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<span class="kw">set </span><font color="Maroon" title="c2">A</font> =  <a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font>;<br/>
<span class="kw">thus </span><a NAME="E1:36_1_1"/>
 <a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font> is  <a href="altcat_1.html#V2" title="ALTCAT_1:attr.2">transitive</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">  not  <a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> </span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="36_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">o1</font>, <font color="Maroon" title="c4">o2</font>, <font color="Maroon" title="c5">o3</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font></span>)</span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="altcat_1.html#D2" onmouseover="rs('altcat_1/D2')" onmouseout="rh()">ALTCAT_1:def 2</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"> ( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c5">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  not <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c5">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">o19</font> = <font color="Maroon" title="c3">o1</font>, <font color="Maroon" title="c7">o29</font> = <font color="Maroon" title="c4">o2</font>, <font color="Maroon" title="c8">o39</font> = <font color="Maroon" title="c5">o3</font> as    <a href="orders_3.html#M1" title="ORDERS_3:mode.1">Element</a> of <font color="Maroon" title="c1">P</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="orders_3.html#D9" target="_self" onmouseover="rs('orders_3/D9')" onmouseout="rh()">Def9</a></span>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:36_1_1_1"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">and </span><br/><a NAME="E3:36_1_1_1"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c5">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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">  not <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c5">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<a NAME="E4:36_1_1_1"/>
 <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> (<font color="Maroon" title="c6">o19</font>,<font color="Maroon" title="c7">o29</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36_1_1_1"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="orders_3.html#D9" target="_self" onmouseover="rs('orders_3/D9')" onmouseout="rh()">Def9</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">f</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:36_1_1_1"/><span class="lab"><font color="Green" title="E16">A3</font></span>: 
<font color="Maroon" title="c9">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> (<font color="Maroon" title="c6">o19</font>,<font color="Maroon" title="c7">o29</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" onmouseover="rs('xboole_0/D1')" onmouseout="rh()">XBOOLE_0:def 1</a></span>;<br/>
<a NAME="E7:36_1_1_1"/>
 <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> (<font color="Maroon" title="c7">o29</font>,<font color="Maroon" title="c8">o39</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36_1_1_1"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="orders_3.html#D9" target="_self" onmouseover="rs('orders_3/D9')" onmouseout="rh()">Def9</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">g</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E9:36_1_1_1"/><span class="lab"><font color="Green" title="E17">A4</font></span>: 
<font color="Maroon" title="c10">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> (<font color="Maroon" title="c7">o29</font>,<font color="Maroon" title="c8">o39</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" onmouseover="rs('xboole_0/D1')" onmouseout="rh()">XBOOLE_0:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">f</font> = <font color="Maroon" title="c9">f</font>, <font color="Maroon" title="c12">g</font> = <font color="Maroon" title="c10">g</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:36_1_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E9:36_1_1_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a></span>;<br/>
<a NAME="E11:36_1_1_1"/>
<font color="Maroon" title="c12">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> (<font color="Maroon" title="c6">o19</font>,<font color="Maroon" title="c8">o39</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:36_1_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E9:36_1_1_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="ref" href="orders_3.html#T6" target="_self" onmouseover="rs('orders_3/T6')" onmouseout="rh()">Th6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:36_1_1_1"/>
 not <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c5">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="orders_3.html#D9" target="_self" onmouseover="rs('orders_3/D9')" onmouseout="rh()">Def9</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/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E2:36_1_1"/>
 not  <a href="orders_3.html#K4" title="ORDERS_3:func.4">POSAltCat</a> <font color="Maroon" title="c1">P</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="orders_3.html#D9" target="_self" onmouseover="rs('orders_3/D9')" onmouseout="rh()">Def9</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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