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<b>let </b><font color="Maroon">R</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="pboole.html#NM2">ManySortedFunction</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>;<br/>



<b>assume </b><a NAME="E1:4"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">F</font> is <a href="osalg_3.html#V1" target="_self">order-sorted</a>
 ;<br/>

<b>let </b><font color="Maroon">w1</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="finseq_1.html#K13">*</a> , <font color="Maroon">w2</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="finseq_1.html#K13">*</a> ;<br/>

<b>assume </b><a NAME="E2:4"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">w1</font> <a href="osalg_1.html#R2">&lt;=</a> <font color="Maroon">w2</font>
 ;<br/>

<a NAME="E3:4"/><i><font color="Green">E34</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">w1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">w2</font> &amp; ( for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">w1</font> holds <br/>for <font color="Olive">s1</font>, <font color="Olive">s2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>  st <font color="Olive">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">s2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> holds <br/><font color="Olive">s1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">s2</font> ) )
 
<b>by </b><i>, <a class="ref" href="osalg_1.html#D7">OSALG_1:def 7</a></i>;<br/>
<a NAME="E4:4"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">w1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">w2</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<b>thus </b><a NAME="E5:4"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w1</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:4_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w1</font>
 ;<br/>

<b>set </b><font color="Maroon">a</font> = <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w1</font>;<br/>
<b>set </b><font color="Maroon">b</font> = <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w2</font>;<br/>
<a NAME="E2:4_1"/><i><font color="Green">E44</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w1</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">w1</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w2</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">w2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="pboole.html#D19">PBOOLE:def 19</a></i>;<br/>
<b>then consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:4_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">w1</font></span>)</span> )
 <b>and </b><br/><a NAME="E5:4_1"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">w1</font></span>)</span> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">w1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font>
 <b>by </b><i>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<a NAME="E6:4_1"/>
( <font color="Maroon">w1</font> is    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> &amp; <font color="Maroon">w2</font> is    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E7:4_1"/><b>then </b><i><font color="Green">E51</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">w1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">w2</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> )
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E8:4_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E9:4_1"/><b>then </b><i><font color="Green">E52</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">w1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">w1</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">w2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">w2</font> )
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E10:4_1"/>
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">w2</font></span>)</span> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">w2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:4_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">w2</font></span>)</span>
 ;<br/>

<a NAME="E2:4_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">w1</font></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E3:4_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">w1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E4:4_1_1"/>
( <font color="Maroon">w1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">w1</font> &amp; <font color="Maroon">w2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">w2</font> )
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">s1</font> = <font color="Maroon">w1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>, <font color="Maroon">s2</font> = <font color="Maroon">w2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/>
<a NAME="E6:4_1_1"/><i><font color="Green">E56</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">w1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">w1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">w2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">w2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span> )
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<a NAME="E7:4_1_1"/>
<font color="Maroon">s1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s2</font>
 
<b>by </b><i>, , , <a class="ref" href="osalg_1.html#D7">OSALG_1:def 7</a></i>;<br/>
<a NAME="E8:4_1_1"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E9:4_1_1"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">w2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:4_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pboole.html#K6">#</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w2</font>
 <b>by </b><i>, , , , <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>


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


</div>
