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<b>let </b><font color="Maroon">G</font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">H</font> be    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">F</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>;<br/>





<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>] <b>means </b>(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">a<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">H</font> implies  <a href="group_4.html#K3" target="_self">Product</a> <font color="Maroon">a<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font> );<br/>
<a NAME="E1:24"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">H</font>
 ;<br/>

<a NAME="E2:24_1"/>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T55">GROUP_2:55</a></i>;<br/>
<b>hence </b><a NAME="E3:24_1"/>
 <a href="group_4.html#K3" target="_self">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font></span>)</span> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E33</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">F</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">d</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/><b>assume </b><a NAME="E1:24_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">F</font>]
 ;<br/><b>thus </b><a NAME="E2:24_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_2_1"/><i><font color="Green">E37</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">H</font>
 ;<br/>

<a NAME="E2:24_2_1"/>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T42">FINSEQ_1:42</a>, <a class="ref" href="finseq_1.html#T43">FINSEQ_1:43</a></i>;<br/>
<a NAME="E3:24_2_1"/><b>then </b>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">H</font> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="tarski.html#R1">c=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">H</font> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d</font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E4:24_2_1"/><b>then </b>
( <font color="Maroon">d</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font> &amp;  <a href="group_4.html#K3" target="_self">Product</a> <font color="Maroon">F</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font> &amp;  <a href="group_4.html#K3" target="_self">Product</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_4.html#K3" target="_self">Product</a> <font color="Maroon">F</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">d</font> )
 
<b>by </b><i>, , <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:24_2_1"/>
 <a href="group_4.html#K3" target="_self">Product</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="group_2.html#T59">GROUP_2:59</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E3:24"/>
for <font color="Olive">F</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">F</font>]
 
<b>from </b><i><a class="ref" href="finseq_2.html#S2">FINSEQ_2:sch 2</a>(, );<br/></i>
<b>hence </b><a NAME="E4:24"/>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">H</font> implies  <a href="group_4.html#K3" target="_self">Product</a> <font color="Maroon">F</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font> )
 ;<br/>


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