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<b>let </b><font color="Maroon" title="c2">A</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font>;<br/>

<b>assume </b><a NAME="E1:20_2_1"/><i><font color="Green" title="E6">A2</font></i>: 
not <font color="Maroon" title="c2">A</font> is <a href="tex_2.html#V1" title="TEX_2:attr.1">proper</a>
 ;<br/>

<b>assume </b><a NAME="E2:20_2_1"/>
<font color="Maroon" title="c2">A</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">s</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">A</font><b> such that </b><br/><a NAME="E4:20_2_1"/><i><font color="Green" title="E7">A3</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">s</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="tex_2.html#D1" target="_self" title="TEX_2:def.1">Def1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">s</font> = <font color="Maroon" title="c3">s</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Y</font> ;<br/>
<a NAME="E6:20_2_1"/>
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">s</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="tex_2.html#E4:20_2_1"><i><font color="Green" title="E7">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:20_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="tex_2.html#E1:20_2_1"><i><font color="Green" title="E6">A2</font></i></a></i>;<br/>


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