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<span class="kw">let </span><font color="Maroon" title="c1">A</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font><br/> for <font color="Olive" title="b3">o</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font>  holds <br/>( <font color="Olive" title="b1">a</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Olive" title="b3">o</font>,<font color="Olive" title="b2">b</font> or <font color="Olive" title="b2">b</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Olive" title="b3">o</font>,<font color="Olive" title="b1">a</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">o</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font>  holds <br/>( <font color="Maroon" title="c2">a</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Olive" title="b1">o</font>,<font color="Maroon" title="c3">b</font> or <font color="Maroon" title="c3">b</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Olive" title="b1">o</font>,<font color="Maroon" title="c2">a</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">o</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">a</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">b</font> or <font color="Maroon" title="c3">b</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">a</font> )</span><br/>







<a NAME="E1:14"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">o</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">o</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="arrow.html#D1" target="_self" title="ARROW:def.1">Def1</a></span>;<br/>
<span class="kw">thus </span><a NAME="E2:14"/>
( <font color="Maroon" title="c2">a</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c3">b</font> or <font color="Maroon" title="c3">b</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="arrow.html#D4" target="_self" title="ARROW:def.4">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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