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<span class="kw">let </span><font color="Maroon" title="c1">T</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">c</font> being   <a href="topalg_6.html#V7" title="TOPALG_6:attr.7">with_endpoints</a>  <a href="topalg_6.html#NM1" title="TOPALG_6:NM.1">Curve</a> of <font color="Maroon" title="c1">T</font> holds <br/> (  <a href="topalg_6.html#K2" title="TOPALG_6:func.2">the_first_point_of</a> <font color="Olive" title="b1">c</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Olive" title="b1">c</font> &amp;  <a href="topalg_6.html#K3" title="TOPALG_6:func.3">the_last_point_of</a> <font color="Olive" title="b1">c</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Olive" title="b1">c</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">c</font> be   <a href="topalg_6.html#V7" title="TOPALG_6:attr.7">with_endpoints</a>  <a href="topalg_6.html#NM1" title="TOPALG_6:NM.1">Curve</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="topalg_6.html#K2" title="TOPALG_6:func.2">the_first_point_of</a> <font color="Maroon" title="c2">c</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c2">c</font> &amp;  <a href="topalg_6.html#K3" title="TOPALG_6:func.3">the_last_point_of</a> <font color="Maroon" title="c2">c</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c2">c</font> )</span><br/>



<a NAME="E1:60"/><span class="lab"><font color="Green" title="E34">A1</font></span>: 
 <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#T40" onmouseover="rs('xxreal_2/T40')" onmouseout="rh()">XXREAL_2:40</a></span>;<br/>
<a NAME="E2:60"/>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_6.html#T27" target="_self" onmouseover="rs('topalg_6/T27')" onmouseout="rh()">Th27</a></span>;<br/>
<a NAME="E3:60"/><span class="kw">then </span>
(  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c</font> &amp;  <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:60"/>
(  <a href="topalg_6.html#K2" title="TOPALG_6:func.2">the_first_point_of</a> <font color="Maroon" title="c2">c</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c2">c</font> &amp;  <a href="topalg_6.html#K3" title="TOPALG_6:func.3">the_last_point_of</a> <font color="Maroon" title="c2">c</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c2">c</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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