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<div class="add">

<b>let </b><font color="Maroon">G1</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="group_1.html#L1">HGrStr</a> ;<br/><b>let </b><font color="Maroon">x1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font>;<br/><b>let </b><font color="Maroon">x2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font>;<br/>





<b>set </b><font color="Maroon">G</font> = <span class="p1"><a href="group_7.html#K4" target="_self">&lt;*</a><span class="default"><font color="Maroon">G1</font></span><a href="group_7.html#K4" target="_self">*&gt;</a></span>;<br/>
<b>reconsider </b><font color="Maroon">l</font> = <span class="p1"><a href="group_7.html#K10" target="_self">&lt;*</a><span class="default"><font color="Maroon">x1</font></span><a href="group_7.html#K10" target="_self">*&gt;</a></span>, <font color="Maroon">p</font> = <span class="p1"><a href="group_7.html#K10" target="_self">&lt;*</a><span class="default"><font color="Maroon">x2</font></span><a href="group_7.html#K10" target="_self">*&gt;</a></span>, <font color="Maroon">lpl</font> = <span class="p1"><a href="group_7.html#K10" target="_self">&lt;*</a><span class="default"><font color="Maroon">x1</font></span><a href="group_7.html#K10" target="_self">*&gt;</a></span> <a href="group_1.html#K1">*</a> <span class="p1"><a href="group_7.html#K10" target="_self">&lt;*</a><span class="default"><font color="Maroon">x2</font></span><a href="group_7.html#K10" target="_self">*&gt;</a></span>, <font color="Maroon">lpp</font> = <span class="p1"><a href="group_7.html#K10" target="_self">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x2</font></span>)</span></span><a href="group_7.html#K10" target="_self">*&gt;</a></span> as    <a href="pralg_2.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2"><a href="group_7.html#K4" target="_self">&lt;*</a><span class="default"><font color="Maroon">G1</font></span><a href="group_7.html#K4" target="_self">*&gt;</a></span></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E2:76"/><i><font color="Green">E29</font></i>: 
1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:76"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> &amp; <span class="p1"><a href="group_7.html#K4" target="_self">&lt;*</a><span class="default"><font color="Maroon">G1</font></span><a href="group_7.html#K4" target="_self">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">lpl</font> = 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2"><a href="group_7.html#K4" target="_self">&lt;*</a><span class="default"><font color="Maroon">G1</font></span><a href="group_7.html#K4" target="_self">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="card_3.html#T18">CARD_3:18</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/>
<a NAME="E5:76"/><b>then </b><i><font color="Green">E48</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">lpl</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:76"/><i><font color="Green">E50</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">lpp</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E7:76"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">lpp</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x2</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/>
<a NAME="E8:76"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 holds <br/><font color="Maroon">lpl</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">lpp</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:76_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/>

<a NAME="E2:76_2"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 
;<br/>
<a NAME="E3:76_2"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:76_2"/>
<font color="Maroon">lpl</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">lpp</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="group_7.html#D1" target="_self">Def1</a>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:76"/>
<span class="p1"><a href="group_7.html#K10" target="_self">&lt;*</a><span class="default"><font color="Maroon">x1</font></span><a href="group_7.html#K10" target="_self">*&gt;</a></span> <a href="group_1.html#K1">*</a> <span class="p1"><a href="group_7.html#K10" target="_self">&lt;*</a><span class="default"><font color="Maroon">x2</font></span><a href="group_7.html#K10" target="_self">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="group_7.html#K10" target="_self">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x2</font></span>)</span></span><a href="group_7.html#K10" target="_self">*&gt;</a></span>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>


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