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by 

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Define the homotopy g ( γ, x ) = γ x + (1 − γ ) ψ $^{1}$( x ) for γ ∈ [0 , 1]. Then the homotopy formula gives 

We define R = g$_{#}$ ([0 , 1] × T ) and Q$_{1}$ = g$_{#}$ ([0 , 1] × ∂T ). Then we get 

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Finally, we map ψ 1 $_{#}$( ∂T ) forward to the ( d − 1)-skeleton of simplicial complex K with φ = φ$_{k}$$_{+1}$ to get ψ 2 $_{#}$( ∂T ) = φ$_{#}$ ( ψ 1 $_{#}$( ∂T )). For this purpose, consider the homotopy h ( γ, x ) from ψ 1 $_{#}$( ∂T ) to ψ 2 $_{#}$( ∂T ), i.e., 

We define 

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P is a d -current whose boundary ∂P is contained in the ( d − 1)-skeleton of K . Define Q$_{2}$ = h$_{#}$ ([0 , 1] × ψ 1 $_{#}$( ∂T )). Using the homotopy formula, we get 

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