X_{P}

is obtained from

X_{U, P}

by filling the

U

-boundary along

μ_{U}

. The filling core represents

ι_{*}^{\partial X_{U}} [λ_{U}]

H_{1} (X_{P})

ι_{*}^{X_{U, P}} τ (X_{U, P}) = (1 - ι_{*}^{\partial X_{U}} [λ_{U}]) τ (X_{P}) .

τ (X_{P}) = \frac{Δ ( X _{P} )}{1 - ι _{*}^{\partial X_{P}} [ μ _{P} ]} .

Substitution proves

ι_{*}^{X_{U, P}} τ (X_{U, P}) = \frac{Δ ( X _{P} ) ( 1 - ι _{*}^{\partial X_{U}} [ λ _{U} ] )}{1 - ι _{*}^{\partial X_{P}} [ μ _{P} ]} .

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