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<span class="kw">assume </span><a NAME="E1:101_1_1"/>
 <a href="simplex0.html#K8" title="SIMPLEX0:func.8">Skeleton_of</a> (<font color="Maroon" title="c2">KX</font>,<font color="Maroon" title="c3">i</font>) is  <a href="simplex0.html#NV9" title="SIMPLEX0:NV.9">with_non-empty_element</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"> contradiction</span><br/>

<a NAME="E2:101_1_1"/><span class="kw">then </span>
 the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K8" title="SIMPLEX0:func.8">Skeleton_of</a> (<font color="Maroon" title="c2">KX</font>,<font color="Maroon" title="c3">i</font>)</span>)</span> is  <a href="setfam_1.html#NV3" title="SETFAM_1:NV.3">with_non-empty_element</a> 
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">x</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E4:101_1_1"/><span class="lab"><font color="Green" title="E41">A1</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K8" title="SIMPLEX0:func.8">Skeleton_of</a> (<font color="Maroon" title="c2">KX</font>,<font color="Maroon" title="c3">i</font>)</span>)</span>
 ;<br/>
<a NAME="E5:101_1_1"/><span class="lab"><font color="Green" title="E42">A2</font></span>: 
 the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">KX</font> is  <a href="setfam_1.html#V2" title="SETFAM_1:attr.2">empty-membered</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#D7" target="_self" onmouseover="rs('simplex0/D7')" onmouseout="rh()">Def7</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E7:101_1_1"/><span class="lab"><font color="Green" title="E43">A3</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">y</font>
 <span class="kw">and </span><br/><a NAME="E8:101_1_1"/><span class="lab"><font color="Green" title="E44">A4</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="simplex0.html#NK4" title="SIMPLEX0:NK.4">the_subsets_with_limited_card</a> (<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1</span>)</span></span>)</span>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">KX</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:101_1_1"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="simplex0.html#T2" target="_self" onmouseover="rs('simplex0/T2')" onmouseout="rh()">Th2</a></span>;<br/>
<a NAME="E9:101_1_1"/>
<font color="Maroon" title="c5">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">KX</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:101_1_1"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="ref" href="simplex0.html#D2" target="_self" onmouseover="rs('simplex0/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<a NAME="E10:101_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">y</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:101_1_1"><span class="lab"><font color="Green" title="E42">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:101_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:101_1_1"><span class="lab"><font color="Green" title="E43">A3</font></span></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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