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<div class="add">

<b>let </b><font color="Maroon" title="c22">C</font> be   <a href="topreal2.html#NM1" title="TOPREAL2:NM.1">Simple_closed_curve</a>;<br/>

<b>assume </b><a NAME="E1:147"/><i><font color="Green" title="E171">A1</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="jordan24.html#R1" title="JORDAN24:pred.1">realize-max-dist-in</a> <font color="Maroon" title="c22">C</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c23">Jc</font> =  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c22">C</font>;<br/>
<b>consider </b><font color="Maroon" title="c24">Pf</font> being    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,3</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 3</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>, <font color="Maroon" title="c25">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,3</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,<span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 3</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:147"/><i><font color="Green" title="E172">A2</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c25">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,3</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 3</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c24">Pf</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c25">f</font> )
 <b>by </b><i><a class="ref" href="jordan.html#T43" target="_self" title="JORDAN:th.43">Th43</a></i>;<br/>
<a NAME="E4:147"/>
( <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c22">C</font> &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <font color="Maroon" title="c22">C</font> )
 
<b>by </b><i><a class="txt" href="jordan.html#E1:147"><i><font color="Green" title="E171">A1</font></i></a>, <a class="ref" href="jordan.html#T79" target="_self" title="JORDAN:th.79">Th79</a>, <a class="ref" href="jordan.html#T80" target="_self" title="JORDAN:th.80">Th80</a></i>;<br/>
<a NAME="E5:147"/><b>then </b>
 <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c22">C</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 
<b>by </b><i><a class="ref" href="jordan6.html#D8" title="JORDAN6:def.8">JORDAN6:def 8</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c26">Pg</font> being    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>, <font color="Maroon" title="c27">g</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c22">C</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E7:147"/><i><font color="Green" title="E173">A3</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c27">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c22">C</font> &amp; <font color="Maroon" title="c26">Pg</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c27">g</font> )
 <b>by </b><i><a class="ref" href="jordan.html#T42" target="_self" title="JORDAN:th.42">Th42</a></i>;<br/>
<a NAME="E8:147"/><i><font color="Green" title="E174">A4</font></i>: 
 <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c22">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c22">C</font>
 
<b>by </b><i><a class="ref" href="jordan6.html#T76" title="JORDAN6:th.76">JORDAN6:76</a></i>;<br/>
<a NAME="E9:147"/><i><font color="Green" title="E175">A5</font></i>: 
<font color="Maroon" title="c22">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 3</span>)</span>,3
 
<b>by </b><i><a class="txt" href="jordan.html#E1:147"><i><font color="Green" title="E171">A1</font></i></a>, <a class="ref" href="jordan.html#T71" target="_self" title="JORDAN:th.71">Th71</a></i>;<br/>
<a NAME="E10:147"/><i><font color="Green" title="E176">A6</font></i>: 
( <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">C</font> &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">C</font> )
 
<b>by </b><i><a class="txt" href="jordan.html#E1:147"><i><font color="Green" title="E171">A1</font></i></a>, <a class="ref" href="jordan24.html#D1" title="JORDAN24:def.1">JORDAN24:def 1</a></i>;<br/>
<a NAME="E11:147"/><i><font color="Green" title="E177">A7</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topreala.html#K1" title="TOPREALA:func.1">Trectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 3</span>)</span>,3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 3</span>)</span>,3
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c28">AR</font> = <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>, <font color="Maroon" title="c29">BR</font> = <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>, <font color="Maroon" title="c30">CR</font> = <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,3</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>, <font color="Maroon" title="c31">DR</font> = <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 3</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topreala.html#K1" title="TOPREALA:func.1">Trectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 3</span>)</span>,3</span>)</span> <b>by </b><i><a class="txt" href="jordan.html#E10:147"><i><font color="Green" title="E176">A6</font></i></a>, <a class="txt" href="jordan.html#E9:147"><i><font color="Green" title="E175">A5</font></i></a>, <a class="txt" href="jordan.html#E141"><i><font color="Green" title="E124">Lm60</font></i></a>, <a class="txt" href="jordan.html#E142"><i><font color="Green" title="E125">Lm61</font></i></a>, <a class="txt" href="jordan.html#E148"><i><font color="Green" title="E131">Lm65</font></i></a>, <a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E13:147"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c26">Pg</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topreala.html#K1" title="TOPREALA:func.1">Trectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 3</span>)</span>,3</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan.html#E7:147"><i><font color="Green" title="E173">A3</font></i></a>, <a class="txt" href="jordan.html#E8:147"><i><font color="Green" title="E174">A4</font></i></a>, <a class="txt" href="jordan.html#E9:147"><i><font color="Green" title="E175">A5</font></i></a>, <a class="txt" href="jordan.html#E11:147"><i><font color="Green" title="E177">A7</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c32">h</font> = <font color="Maroon" title="c26">Pg</font> as    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c28">AR</font>,<font color="Maroon" title="c29">BR</font> <b>by </b><i><a class="ref" href="jordan.html#T30" target="_self" title="JORDAN:th.30">Th30</a></i>;<br/>
<a NAME="E15:147"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,3</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 3</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jgraph_6.html#K2" title="JGRAPH_6:func.2">closed_inside_of_rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 3</span>)</span>,3
 
<b>by </b><i><a class="txt" href="jordan.html#E141"><i><font color="Green" title="E124">Lm60</font></i></a>, <a class="txt" href="jordan.html#E148"><i><font color="Green" title="E131">Lm65</font></i></a>, <a class="txt" href="jordan.html#E142"><i><font color="Green" title="E125">Lm61</font></i></a>, <a class="ref" href="jordan1.html#D1" title="JORDAN1:def.1">JORDAN1:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c33">v</font> = <font color="Maroon" title="c24">Pf</font> as    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c30">CR</font>,<font color="Maroon" title="c31">DR</font> <b>by </b><i><a class="txt" href="jordan.html#E3:147"><i><font color="Green" title="E172">A2</font></i></a>, <a class="txt" href="jordan.html#E11:147"><i><font color="Green" title="E177">A7</font></i></a>, <a class="ref" href="jordan.html#T30" target="_self" title="JORDAN:th.30">Th30</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c34">s</font>, <font color="Maroon" title="c35">t</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <b> such that </b><br/><a NAME="E18:147"/><i><font color="Green" title="E178">A8</font></i>: 
<font color="Maroon" title="c32">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c34">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c35">t</font>
 <b>by </b><i><a class="txt" href="jordan.html#E95"><i><font color="Green" title="E82">Lm18</font></i></a>, <a class="txt" href="jordan.html#E96"><i><font color="Green" title="E83">Lm19</font></i></a>, <a class="txt" href="jordan.html#E100"><i><font color="Green" title="E87">Lm23</font></i></a>, <a class="txt" href="jordan.html#E102"><i><font color="Green" title="E89">Lm25</font></i></a>, <a class="ref" href="jgraph_8.html#T6" title="JGRAPH_8:th.6">JGRAPH_8:6</a></i>;<br/>
<a NAME="E19:147"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c32">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c33">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E20:147"/><b>then </b>
( <font color="Maroon" title="c33">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c35">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c24">Pf</font> &amp; <font color="Maroon" title="c32">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c34">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c26">Pg</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E21:147"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,3</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 3</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a> <font color="Maroon" title="c22">C</font>
 <b>by </b><i><a class="txt" href="jordan.html#E8:147"><i><font color="Green" title="E174">A4</font></i></a>, <a class="txt" href="jordan.html#E3:147"><i><font color="Green" title="E172">A2</font></i></a>, <a class="txt" href="jordan.html#E7:147"><i><font color="Green" title="E173">A3</font></i></a>, <a class="txt" href="jordan.html#E18:147"><i><font color="Green" title="E178">A8</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>


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