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<span class="kw">assume </span><a NAME="E1:31_1_1"/>
 not  <a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> (<font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c3">r</font>) is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:31_1_1"/><span class="lab"><font color="Green" title="E11">A1</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> (<font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c3">r</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">e1</font> = <font color="Maroon" title="c2">e</font> as   <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">f</font> =  <a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c3">r</font>);<br/>
<a NAME="E5:31_1_1"/><span class="lab"><font color="Green" title="E12">A2</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c3">r</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_9.html#D3" target="_self" title="EUCLID_9:def.3">Def3</a></span>;<br/>
<a NAME="E6:31_1_1"/><span class="lab"><font color="Green" title="E13">A3</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T89" title="FINSEQ_1:th.89">FINSEQ_1:89</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">N</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E8:31_1_1"/><span class="lab"><font color="Green" title="E14">A4</font></span>: 
<font color="Maroon" title="c7">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<a NAME="E9:31_1_1"/>
<span class="p1">(<span class="default"><a href="euclid_9.html#K3" title="EUCLID_9:func.3">Intervals</a> (<font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c3">r</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">e1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">N</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">e1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">N</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:31_1_1"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="txt" href="#E6:31_1_1"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="ref" href="euclid_9.html#D3" target="_self" title="EUCLID_9:def.3">Def3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:31_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31_1_1"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E5:31_1_1"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="txt" href="#E6:31_1_1"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="txt" href="#E8:31_1_1"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="card_3.html#T9" title="CARD_3:th.9">CARD_3:9</a></span>; <a class="txt" 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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