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<b>let </b><font color="Maroon" title="c3">P</font>, <font color="Maroon" title="c4">Q</font> be    <a href="pcs_0.html#L2" title="PCS_0:struct.2">pcs-Str</a> ;<br/><b>let </b><font color="Maroon" title="c5">p</font>, <font color="Maroon" title="c6">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="pcs_0.html#K22" title="PCS_0:func.22">pcs-times</a> <font color="Maroon" title="c4">Q</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c7">p1</font>, <font color="Maroon" title="c8">p2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">P</font>;<br/><b>let </b><font color="Maroon" title="c9">q1</font>, <font color="Maroon" title="c10">q2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">Q</font>;<br/>







<b>assume </b><a NAME="E1:173"/><i><font color="Green" title="E63">A1</font></i>: 
( <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c9">q1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c6">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c10">q2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 ;<br/>

<b>thus </b><a NAME="E2:173"/>
( <font color="Maroon" title="c5">p</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">q</font> implies ( <font color="Maroon" title="c7">p1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">p2</font> &amp; <font color="Maroon" title="c9">q1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c10">q2</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="173_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:173_1"/>
<font color="Maroon" title="c5">p</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">q</font>
 ;<br/>

<a NAME="E2:173_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="yellow_3.html#K2" title="YELLOW_3:func.2">["</a><span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">P</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">Q</font></span><a href="yellow_3.html#K2" title="YELLOW_3:func.2">"]</a></span>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">a</font>, <font color="Maroon" title="c12">b</font>, <font color="Maroon" title="c13">s</font>, <font color="Maroon" title="c14">t</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:173_1"/><i><font color="Green" title="E64">A2</font></i>: 
( <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c6">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">s</font>,<font color="Maroon" title="c14">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>and </b><br/><a NAME="E5:173_1"/><i><font color="Green" title="E65">A3</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c13">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">P</font>
 <b>and </b><br/><a NAME="E6:173_1"/><i><font color="Green" title="E66">A4</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">b</font>,<font color="Maroon" title="c14">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">Q</font>
 <b>by </b><i><a class="ref" href="yellow_3.html#D1" title="YELLOW_3:def.1">YELLOW_3:def 1</a></i>;<br/>
<a NAME="E7:173_1"/><i><font color="Green" title="E67">A5</font></i>: 
( <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p1</font> &amp; <font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q1</font> &amp; <font color="Maroon" title="c13">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p2</font> &amp; <font color="Maroon" title="c14">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">q2</font> )
 
<b>by </b><i><a class="txt" href="pcs_0.html#E1:173"><i><font color="Green" title="E63">A1</font></i></a>, <a class="txt" href="pcs_0.html#E4:173_1"><i><font color="Green" title="E64">A2</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E8:173_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">P</font>
 <b>by </b><i><a class="txt" href="pcs_0.html#E5:173_1"><i><font color="Green" title="E65">A3</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a><br/>

<b>thus </b><a NAME="E9:173_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">q1</font>,<font color="Maroon" title="c10">q2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">Q</font>
 <b>by </b><i><a class="txt" href="pcs_0.html#E6:173_1"><i><font color="Green" title="E66">A4</font></i></a>, <a class="txt" href="pcs_0.html#E7:173_1"><i><font color="Green" title="E67">A5</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a><br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:173"/>
( <font color="Maroon" title="c7">p1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">p2</font> &amp; <font color="Maroon" title="c9">q1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c10">q2</font> )
 ;<br/>

<a NAME="E4:173"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">P</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">q1</font>,<font color="Maroon" title="c10">q2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">Q</font> )
 
<b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>
<b>hence </b><a NAME="E5:173"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="pcs_0.html#K22" title="PCS_0:func.22">pcs-times</a> <font color="Maroon" title="c4">Q</font></span>)</span>
 <b>by </b><i><a class="txt" href="pcs_0.html#E1:173"><i><font color="Green" title="E63">A1</font></i></a>, <a class="ref" href="yellow_3.html#D1" title="YELLOW_3:def.1">YELLOW_3:def 1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a><br/>


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