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<b>let </b><font color="Maroon" title="c3">P</font> be  <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a>  <a href="pcs_0.html#L2" title="PCS_0:struct.2">pcs-Str</a> ;<br/><b>let </b><font color="Maroon" title="c4">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:154"/><i><font color="Green" title="E58">a0</font></i>: 
not <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">P</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c5">R</font> =  <a href="pcs_0.html#K20" title="PCS_0:func.20">pcs-extension</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">a</font>;<br/>
<a NAME="E2:154"/><i><font color="Green" title="E59">a3</font></i>: 
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="pcs_0.html#K20" title="PCS_0:func.20">pcs-extension</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="pcs_0.html#K20" title="PCS_0:func.20">pcs-extension</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">a</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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="ref" href="pcs_0.html#D39" target="_self" title="PCS_0:def.39">Def91</a></i>;<br/>
<b>let </b><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font>, <font color="Maroon" title="c8">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D8" title="RELAT_2:def.8">RELAT_2:def 8</a>,<a class="ref" href="orders_2.html#D5" title="ORDERS_2:def.5">ORDERS_2:def 5</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E3:154"/><i><font color="Green" title="E60">a2</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pcs_0.html#K20" title="PCS_0:func.20">pcs-extension</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">a</font></span>)</span>
 <b>and </b><br/><a NAME="E4:154"/><i><font color="Green" title="E61">a6</font></i>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pcs_0.html#K20" title="PCS_0:func.20">pcs-extension</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">a</font></span>)</span>
 <b>and </b><br/><a NAME="E5:154"/><i><font color="Green" title="E62">a7</font></i>: 
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pcs_0.html#K20" title="PCS_0:func.20">pcs-extension</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">a</font></span>)</span>
 <b>and </b><br/><a NAME="E6:154"/><i><font color="Green" title="E63">a4</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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"><a href="pcs_0.html#K20" title="PCS_0:func.20">pcs-extension</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">a</font></span>)</span>
 <b>and </b><br/><a NAME="E7:154"/><i><font color="Green" title="E64">a5</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c8">z</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"><a href="pcs_0.html#K20" title="PCS_0:func.20">pcs-extension</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">a</font></span>)</span>
 ;<br/>

<a NAME="E8:154"/><i><font color="Green" title="E65">a8</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c8">z</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="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="pcs_0.html#K20" title="PCS_0:func.20">pcs-extension</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">a</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="pcs_0.html#E5:154"><i><font color="Green" title="E62">a7</font></i></a>, <a class="ref" href="zfmisc_1.html#T128" title="ZFMISC_1:th.128">ZFMISC_1:128</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c10">y</font> = <font color="Maroon" title="c7">y</font>, <font color="Maroon" title="c11">z</font> = <font color="Maroon" title="c8">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="pcs_0.html#K20" title="PCS_0:func.20">pcs-extension</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">a</font></span>)</span> <b>by </b><i><a class="txt" href="pcs_0.html#E3:154"><i><font color="Green" title="E60">a2</font></i></a>, <a class="txt" href="pcs_0.html#E4:154"><i><font color="Green" title="E61">a6</font></i></a>, <a class="txt" href="pcs_0.html#E5:154"><i><font color="Green" title="E62">a7</font></i></a></i>;<br/>
<a NAME="E10:154"/><i><font color="Green" title="E66">b1</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c10">y</font>
 
<b>by </b><i><a class="txt" href="pcs_0.html#E6:154"><i><font color="Green" title="E63">a4</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E11:154"/><i><font color="Green" title="E67">b4</font></i>: 
<font color="Maroon" title="c10">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c11">z</font>
 
<b>by </b><i><a class="txt" href="pcs_0.html#E7:154"><i><font color="Green" title="E64">a5</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="154_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:154_1"/>
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> or <font color="Maroon" title="c9">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="154_1_1"><b>suppose </b></a><a NAME="E1:154_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:154_1_1"/>
<span class="p1"><a href="pcs_0.html#NK13" title="PCS_0:NK.13">[^</a><span class="default">,</span><a href="pcs_0.html#NK13" title="PCS_0:NK.13">^]</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"><a href="pcs_0.html#K20" title="PCS_0:func.20">pcs-extension</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="pcs_0.html#E2:154"><i><font color="Green" title="E59">a3</font></i></a>, <a class="txt" href="pcs_0.html#E8:154"><i><font color="Green" title="E65">a8</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="154_1_2"><b>suppose </b></a><a NAME="E1:154_1_2"/><i><font color="Green" title="E68">b2</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c12">x0</font> = <font color="Maroon" title="c9">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">P</font> <b>by </b><i><a class="ref" href="pcs_0.html#T25" target="_self" title="PCS_0:th.25">Th10</a></i>;<br/><a NAME="E3:154_1_2"/><i><font color="Green" title="E69">b3</font></i>: 
<font color="Maroon" title="c12">x0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a</font>
 <b>by </b><i><a class="txt" href="pcs_0.html#E1:154_1_2"><i><font color="Green" title="E68">b2</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">y0</font> = <font color="Maroon" title="c10">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">P</font> <b>by </b><i><a class="txt" href="pcs_0.html#E1:154"><i><font color="Green" title="E58">a0</font></i></a>, <a class="txt" href="pcs_0.html#E10:154"><i><font color="Green" title="E66">b1</font></i></a>, <a class="txt" href="pcs_0.html#E3:154_1_2"><i><font color="Green" title="E69">b3</font></i></a>, <a class="ref" href="pcs_0.html#T24" target="_self" title="PCS_0:th.24">Th11</a></i>;<br/><a NAME="E5:154_1_2"/>
<font color="Maroon" title="c13">y0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a</font>
 <b>by </b><i><a class="txt" href="pcs_0.html#E3:154_1_2"><i><font color="Green" title="E69">b3</font></i></a>, <a class="txt" href="pcs_0.html#E1:154"><i><font color="Green" title="E58">a0</font></i></a>, <a class="txt" href="pcs_0.html#E10:154"><i><font color="Green" title="E66">b1</font></i></a>, <a class="ref" href="pcs_0.html#T24" target="_self" title="PCS_0:th.24">Th11</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c14">z0</font> = <font color="Maroon" title="c11">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">P</font> <b>by </b><i><a class="txt" href="pcs_0.html#E1:154"><i><font color="Green" title="E58">a0</font></i></a>, <a class="txt" href="pcs_0.html#E11:154"><i><font color="Green" title="E67">b4</font></i></a>, <a class="ref" href="pcs_0.html#T24" target="_self" title="PCS_0:th.24">Th11</a></i>;<br/><a NAME="E7:154_1_2"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a</font>
 <b>by </b><i><a class="txt" href="pcs_0.html#E1:154"><i><font color="Green" title="E58">a0</font></i></a>, <a class="txt" href="pcs_0.html#E10:154"><i><font color="Green" title="E66">b1</font></i></a>, <a class="txt" href="pcs_0.html#E3:154_1_2"><i><font color="Green" title="E69">b3</font></i></a>, <a class="ref" href="pcs_0.html#T24" target="_self" title="PCS_0:th.24">Th11</a></i>;<br/><a NAME="E8:154_1_2"/><b>then </b>
( <font color="Maroon" title="c12">x0</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c13">y0</font> &amp; <font color="Maroon" title="c13">y0</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c14">z0</font> )
 <b>by </b><i><a class="txt" href="pcs_0.html#E10:154"><i><font color="Green" title="E66">b1</font></i></a>, <a class="txt" href="pcs_0.html#E11:154"><i><font color="Green" title="E67">b4</font></i></a>, <a class="txt" href="pcs_0.html#E1:154_1_2"><i><font color="Green" title="E68">b2</font></i></a>, <a class="ref" href="pcs_0.html#T26" target="_self" title="PCS_0:th.26">Th12</a></i>;<br/><a NAME="E9:154_1_2"/><b>then </b>
<font color="Maroon" title="c12">x0</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c14">z0</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#D2" title="YELLOW_0:def.2">YELLOW_0:def 2</a></i>;<br/><a NAME="E10:154_1_2"/><b>then </b>
<font color="Maroon" title="c9">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c11">z</font>
 <b>by </b><i><a class="ref" href="pcs_0.html#T23" target="_self" title="PCS_0:th.23">Th13</a></i>;<br/><b>hence </b><a NAME="E11:154_1_2"/>
<span class="p1"><a href="pcs_0.html#NK13" title="PCS_0:NK.13">[^</a><span class="default">,</span><a href="pcs_0.html#NK13" title="PCS_0:NK.13">^]</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"><a href="pcs_0.html#K20" title="PCS_0:func.20">pcs-extension</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">a</font></span>)</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/></div><b>end;</b></div></div><b>end;</b></div>

</div>
