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<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="zfmisc_1.html#K1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>2</sub></font> is   <a href="orders_1.html#NM2">Order</a> of <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">F</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E2:2_1_2"/><i><font color="Green">E40</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font>
 <b>and </b><br/><a NAME="E3:2_1_2"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> holds <br/>for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> iff ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>) &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] ) )
 <b>from </b><i><a class="ref" href="card_3.html#S4">CARD_3:sch 4</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">P</font> being <a href="orders_2.html#V1">strict</a> <a href="orders_2.html#NM1">Poset</a> st <br/>( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">P</font> &amp; the <a href="orders_2.html#U1">InternalRel</a> of <font color="Olive">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<div><i><font color="Green">E43</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">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:2_1_2_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>]
 ;<br/><b>then consider </b><font color="Maroon">P1</font> being  <a href="orders_2.html#V1">strict</a> <a href="orders_2.html#NM1">Poset</a><b> such that </b><br/><a NAME="E3:2_1_2_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font> &amp; the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 ;<br/><b>assume </b><a NAME="E4:2_1_2_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">z</font>]
 ;<br/><b>then consider </b><font color="Maroon">P2</font> being  <a href="orders_2.html#V1">strict</a> <a href="orders_2.html#NM1">Poset</a><b> such that </b><br/><a NAME="E6:2_1_2_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P2</font> &amp; the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 ;<br/><a NAME="E7:2_1_2_1"/>
 <a href="relset_1.html#K4">dom</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P1</font>
 <b>by </b><i><a class="ref" href="orders_1.html#T99">ORDERS_1:99</a></i>;<br/><b>hence </b><a NAME="E8:2_1_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>by </b><i>, , <a class="ref" href="orders_1.html#T99">ORDERS_1:99</a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">A</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:2_1_2"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> iff  ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> <font color="Maroon">F</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">y</font>,<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="tarski.html#S1">TARSKI:sch 1</a>();<br/></i>
<b>take </b>
<font color="Maroon">A</font>
;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:2_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:2_1_2_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E4:2_1_2_2"/><i><font color="Green">E51</font></i>: 
 ex <font color="Olive">P</font> being <a href="orders_2.html#V1">strict</a> <a href="orders_2.html#NM1">Poset</a> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">P</font> &amp; the <a href="orders_2.html#U1">InternalRel</a> of <font color="Olive">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">P</font> being  <a href="orders_2.html#V1">strict</a> <a href="orders_2.html#NM1">Poset</a><b> such that </b><br/><a NAME="E6:2_1_2_2"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> &amp; the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:2_1_2_2"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> )
 <b>by </b><i>, , <a class="ref" href="card_5.html#T10">CARD_5:10</a></i>;<br/><b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="orders_1.html#NM2">Order</a> of <font color="Maroon">z</font> <b>by </b><i>, </i>;<br/><b>thus </b><a NAME="E10:2_1_2_2"/>
<font color="Maroon">x</font> is  <a href="orders_2.html#V1">strict</a> <a href="orders_2.html#NM1">Poset</a>
 <b>by </b><i/>;<br/><a NAME="E11:2_1_2_2"/>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P</font> &amp; <font color="Maroon">x</font> <a href="yellow21.html#K1" target="_self">as_1-sorted</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> )
 <b>by </b><i>, , <a class="ref" href="orders_1.html#T99">ORDERS_1:99</a></i>;<br/><b>hence </b><a NAME="E12:2_1_2_2"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="yellow21.html#K1" target="_self">as_1-sorted</a> </span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E8:2_1_2"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">x</font> is  <a href="orders_2.html#V1">strict</a> <a href="orders_2.html#NM1">Poset</a> &amp; the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="yellow21.html#K1" target="_self">as_1-sorted</a> </span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">P</font> = <font color="Maroon">x</font> as  <a href="orders_2.html#V1">strict</a> <a href="orders_2.html#NM1">Poset</a> ;<br/>
<a NAME="E10:2_1_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">x</font> <a href="yellow21.html#K1" target="_self">as_1-sorted</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">P</font>
 
<b>by </b><i/>;<br/>
<a NAME="E11:2_1_2"/>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">P</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
;<br/>
<a NAME="E12:2_1_2"/><b>then </b>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E13:2_1_2"/><b>then </b>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">P</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K3">Union</a> <font color="Maroon">F</font>
 
<b>by </b><i>, , , <a class="ref" href="card_5.html#T10">CARD_5:10</a></i>;<br/>
<b>hence </b><a NAME="E14:2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/>


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