`∀[C:SmallCategory]. ∀[M:Monad(C)]. ∀[x,y:cat-ob(C)]. ∀[f:cat-arrow(C) x M(y)].`
`  (monad-extend(C;M;x;y;f) ∈ cat-arrow(C) M(x) M(y))`

Proof

Definitions occuring in Statement :  monad-extend: `monad-extend(C;M;x;y;f)` monad-fun: `M(x)` cat-monad: `Monad(C)` cat-arrow: `cat-arrow(C)` cat-ob: `cat-ob(C)` small-category: `SmallCategory` uall: `∀[x:A]. B[x]` member: `t ∈ T` apply: `f a`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` cat-monad: `Monad(C)` spreadn: spread3 monad-fun: `M(x)` monad-functor: `monad-functor(M)` pi1: `fst(t)` monad-extend: `monad-extend(C;M;x;y;f)` monad-op: `monad-op(M;x)` pi2: `snd(t)` and: `P ∧ Q` nat-trans: `nat-trans(C;D;F;G)` subtype_rel: `A ⊆r B` uimplies: `b supposing a` functor-ob: `ob(F)` functor-comp: `functor-comp(F;G)` mk-functor: mk-functor
Lemmas referenced :  cat-comp_wf functor-ob_wf functor-arrow_wf subtype_rel-equal cat-arrow_wf functor-comp_wf monad-fun_wf cat-ob_wf cat-monad_wf small-category_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation cut sqequalHypSubstitution setElimination thin rename productElimination sqequalRule applyEquality introduction extract_by_obid isectElimination hypothesisEquality hypothesis because_Cache independent_isectElimination

Latex:
\mforall{}[C:SmallCategory].  \mforall{}[M:Monad(C)].  \mforall{}[x,y:cat-ob(C)].  \mforall{}[f:cat-arrow(C)  x  M(y)].