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<b>let </b><font color="Maroon" title="c1">Y</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Y</font>  st <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is <a href="tex_2.html#V1" title="TEX_2:attr.1">proper</a> holds <br/> not <font color="Maroon" title="c1">Y</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a></span><br/><b>let </b><font color="Maroon" title="c2">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is <a href="tex_2.html#V1" title="TEX_2:attr.1">proper</a> implies  not <font color="Maroon" title="c1">Y</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> )</span><br/>



<b>assume </b><a NAME="E1:17"/><i><font color="Green" title="E5">A1</font></i>: 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is <a href="tex_2.html#V1" title="TEX_2:attr.1">proper</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c1">Y</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a></span><br/>

<b>assume </b><a NAME="E2:17"/>
<font color="Maroon" title="c1">Y</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then consider </b><font color="Maroon" title="c3">s</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Y</font><b> such that </b><br/><a NAME="E4:17"/><i><font color="Green" title="E6">A2</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Y</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_1.html#D1" title="TEX_1:def.1">TEX_1:def 1</a></i>;<br/>
<a NAME="E5:17"/>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <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 <font color="Maroon" title="c1">Y</font>
 
<b>by </b><i><a class="txt" href="tex_2.html#E4:17"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="zfmisc_1.html#T24" title="ZFMISC_1:th.24">ZFMISC_1:24</a></i>;<br/>
<b>hence </b><a NAME="E6:17"/>
contradiction
 <b>by </b><i><a class="txt" href="tex_2.html#E1:17"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="tex_2.html#T5" target="_self" title="TEX_2:th.5">Th5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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