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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">x1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font> ex <font color="Olive">x2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G2</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="group_7.html#K17">&lt;*</a><span class="default"><font color="Olive">x1</font>,<font color="Olive">x2</font></span><a href="group_7.html#K17">*&gt;</a></span> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="group_7.html#K17">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x2</font></span>)</span></span><a href="group_7.html#K17">*&gt;</a></span> );<br/>
<a NAME="E1:2_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="group_7.html#K2">product</a> <span class="p2"><a href="group_7.html#K11">&lt;*</a><span class="default"><font color="Maroon">G1</font>,<font color="Maroon">G2</font></span><a href="group_7.html#K11">*&gt;</a></span></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="group_7.html#K2">product</a> <span class="p2"><a href="group_7.html#K11">&lt;*</a><span class="default"><font color="Maroon">H1</font>,<font color="Maroon">H2</font></span><a href="group_7.html#K11">*&gt;</a></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="2_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="group_7.html#K2">product</a> <span class="p2"><a href="group_7.html#K11">&lt;*</a><span class="default"><font color="Maroon">G1</font>,<font color="Maroon">G2</font></span><a href="group_7.html#K11">*&gt;</a></span></span>)</span>;<br/>

<b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font>, <font color="Maroon">h</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G2</font><b> such that </b><br/><a NAME="E2:2_1_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="group_7.html#K17">&lt;*</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">h</font></span><a href="group_7.html#K17">*&gt;</a></span>
 <b>by </b><i/>;<br/>
<b>take </b>
<span class="p1"><a href="group_7.html#K17">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">h</font></span>)</span></span><a href="group_7.html#K17">*&gt;</a></span>
;<br/>

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

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

<b>thus </b><a NAME="E3:2_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="group_7.html#K17">&lt;*</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">h</font></span><a href="group_7.html#K17">*&gt;</a></span> &amp; <span class="p1"><a href="group_7.html#K17">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">h</font></span>)</span></span><a href="group_7.html#K17">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="group_7.html#K17">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">h</font></span>)</span></span><a href="group_7.html#K17">*&gt;</a></span> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:2_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="group_7.html#K2">product</a> <span class="p2"><a href="group_7.html#K11">&lt;*</a><span class="default"><font color="Maroon">G1</font>,<font color="Maroon">G2</font></span><a href="group_7.html#K11">*&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="group_7.html#K2">product</a> <span class="p2"><a href="group_7.html#K11">&lt;*</a><span class="default"><font color="Maroon">H1</font>,<font color="Maroon">H2</font></span><a href="group_7.html#K11">*&gt;</a></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="group_7.html#K2">product</a> <span class="p2"><a href="group_7.html#K11">&lt;*</a><span class="default"><font color="Maroon">G1</font>,<font color="Maroon">G2</font></span><a href="group_7.html#K11">*&gt;</a></span></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="E3:2_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="group_7.html#K2">product</a> <span class="p2"><a href="group_7.html#K11">&lt;*</a><span class="default"><font color="Maroon">G1</font>,<font color="Maroon">G2</font></span><a href="group_7.html#K11">*&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="group_7.html#K2">product</a> <span class="p2"><a href="group_7.html#K11">&lt;*</a><span class="default"><font color="Maroon">H1</font>,<font color="Maroon">H2</font></span><a href="group_7.html#K11">*&gt;</a></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="group_7.html#K2">product</a> <span class="p2"><a href="group_7.html#K11">&lt;*</a><span class="default"><font color="Maroon">G1</font>,<font color="Maroon">G2</font></span><a href="group_7.html#K11">*&gt;</a></span></span>)</span>  ex <font color="Olive">x1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font> ex <font color="Olive">x2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G2</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="group_7.html#K17">&lt;*</a><span class="default"><font color="Olive">x1</font>,<font color="Olive">x2</font></span><a href="group_7.html#K17">*&gt;</a></span> &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> <span class="p1"><a href="group_7.html#K17">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x2</font></span>)</span></span><a href="group_7.html#K17">*&gt;</a></span> )
 ;<br/>


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