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<div class="add">

<b>let </b><font color="Maroon" title="c1">G1</font>, <font color="Maroon" title="c2">G2</font>, <font color="Maroon" title="c3">G3</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a> ;<br/><b>let </b><font color="Maroon" title="c4">x1</font>, <font color="Maroon" title="c5">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>let </b><font color="Maroon" title="c6">y1</font>, <font color="Maroon" title="c7">y2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G2</font>;<br/><b>let </b><font color="Maroon" title="c8">z1</font>, <font color="Maroon" title="c9">z2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">G3</font>;<br/>

















<b>set </b><font color="Maroon" title="c10">G</font> = <span class="p1"><a href="group_7.html#K18" title="GROUP_7:func.18">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c3">G3</font></span><a href="group_7.html#K18" title="GROUP_7:func.18">*&gt;</a></span>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">l</font> = <span class="p1"><a href="group_7.html#K24" title="GROUP_7:func.24">&lt;*</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">y1</font>,<font color="Maroon" title="c8">z1</font></span><a href="group_7.html#K24" title="GROUP_7:func.24">*&gt;</a></span>, <font color="Maroon" title="c12">p</font> = <span class="p1"><a href="group_7.html#K24" title="GROUP_7:func.24">&lt;*</a><span class="default"><font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c7">y2</font>,<font color="Maroon" title="c9">z2</font></span><a href="group_7.html#K24" title="GROUP_7:func.24">*&gt;</a></span>, <font color="Maroon" title="c13">lpl</font> = <span class="p1"><a href="group_7.html#K24" title="GROUP_7:func.24">&lt;*</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">y1</font>,<font color="Maroon" title="c8">z1</font></span><a href="group_7.html#K24" title="GROUP_7:func.24">*&gt;</a></span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1"><a href="group_7.html#K24" title="GROUP_7:func.24">&lt;*</a><span class="default"><font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c7">y2</font>,<font color="Maroon" title="c9">z2</font></span><a href="group_7.html#K24" title="GROUP_7:func.24">*&gt;</a></span>, <font color="Maroon" title="c14">lpp</font> = <span class="p1"><a href="group_7.html#K24" title="GROUP_7:func.24">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c5">x2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">y1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c7">y2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">z1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c9">z2</font></span>)</span></span><a href="group_7.html#K24" title="GROUP_7:func.24">*&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#K18" title="GROUP_7:func.18">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c3">G3</font></span><a href="group_7.html#K18" title="GROUP_7:func.18">*&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="E2:78"/><i><font color="Green" title="E35">A1</font></i>: 
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default">1,2,3</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> &amp; 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default">1,2,3</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> &amp; 3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default">1,2,3</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> )
 
<b>by </b><i><a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/>
<a NAME="E3:78"/><i><font color="Green" title="E36">A2</font></i>: 
( <font color="Maroon" title="c11">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="c4">x1</font> &amp; <font color="Maroon" title="c11">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y1</font> &amp; <font color="Maroon" title="c11">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">z1</font> &amp; <font color="Maroon" title="c12">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="c5">x2</font> &amp; <font color="Maroon" title="c12">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y2</font> &amp; <font color="Maroon" title="c12">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">z2</font> &amp; <span class="p1"><a href="group_7.html#K18" title="GROUP_7:func.18">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c3">G3</font></span><a href="group_7.html#K18" title="GROUP_7:func.18">*&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> &amp; <span class="p1"><a href="group_7.html#K18" title="GROUP_7:func.18">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c3">G3</font></span><a href="group_7.html#K18" title="GROUP_7:func.18">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> &amp; <span class="p1"><a href="group_7.html#K18" title="GROUP_7:func.18">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c3">G3</font></span><a href="group_7.html#K18" title="GROUP_7:func.18">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">G3</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></i>;<br/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">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#K18" title="GROUP_7:func.18">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c3">G3</font></span><a href="group_7.html#K18" title="GROUP_7:func.18">*&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> 3
<b>by </b><i><a class="ref" href="finseq_3.html#T1" title="FINSEQ_3:th.1">FINSEQ_3:1</a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
;<br/>
<a NAME="E5:78"/><b>then </b><i><font color="Green" title="E37">A3</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">lpl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 
<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="E6:78"/><i><font color="Green" title="E38">A4</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">lpp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 
<b>by </b><i><a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></i>;<br/>
<a NAME="E7:78"/><i><font color="Green" title="E39">A5</font></i>: 
( <font color="Maroon" title="c14">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="c4">x1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c5">x2</font> &amp; <font color="Maroon" title="c14">lpp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c7">y2</font> &amp; <font color="Maroon" title="c14">lpp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">z1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c9">z2</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></i>;<br/>
<a NAME="E8:78"/>
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> 3 holds <br/><font color="Maroon" title="c13">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="c14">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="78_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">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="E1:78_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">k</font> &amp; <font color="Maroon" title="c15">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3 )
 ;<br/>

<a NAME="E2:78_2"/><b>then </b><i><font color="Green" title="E40">A6</font></i>: 
<font color="Maroon" title="c15">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> 3
 
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:78_2_1"/>
( <font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 </span><b>by </b><i><a class="txt" href="group_7.html#E2:78_2"><i><font color="Green" title="E40">A6</font></i></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a>, <a class="ref" href="finseq_3.html#T1" title="FINSEQ_3:th.1">FINSEQ_3:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_2_1_1"><b>suppose </b></a><a NAME="E1:78_2_1_1"/>
<font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:78_2_1_1"/>
<font color="Maroon" title="c13">lpl</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">lpp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">k</font>
 <b>by </b><i><a class="txt" href="group_7.html#E2:78"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="group_7.html#E3:78"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="group_7.html#E7:78"><i><font color="Green" title="E39">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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_2_1_2"><b>suppose </b></a><a NAME="E1:78_2_1_2"/>
<font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:78_2_1_2"/>
<font color="Maroon" title="c13">lpl</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">lpp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">k</font>
 <b>by </b><i><a class="txt" href="group_7.html#E2:78"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="group_7.html#E3:78"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="group_7.html#E7:78"><i><font color="Green" title="E39">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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_2_1_3"><b>suppose </b></a><a NAME="E1:78_2_1_3"/>
<font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ;<br/><div class="add"><b>hence </b><a NAME="E2:78_2_1_3"/>
<font color="Maroon" title="c13">lpl</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">lpp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">k</font>
 <b>by </b><i><a class="txt" href="group_7.html#E2:78"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="group_7.html#E3:78"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="group_7.html#E7:78"><i><font color="Green" title="E39">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></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E9:78"/>
<span class="p1"><a href="group_7.html#K24" title="GROUP_7:func.24">&lt;*</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">y1</font>,<font color="Maroon" title="c8">z1</font></span><a href="group_7.html#K24" title="GROUP_7:func.24">*&gt;</a></span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1"><a href="group_7.html#K24" title="GROUP_7:func.24">&lt;*</a><span class="default"><font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c7">y2</font>,<font color="Maroon" title="c9">z2</font></span><a href="group_7.html#K24" title="GROUP_7:func.24">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_7.html#K24" title="GROUP_7:func.24">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c5">x2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">y1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c7">y2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">z1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c9">z2</font></span>)</span></span><a href="group_7.html#K24" title="GROUP_7:func.24">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="group_7.html#E5:78"><i><font color="Green" title="E37">A3</font></i></a>, <a class="txt" href="group_7.html#E6:78"><i><font color="Green" title="E38">A4</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/>


</div>
