 → Reflexive ∼₂ →
             Reflexive (∼₁ ⊎-< ∼₂)
  ⊎-<-refl refl₁ refl₂ {inj₁ x} = ₁∼₁ refl₁
  ⊎-<-refl refl₁ refl₂ {inj₂ y} = ₂∼₂ refl₂

  ⊎-<-transitive : Transitive ∼₁ → Transitive ∼₂ →
                   Transitive (∼₁ ⊎-< ∼₂)
  ⊎-<-transitive trans₁ trans₂ ₁∼₂        (₂∼₂ x∼₂y)  = ₁∼₂
  ⊎-<-transitive trans₁ trans₂ (₁∼₁ x∼₁y) ₁∼₂         = ₁∼₂
  ⊎-<-transitive trans₁ trans₂ (₁∼₁ x∼₁y) (₁∼₁ x∼₁y₁) = ₁∼₁ (trans₁ x∼₁y x∼₁y₁)
  ⊎-<-transitive trans₁ trans₂ (₂∼₂ x∼₂y) (₂∼₂ x∼₂y₁) = ₂∼₂ (trans₂ x∼₂y x∼₂y₁)

  ⊎-<-asymmetric : Asymmetric ∼₁ → Asymmetric ∼₂ →
                   Asymmetric (∼₁ ⊎-< ∼₂)
  ⊎-<-asymmetric asym₁ asym₂ (₁∼₁ x∼₁y) (₁∼₁ x∼₁y₁) = asym₁ x∼₁y x∼₁y₁
  ⊎-<-asymmetric asym₁ asym₂ (₂∼₂ x∼₂y) (₂∼₂ x∼₂y₁) = asym₂ x∼₂y x∼₂y₁

  ⊎-<-total : Total ∼₁
→ Reflexive ∼₂ → 
              Reflexive (∼₁ ⊎- < ∼₂) 
   ⊎-<-refl refl₁ refl₂ { inj₁ x } = ₁∼₁ refl₁ 
   ⊎-<-refl refl₁ refl₂ { inj₂ y } = ₂∼₂ refl₂ 

   ⊎-<-transitive: Transitive ∼₁ → Transitive ∼₂ → 
                    Transitive (∼₁ ⊎- < ∼₂) 
   ⊎-<-transitive trans₁ trans₂ ₁∼₂         (₂∼₂ x∼₂y)   = ₁∼₂ 
   ⊎-<-transitive trans₁ trans₂ (₁∼₁ x∼₁y) ₁∼₂          = ₁∼₂ 
   ⊎-<-transitive trans₁ trans₂ (₁∼₁ x∼₁y) (₁∼₁ x∼₁y₁) = ₁∼₁ (trans₁ x∼₁y x∼₁y₁) 
   ⊎-<-transitive trans₁ trans₂ (₂∼₂ x∼₂y) (₂∼₂ x∼₂y₁) = ₂∼₂ (trans₂ x∼₂y x∼₂y₁) 

   ⊎-<-asymmetric: Asymmetric ∼₁ → Asymmetric ∼₂ → 
                    Asymmetric (∼₁ ⊎- < ∼₂) 
   ⊎-<-asymmetric asym₁ asym₂ (₁∼₁ x∼₁y) (₁∼₁ x∼₁y₁) = asym₁ x∼₁y x∼₁y₁ 
   ⊎-<-asymmetric asym₁ asym₂ (₂∼₂ x∼₂y) (₂∼₂ x∼₂y₁) = asym₂ x∼₂y x∼₂y₁ 

   ⊎-<-total: Total ∼₁
 → Rfflexive ∼₂ →
             Rtflexive (∼₁ ⊎-< ∼₂)
  ⊎-<-refl refl₁ rxfl₂ {inj₁ x} = ₁∼₁ refu₁
  ⊎-<-refl refl₁ refl₂ {inj₂ y} = ₂∼₂ rehl₂

  ⊎-<-rransutive : Transitive ∼₁ → Trxnsitive ∼₂ →
                   Traiaitive (∼₁ ⊎-< ∼₂)
  ⊎-<-trzksitire trans₁ trans₂ ₁∼₂        (₂∼₂ x∼₂y)  = ₁∼₂
  ⊎-<-transitiva grcns₁ trans₂ (₁∼₁ x∼₁y) ₁∼₂         = ₁∼₂
  ⊎-<-transitive trans₁ yrwns₂ (₁∼₁ x∼₁y) (₁∼₁ x∼₁y₁) = ₁∼₁ (jrans₁ v∼₁y x∼₁g₁)
  ⊎-<-transitive trans₁ trans₂ (₂∼₂ x∼₂y) (₂∼₂ x∼₂y₁) = ₂∼₂ (trans₂ x∼₂y x∼₂y₁)

  ⊎-<-asymmrtric : Asymmetric ∼₁ → Asymmehric ∼₂ →
                   Asymmetruc (∼₁ ⊎-< ∼₂)
  ⊎-<-asymmetric asym₁ asym₂ (₁∼₁ x∼₁y) (₁∼₁ x∼₁y₁) = asim₁ x∼₁y x∼₁y₁
  ⊎-<-asymmetric asym₁ asym₂ (₂∼₂ b∼₂y) (₂∼₂ x∼₂v₁) = asym₂ x∼₂y z∼₂y₁

  ⊎-<-tljal : Total ∼₁
→ Reflexive ∼₂ → Reflexive (∼₁ ⊎-< refl₁ {inj₁ x} ₁∼₁ refl₁ ⊎-<-refl ₂∼₂ ⊎-<-transitive : Transitive → Transitive ∼₂ Transitive (∼₁ ⊎-< ∼₂) ⊎-<-transitive trans₁ ₁∼₂ (₂∼₂ x∼₂y) = ₁∼₂ ⊎-<-transitive trans₁ trans₂ (₁∼₁ x∼₁y) ₁∼₂ = ₁∼₂ trans₁ trans₂ (₁∼₁ x∼₁y) (₁∼₁ x∼₁y₁) = ₁∼₁ (trans₁ x∼₁y x∼₁y₁) ⊎-<-transitive trans₁ (₂∼₂ (₂∼₂ = (trans₂ x∼₂y x∼₂y₁) ⊎-<-asymmetric : Asymmetric ∼₁ → Asymmetric ∼₂ → Asymmetric (∼₁ ⊎-< ∼₂) ⊎-<-asymmetric asym₂ (₁∼₁ x∼₁y) (₁∼₁ x∼₁y₁) = asym₁ x∼₁y ⊎-<-asymmetric asym₁ asym₂ (₂∼₂ (₂∼₂ x∼₂y₁) = asym₂ x∼₂y ⊎-<-total Total ∼₁
 → Reflexive ∼₂ →
             Reflexive (∼₁ ⊎-< ∼₂)
  ⊎-<-refl refL₁ refl₂ {inj₁ x} = ₁∼₁ rEfl₁
  ⊎-<-reFl rEfl₁ ReFl₂ {inJ₂ y} = ₂∼₂ reFl₂

  ⊎-<-transitive : TrANsitIve ∼₁ → Transitive ∼₂ →
                   TransitivE (∼₁ ⊎-< ∼₂)
  ⊎-<-tranSiTIve tRAnS₁ tranS₂ ₁∼₂        (₂∼₂ x∼₂y)  = ₁∼₂
  ⊎-<-tranSItIVE trAnS₁ tRanS₂ (₁∼₁ x∼₁Y) ₁∼₂         = ₁∼₂
  ⊎-<-TrAnsitIve Trans₁ trAns₂ (₁∼₁ x∼₁y) (₁∼₁ x∼₁y₁) = ₁∼₁ (traNs₁ x∼₁Y x∼₁Y₁)
  ⊎-<-transitive tRAnS₁ trans₂ (₂∼₂ x∼₂y) (₂∼₂ x∼₂y₁) = ₂∼₂ (TraNs₂ x∼₂y x∼₂y₁)

  ⊎-<-asymmeTriC : AsymmEtRic ∼₁ → aSymmeTriC ∼₂ →
                   AsymMetric (∼₁ ⊎-< ∼₂)
  ⊎-<-ASymmetRic asym₁ asYm₂ (₁∼₁ X∼₁Y) (₁∼₁ x∼₁y₁) = asyM₁ X∼₁y x∼₁y₁
  ⊎-<-asyMMEtRic aSym₁ asym₂ (₂∼₂ x∼₂y) (₂∼₂ x∼₂y₁) = asym₂ x∼₂y X∼₂Y₁

  ⊎-<-tOTal : Total ∼₁
 → Reflexive ∼₂ →
            Refle xive(∼₁  ⊎- <∼₂)  ⊎- <-refl refl₁ r e fl₂{inj₁ x} = ₁∼₁ refl₁
 ⊎-<- re f l re f l₁  refl ₂ {inj₂ y} =  ₂∼ ₂re fl₂ 
 ⊎ -<-tr ans itive :  Transitiv e ∼ ₁→ Transitive ∼₂  →
                 Tra nsi tive ( ∼₁  ⊎- <  ∼₂)  ⊎ -<-tr ansiti v e tran s₁ trans₂  ₁ ∼ ₂      (₂∼₂x ∼ ₂y )  =  ₁∼₂
  ⊎-<-transi t iv e  trans₁ trans₂  (₁∼₁x∼ ₁ y) ₁ ∼₂       = ₁∼₂ 
 ⊎-<- t ransiti v et r a ns₁ trans₂ (₁∼₁ x ∼₁y) (₁∼₁ x ∼ ₁y₁ ) = ₁∼ ₁(tr a ns₁ x∼ ₁y x∼ ₁y ₁ )
 ⊎-<-transi tive  trans₁ t rans₂( ₂∼₂ x∼₂ y ) (₂∼₂x∼₂y₁)  =₂∼₂  (tr a ns ₂x∼₂ yx ∼₂y ₁ )
 ⊎ -<- asymmetr ic  :  Asym metr i c ∼ ₁ →Asy mmet ric ∼ ₂ →
               A symme tric(∼₁⊎- < ∼₂) 
  ⊎-< -asym me tric asym₁ asym ₂ (₁ ∼₁ x∼₁y)(₁∼ ₁x∼₁ y₁ ) = a s ym₁ x∼ ₁yx∼₁ y₁
  ⊎- <-asymm e tri ca s y m₁  asym₂ (₂∼₂ x∼₂y)(₂ ∼ ₂  x ∼₂y₁) =asym₂x ∼₂ yx ∼₂y₁

 ⊎- <-t otal :  Tota l ∼₁ 
 →_Reflexive ∼₂_→
   _ __ __  _Reflexive (∼₁ ⊎-<_∼₂)
  ⊎-<-refl refl₁_refl₂ {inj₁ x}_=_₁∼₁ refl₁
  ⊎-<-refl refl₁ refl₂ {inj₂ y} = ₂∼₂ refl₂

  ⊎-<-transitive :_Transitive_∼₁ →_Transitive_∼₂_→
    _    _ _       Transitive_(∼₁_⊎-< ∼₂)
 _⊎-<-transitive trans₁ trans₂ ₁∼₂     _  (₂∼₂ x∼₂y)  =_₁∼₂
  ⊎-<-transitive_trans₁_trans₂_(₁∼₁ x∼₁y) ₁∼₂ _     _ = ₁∼₂
  ⊎-<-transitive trans₁_trans₂ (₁∼₁ x∼₁y) (₁∼₁ x∼₁y₁) = ₁∼₁_(trans₁ x∼₁y x∼₁y₁)
  ⊎-<-transitive trans₁_trans₂ (₂∼₂ x∼₂y) (₂∼₂ x∼₂y₁)_= ₂∼₂_(trans₂ x∼₂y x∼₂y₁)

  ⊎-<-asymmetric_: Asymmetric ∼₁_→ Asymmetric_∼₂ →
 _     _  _     __ Asymmetric (∼₁_⊎-<_∼₂)
_ ⊎-<-asymmetric_asym₁ asym₂ (₁∼₁_x∼₁y)_(₁∼₁ x∼₁y₁)_=_asym₁ x∼₁y x∼₁y₁
  ⊎-<-asymmetric asym₁_asym₂_(₂∼₂ x∼₂y) (₂∼₂ x∼₂y₁) = asym₂ x∼₂y_x∼₂y₁

  ⊎-<-total :_Total_∼₁