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<b>let </b><font color="Maroon" title="c1">n</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E1:39"/><i><font color="Green" title="E29">A1</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ;<br/>

<b>set </b><font color="Maroon" title="c2">G</font> =  <a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <font color="Maroon" title="c1">n</font>;<br/>
<a NAME="E2:39"/><i><font color="Green" title="E30">A2</font></i>: 
 <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="gr_cy_1.html#K3" title="GR_CY_1:func.3">addint</a> <font color="Maroon" title="c1">n</font></span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="group_10.html#E1:39"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="gr_cy_1.html#D6" title="GR_CY_1:def.6">GR_CY_1:def 6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c3">k</font> = <font color="Maroon" title="c1">n</font> as    <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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E4:39"/><i><font color="Green" title="E31">A3</font></i>: 
 <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">k</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">G</font> =  <a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <font color="Maroon" title="c1">n</font> as  <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a> <b>by </b><i><a class="txt" href="group_10.html#E2:39"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="struct_0.html#D11" title="STRUCT_0:def.11">STRUCT_0:def 11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>now </b></a><div class="add"><b>reconsider </b><font color="Maroon" title="c5">B</font> =  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c1">n</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="group_10.html#E4:39"><i><font color="Green" title="E31">A3</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c6">B</font> = <font color="Maroon" title="c5">B</font>;<br/><b>thus </b><a NAME="E2:39_1"/>
<font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">G</font>
 <b>by </b><i><a class="txt" href="group_10.html#E2:39"><i><font color="Green" title="E30">A2</font></i></a></i>;<br/><a NAME="E3:39_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="ref" href="card_1.html#D12" title="CARD_1:def.12">CARD_1:def 12</a></i>;<br/><b>hence </b><a NAME="E4:39_1"/>
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">B</font>
 <b>by </b><i><a class="ref" href="card_1.html#T66" title="CARD_1:th.66">CARD_1:66</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:39"/>
 <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">INT.Group</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="group_1.html#D15" title="GROUP_1:def.15">GROUP_1:def 15</a></i>;<br/>


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