forall (a : R), forall (b : R), R), forall (mulR : forall (a : R), forall (b : R), R), forall (S : Sort succ v), forall (zeroS : S), forall (oneS : S), forall (addS : forall (a : S), forall (b : S), S), forall (negS : forall (a : S), S), forall (subS : forall (a : S), forall (b : S), S), forall (mulS : forall (a : S), forall (b : S), S), forall (f : forall (x : R), S), forall (hom_args : @RingHomLawArgs.{succ u,succ v} R zeroR oneR addR negR subR mulR S zeroS oneS addS negS subS mulS f), forall (a : R), forall (a2 : R), forall (b : R), forall (b2 : R), forall (ha : @KerRel.{succ u,succ v} R S f a a2), forall (hb : @KerRel.{succ u,succ v} R S f b b2), @Eq.{succ u} (@RingKerQuot.{u,v} R S f) (@RingKerQuotMulRep.{u,v} R mulR S f a b) (@RingKerQuotMulRep.{u,v} R mulR S f a2 b2)

theorem ring_ker_quot_mul_rep_compat.{u,v} :
  forall (R : Sort succ u), forall (zeroR : R), forall (oneR : R), forall (addR : forall (a : R), forall (b : R), R), forall (negR : forall (a : R), R), forall (subR : forall (a : R), forall (b : R), R), forall (mulR : forall (a : R), forall (b : R), R), forall (S : Sort succ v), forall (zeroS : S), forall (oneS : S), forall (addS : forall (a : S), forall (b : S), S), forall (negS : forall (a : S), S), forall (subS : forall (a : S), forall (b : S), S), forall (mulS : forall (a : S), forall (b : S), S), forall (f : forall (x : R), S), forall (hom_args : @RingHomLawArgs.{succ u,succ v} R zeroR oneR addR negR subR mulR S zeroS oneS addS negS subS mulS f), forall (a : R), forall (a2 : R), forall (b : R), forall (b2 : R), forall (ha : @KerRel.{succ u,succ v} R S f a a2), forall (hb : @KerRel.{succ u,succ v} R S f b b2), @Eq.{succ u} (@RingKerQuot.{u,v} R S f) (@RingKerQuotMulRep.{u,v} R mulR S f a b) (@RingKerQuotMulRep.{u,v} R mulR S f a2 b2) :=
  fun R => fun zeroR => fun oneR => fun addR => fun negR => fun subR => fun mulR => fun S => fun zeroS => fun oneS => fun addS => fun negS => fun subS => fun mulS => fun f => fun hom_args => fun a => fun a2 => fun b => fun b2 => fun ha => fun hb => @ker_quot_sound.{u,v} R S f (mulR a b) (mulR a2 b2) (@eq_trans.{succ v} S (f (mulR a b)) (mulS (f a) (f b)) (f (mulR a2 b2)) (@ring_hom_mul.{succ u,succ v} R zeroR oneR addR negR subR mulR S zeroS oneS addS negS subS mulS f hom_args a b) (@eq_trans.{succ v} S (mulS (f a) (f b)) (mulS (f a2) (f b2)) (f (mulR a2 b2)) (@eq_congr2.{succ v,succ v,succ v} S S S mulS (f a) (f a2) (f b) (f b2) ha hb) (@eq_symm.{succ v} S (f (mulR a2 b2)) (mulS (f a2) (f b2)) (@ring_hom_mul.{succ u,succ v} R zeroR oneR addR negR subR mulR S zeroS oneS addS negS subS mulS f hom_args a2 b2))))