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<b>let </b><font color="Maroon" title="c1">G1</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">x1</font>, <font color="Maroon" title="c3">x2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G1</font>;<br/>





<b>set </b><font color="Maroon" title="c4">G</font> = <span class="p1"><a href="group_7.html#K4" title="GROUP_7:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font></span><a href="group_7.html#K4" title="GROUP_7:func.4">*&gt;</a></span>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">l</font> = <span class="p1"><a href="group_7.html#K10" title="GROUP_7:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">x1</font></span><a href="group_7.html#K10" title="GROUP_7:func.10">*&gt;</a></span>, <font color="Maroon" title="c6">p</font> = <span class="p1"><a href="group_7.html#K10" title="GROUP_7:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c3">x2</font></span><a href="group_7.html#K10" title="GROUP_7:func.10">*&gt;</a></span>, <font color="Maroon" title="c7">lpl</font> = <span class="p1"><a href="group_7.html#K10" title="GROUP_7:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">x1</font></span><a href="group_7.html#K10" title="GROUP_7:func.10">*&gt;</a></span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1"><a href="group_7.html#K10" title="GROUP_7:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c3">x2</font></span><a href="group_7.html#K10" title="GROUP_7:func.10">*&gt;</a></span>, <font color="Maroon" title="c8">lpp</font> = <span class="p1"><a href="group_7.html#K10" title="GROUP_7:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c3">x2</font></span>)</span></span><a href="group_7.html#K10" title="GROUP_7:func.10">*&gt;</a></span> as    <a href="pralg_2.html#M1" title="PRALG_2:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2"><a href="group_7.html#K4" title="GROUP_7:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font></span><a href="group_7.html#K4" title="GROUP_7:func.4">*&gt;</a></span></span>)</span> <b>by </b><i><a class="ref" href="group_7.html#D2" target="_self" title="GROUP_7:def.2">Def2</a></i>;<br/>
<a NAME="E2:73"/><i><font color="Green" title="E34">A1</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:73"/><i><font color="Green" title="E35">A2</font></i>: 
( <font color="Maroon" title="c5">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x1</font> &amp; <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x2</font> &amp; <span class="p1"><a href="group_7.html#K4" title="GROUP_7:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font></span><a href="group_7.html#K4" title="GROUP_7:func.4">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G1</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a></i>;<br/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">lpl</font> = 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2"><a href="group_7.html#K4" title="GROUP_7:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font></span><a href="group_7.html#K4" title="GROUP_7:func.4">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
;<br/>
<a NAME="E5:73"/><b>then </b><i><font color="Green" title="E36">A3</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">lpl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:73"/><i><font color="Green" title="E37">A4</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">lpp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E7:73"/><i><font color="Green" title="E38">A5</font></i>: 
<font color="Maroon" title="c8">lpp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c3">x2</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a></i>;<br/>
<a NAME="E8:73"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 holds <br/><font color="Maroon" title="c7">lpl</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">lpp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:73_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k</font> &amp; <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ;<br/>

<a NAME="E2:73_2"/><b>then </b>
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1
 
;<br/>
<a NAME="E3:73_2"/><b>then </b>
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:73_2"/>
<font color="Maroon" title="c7">lpl</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">lpp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
 <b>by </b><i><a class="txt" href="group_7.html#E2:73"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="group_7.html#E3:73"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="group_7.html#E7:73"><i><font color="Green" title="E38">A5</font></i></a>, <a class="ref" href="group_7.html#T4" target="_self" title="GROUP_7:th.4">Th4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:73"/>
<span class="p1"><a href="group_7.html#K10" title="GROUP_7:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">x1</font></span><a href="group_7.html#K10" title="GROUP_7:func.10">*&gt;</a></span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1"><a href="group_7.html#K10" title="GROUP_7:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c3">x2</font></span><a href="group_7.html#K10" title="GROUP_7:func.10">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_7.html#K10" title="GROUP_7:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c3">x2</font></span>)</span></span><a href="group_7.html#K10" title="GROUP_7:func.10">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="group_7.html#E5:73"><i><font color="Green" title="E36">A3</font></i></a>, <a class="txt" href="group_7.html#E6:73"><i><font color="Green" title="E37">A4</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/>


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