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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">ls</font> being  <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">s</font> ex <font color="Olive">lt</font> being  <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <font color="Maroon">S</font>,<font color="Maroon">s</font></span>)</span>,<font color="Olive">ls</font> &amp; <font color="Olive">lt</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Olive">ls</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <font color="Maroon">T</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span></span>)</span>,<font color="Olive">lt</font> );<br/>
<a NAME="E1:36_1_1"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" target="_self">pi_1</a> <font color="Maroon">S</font>,<font color="Maroon">s</font></span>)</span>  ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" target="_self">pi_1</a> <font color="Maroon">T</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" target="_self">pi_1</a> <font color="Maroon">S</font>,<font color="Maroon">s</font></span>)</span>;<br/>

<b>consider </b><font color="Maroon">ls</font> being   <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">s</font><b> such that </b><br/><a NAME="E2:36_1_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <font color="Maroon">S</font>,<font color="Maroon">s</font></span>)</span>,<font color="Maroon">ls</font>
 <b>by </b><i><a class="ref" href="topalg_1.html#T48">TOPALG_1:48</a></i>;<br/>
<b>reconsider </b><font color="Maroon">lt</font> = <font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">ls</font> as   <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> =  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <font color="Maroon">T</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span></span>)</span>,<font color="Maroon">lt</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" target="_self">pi_1</a> <font color="Maroon">T</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="topalg_1.html#T48">TOPALG_1:48</a></i>;<br/>
<b>take </b>
<font color="Maroon">y</font>
;<br/>

<b>take </b>
<font color="Maroon">ls</font>
;<br/>

<b>take </b>
<font color="Maroon">lt</font>
;<br/>

<b>thus </b><a NAME="E5:36_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <font color="Maroon">S</font>,<font color="Maroon">s</font></span>)</span>,<font color="Maroon">ls</font> &amp; <font color="Maroon">lt</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">ls</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <font color="Maroon">T</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span></span>)</span>,<font color="Maroon">lt</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:36_1_1"/>
 ex <font color="Olive">h</font> being  <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" target="_self">pi_1</a> <font color="Maroon">S</font>,<font color="Maroon">s</font></span>)</span>,<span class="p1">(<span class="default"><a href="topalg_1.html#NK4" target="_self">pi_1</a> <font color="Maroon">T</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span></span>)</span> st <br/>for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" target="_self">pi_1</a> <font color="Maroon">S</font>,<font color="Maroon">s</font></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>]
 
<b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<b>hence </b><a NAME="E3:36_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" target="_self">pi_1</a> <font color="Maroon">S</font>,<font color="Maroon">s</font></span>)</span>,<span class="p1">(<span class="default"><a href="topalg_1.html#NK4" target="_self">pi_1</a> <font color="Maroon">T</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span></span>)</span> st <br/>for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK4" target="_self">pi_1</a> <font color="Maroon">S</font>,<font color="Maroon">s</font></span>)</span>  ex <font color="Olive">ls</font> being  <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">s</font> ex <font color="Olive">lt</font> being  <a href="topalg_1.html#NM1">Loop</a> of <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <font color="Maroon">S</font>,<font color="Maroon">s</font></span>)</span>,<font color="Olive">ls</font> &amp; <font color="Olive">lt</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Olive">ls</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="topalg_1.html#K2">EqRel</a> <font color="Maroon">T</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span></span>)</span>,<font color="Olive">lt</font> )
 ;<br/>


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