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<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">$1,$1</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$2 is   <a href="orders_1.html#NM2" title="ORDERS_1:NM.2">Order</a> of $1;<br/>
<b>consider </b><font color="Maroon" title="c2">F</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E2:2_1_2"/><i><font color="Green" title="E2">A3</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">W</font>
 <b>and </b><br/><a NAME="E3:2_1_2"/><i><font color="Green" title="E3">A4</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> holds <br/>for <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> iff ( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>) &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] ) )
 <b>from </b><i><a class="ref" href="card_3.html#S4" title="CARD_3:sch.4">CARD_3:sch 4</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">P</font> being <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a> st <br/>( $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">P</font> &amp; the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 );<br/>
<div><i><font color="Green" title="E4">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c5">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:2_1_2_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>]
 ;<br/><b>then consider </b><font color="Maroon" title="c6">P1</font> being  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a><b> such that </b><br/><a NAME="E3:2_1_2_1"/><i><font color="Green" title="E5">A6</font></i>: 
( <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P1</font> &amp; the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c6">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> )
 ;<br/><b>assume </b><a NAME="E4:2_1_2_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font>]
 ;<br/><b>then consider </b><font color="Maroon" title="c7">P2</font> being  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a><b> such that </b><br/><a NAME="E6:2_1_2_1"/><i><font color="Green" title="E6">A7</font></i>: 
( <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">P2</font> &amp; the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c7">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> )
 ;<br/><a NAME="E7:2_1_2_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c6">P1</font> <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="c6">P1</font>
 <b>by </b><i><a class="ref" href="orders_1.html#T99" title="ORDERS_1:th.99">ORDERS_1:99</a></i>;<br/><b>hence </b><a NAME="E8:2_1_2_1"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z</font>
 <b>by </b><i><a class="txt" href="yellow21.html#E3:2_1_2_1"><i><font color="Green" title="E5">A6</font></i></a>, <a class="txt" href="yellow21.html#E6:2_1_2_1"><i><font color="Green" title="E6">A7</font></i></a>, <a class="ref" href="orders_1.html#T99" title="ORDERS_1:th.99">ORDERS_1:99</a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c3">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:2_1_2"/><i><font color="Green" title="E5">A8</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> iff  ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="tarski.html#S1" title="TARSKI:sch.1">TARSKI:sch 1</a>(<a class="txt" href="yellow21.html#E4:2_1_2"><i><font color="Green" title="E4">A5</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c3">A</font>
;<br/>

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:2_1_2_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c5">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:2_1_2_2"/><i><font color="Green" title="E6">A9</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c2">F</font>
 <b>and </b><br/><a NAME="E4:2_1_2_2"/><i><font color="Green" title="E7">A10</font></i>: 
 ex <font color="Olive" title="b1">P</font> being <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a> st <br/>( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">P</font> &amp; the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> )
 <b>by </b><i><a class="txt" href="yellow21.html#E6:2_1_2"><i><font color="Green" title="E5">A8</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c6">P</font> being  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a><b> such that </b><br/><a NAME="E6:2_1_2_2"/><i><font color="Green" title="E8">A11</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P</font> &amp; the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c6">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> )
 <b>by </b><i><a class="txt" href="yellow21.html#E4:2_1_2_2"><i><font color="Green" title="E7">A10</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c7">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:2_1_2_2"/><i><font color="Green" title="E9">A12</font></i>: 
( <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">z</font> )
 <b>by </b><i><a class="txt" href="yellow21.html#E2:2_1_2"><i><font color="Green" title="E2">A3</font></i></a>, <a class="txt" href="yellow21.html#E3:2_1_2_2"><i><font color="Green" title="E6">A9</font></i></a>, <a class="ref" href="card_5.html#T10" title="CARD_5:th.10">CARD_5:10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">y</font> = <font color="Maroon" title="c5">y</font> as   <a href="orders_1.html#NM2" title="ORDERS_1:NM.2">Order</a> of <font color="Maroon" title="c7">z</font> <b>by </b><i><a class="txt" href="yellow21.html#E3:2_1_2"><i><font color="Green" title="E3">A4</font></i></a>, <a class="txt" href="yellow21.html#E8:2_1_2_2"><i><font color="Green" title="E9">A12</font></i></a></i>;<br/><b>thus </b><a NAME="E10:2_1_2_2"/>
<font color="Maroon" title="c4">x</font> is  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>
 <b>by </b><i><a class="txt" href="yellow21.html#E6:2_1_2_2"><i><font color="Green" title="E8">A11</font></i></a></i>;<br/><a NAME="E11:2_1_2_2"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">y</font> <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="c6">P</font> &amp; <font color="Maroon" title="c4">x</font> <a href="yellow21.html#K1" title="YELLOW21:func.1">as_1-sorted</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P</font> )
 <b>by </b><i><a class="txt" href="yellow21.html#E6:2_1_2_2"><i><font color="Green" title="E8">A11</font></i></a>, <a class="ref" href="yellow21.html#D1" target="_self" title="YELLOW21:def.1">Def1</a>, <a class="ref" href="orders_1.html#T99" title="ORDERS_1:th.99">ORDERS_1:99</a></i>;<br/><b>hence </b><a NAME="E12:2_1_2_2"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="yellow21.html#K1" title="YELLOW21:func.1">as_1-sorted</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font>
 <b>by </b><i><a class="txt" href="yellow21.html#E8:2_1_2_2"><i><font color="Green" title="E9">A12</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E8:2_1_2"/><i><font color="Green" title="E6">A13</font></i>: 
( <font color="Maroon" title="c4">x</font> is  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="yellow21.html#K1" title="YELLOW21:func.1">as_1-sorted</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">P</font> = <font color="Maroon" title="c4">x</font> as  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a> ;<br/>
<a NAME="E10:2_1_2"/><i><font color="Green" title="E7">A14</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="yellow21.html#K1" title="YELLOW21:func.1">as_1-sorted</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font>
 
<b>by </b><i><a class="ref" href="yellow21.html#D1" target="_self" title="YELLOW21:def.1">Def1</a></i>;<br/>
<a NAME="E11:2_1_2"/>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c5">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">P</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">P</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
;<br/>
<a NAME="E12:2_1_2"/><b>then </b>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c5">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">P</font>
 
<b>by </b><i><a class="txt" href="yellow21.html#E3:2_1_2"><i><font color="Green" title="E3">A4</font></i></a>, <a class="txt" href="yellow21.html#E8:2_1_2"><i><font color="Green" title="E6">A13</font></i></a>, <a class="txt" href="yellow21.html#E10:2_1_2"><i><font color="Green" title="E7">A14</font></i></a></i>;<br/>
<a NAME="E13:2_1_2"/><b>then </b>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c5">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="txt" href="yellow21.html#E2:2_1_2"><i><font color="Green" title="E2">A3</font></i></a>, <a class="txt" href="yellow21.html#E8:2_1_2"><i><font color="Green" title="E6">A13</font></i></a>, <a class="txt" href="yellow21.html#E10:2_1_2"><i><font color="Green" title="E7">A14</font></i></a>, <a class="ref" href="card_5.html#T10" title="CARD_5:th.10">CARD_5:10</a></i>;<br/>
<b>hence </b><a NAME="E14:2_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="yellow21.html#E6:2_1_2"><i><font color="Green" title="E5">A8</font></i></a></i>;<br/>


</div>
