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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>for <font color="Olive">h</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Olive">h</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font>;<br/>
<a NAME="E1:45_1_1"/><i><font color="Green">E39</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="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S2</font>,<font color="Maroon">T1</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="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S1</font>,<font color="Maroon">T2</font></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="45_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="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S2</font>,<font color="Maroon">T1</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">h</font> = <font color="Maroon">x</font> as  <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S2</font>,<font color="Maroon">T1</font> <b>by </b><i/>;<br/>
<a NAME="E2:45_1_1_1"/>
<font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> is  <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S1</font>,<font color="Maroon">T1</font>
 
<b>by </b><i>, <a class="ref" href="waybel20.html#T29">WAYBEL20:29</a></i>;<br/>
<a NAME="E3:45_1_1_1"/><b>then </b>
<font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> is  <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S1</font>,<font color="Maroon">T2</font>
 
<b>by </b><i><a class="ref" href="waybel27.html#D1" target="_self">Def1</a>, <a class="ref" href="waybel20.html#T29">WAYBEL20:29</a></i>;<br/>
<a NAME="E4:45_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> is  <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S1</font>,<font color="Maroon">T2</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">y</font> = <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S1</font>,<font color="Maroon">T2</font></span>)</span> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">y</font>
;<br/>

<b>thus </b><a NAME="E6:45_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">j</font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S2</font>,<font color="Maroon">T1</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S1</font>,<font color="Maroon">T2</font></span>)</span><b> such that </b><br/><a NAME="E3:45_1_1"/><i><font color="Green">E48</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="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S2</font>,<font color="Maroon">T1</font></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">j</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>reconsider </b><font color="Maroon">F</font> = <font color="Maroon">j</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S2</font>,<font color="Maroon">T1</font></span>)</span>,<span class="p1">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S1</font>,<font color="Maroon">T2</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<b>let </b><font color="Maroon">h</font> be  <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S2</font>,<font color="Maroon">T1</font>;<br/>

<a NAME="E5:45_1_1"/>
<font color="Maroon">h</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S2</font>,<font color="Maroon">T1</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:45_1_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/>


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