<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">p</font> be   <a href="polyform.html#NM2" title="POLYFORM:NM.2">polyhedron</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">p</font> is  <a href="polyform.html#V5" title="POLYFORM:attr.5">simply-connected</a>  iff 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> <font color="Maroon" title="c1">p</font> <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> <font color="Maroon" title="c1">p</font> )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="polyform.html#NM2" title="POLYFORM:NM.2">polyhedron</a>] <span class="kw">means </span>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/>
<span class="kw">thus </span><a NAME="E1:81"/>
( <font color="Maroon" title="c1">p</font> is  <a href="polyform.html#V5" title="POLYFORM:attr.5">simply-connected</a>  implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">p</font>] )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( 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> <font color="Maroon" title="c1">p</font> <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> <font color="Maroon" title="c1">p</font> ) implies <font color="Maroon" title="c1">p</font> is  <a href="polyform.html#V5" title="POLYFORM:attr.5">simply-connected</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:81_1"/><span class="lab"><font color="Green" title="E60">A1</font></span>: 
<font color="Maroon" title="c1">p</font> is  <a href="polyform.html#V5" title="POLYFORM:attr.5">simply-connected</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">p</font>]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">n</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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></span><br/>

<a NAME="E2:81_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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81_1"><span class="lab"><font color="Green" title="E60">A1</font></span></a>, <a class="ref" href="polyform.html#D25" target="_self" title="POLYFORM:def.25">Def25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:81_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#T37" title="VECTSP_4:th.37">VECTSP_4:37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E2:81"/>
( <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#V5" title="POLYFORM:attr.5">simply-connected</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:81_2"/><span class="lab"><font color="Green" title="E60">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">p</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">p</font> is  <a href="polyform.html#V5" title="POLYFORM:attr.5">simply-connected</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">n</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="polyform.html#D25" target="_self" title="POLYFORM:def.25">POLYFORM:def 25</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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></span><br/>

<span class="kw">thus </span><a NAME="E2:81_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81_2"><span class="lab"><font color="Green" title="E60">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div>
