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<b>let </b><font color="Maroon" title="c1">p</font> be    <a href="polyform.html#M1" title="POLYFORM:mode.1">polyhedron</a> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="polyform.html#M1" title="POLYFORM:mode.1">polyhedron</a> ] <b>means </b>for <font color="Olive" title="b1">n</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> holds  <font color="Olive" title="b1">n</font> <a href="polyform.html#K23" title="POLYFORM:func.23">-circuit-space</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> <a href="polyform.html#K25" title="POLYFORM:func.25">-bounding-chain-space</a> $1;<br/>
<b>thus </b><a NAME="E1:79"/>
( <font color="Maroon" title="c1">p</font> is <a href="polyform.html#V1" title="POLYFORM:attr.1">simply-connected</a> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">p</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:79_1"/><i><font color="Green" title="E61">A1</font></i>: 
<font color="Maroon" title="c1">p</font> is <a href="polyform.html#V1" title="POLYFORM:attr.1">simply-connected</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c2">n</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/>

<a NAME="E2:79_1"/>
<font color="Maroon" title="c2">n</font> <a href="polyform.html#K24" title="POLYFORM:func.24">-circuits</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="polyform.html#K26" title="POLYFORM:func.26">-bounding-chains</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="txt" href="polyform.html#E1:79_1"><i><font color="Green" title="E61">A1</font></i></a>, <a class="ref" href="polyform.html#D23" target="_self" title="POLYFORM:def.23">Def23</a></i>;<br/>
<b>hence </b><a NAME="E3:79_1"/>
<font color="Maroon" title="c2">n</font> <a href="polyform.html#K23" title="POLYFORM:func.23">-circuit-space</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="polyform.html#K25" title="POLYFORM:func.25">-bounding-chain-space</a> <font color="Maroon" title="c1">p</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T37" title="VECTSP_4:th.37">VECTSP_4:37</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:79"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">p</font>] implies <font color="Maroon" title="c1">p</font> is <a href="polyform.html#V1" title="POLYFORM:attr.1">simply-connected</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:79_2"/><i><font color="Green" title="E61">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">p</font>]
 ;<br/>

<b>let </b><font color="Maroon" title="c2">n</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="polyform.html#D23" target="_self" title="POLYFORM:def.23">POLYFORM:def 23</a><br/>

<b>thus </b><a NAME="E2:79_2"/>
<font color="Maroon" title="c2">n</font> <a href="polyform.html#K24" title="POLYFORM:func.24">-circuits</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="polyform.html#K26" title="POLYFORM:func.26">-bounding-chains</a> <font color="Maroon" title="c1">p</font>
 <b>by </b><i><a class="txt" href="polyform.html#E1:79_2"><i><font color="Green" title="E61">A2</font></i></a></i>;<br/>


</div><b>end;</b></div>


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