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<b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="setfam_1.html#V1">with_non-empty_elements</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  non <a href="struct_0.html#V3">empty</a> non <a href="ami_1.html#V2">void</a> <a href="ami_1.html#V5">IC-Ins-separated</a> <a href="ami_1.html#V8">definite</a> <a href="amistd_1.html#V4" target="_self">standard</a>  <a href="ami_1.html#L1">AMI-Struct</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="ami_3.html#V1">programmed</a> <a href="amistd_1.html#V9" target="_self">lower</a>  <a href="ami_1.html#M1">FinPartState</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>





<a NAME="E1:89"/><i><font color="Green">E48</font></i>: 
 <a href="amistd_1.html#K10" target="_self">LastLoc</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="amistd_1.html#T51" target="_self">Th51</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E3:89"/><i><font color="Green">E49</font></i>: 
 <a href="amistd_1.html#K10" target="_self">LastLoc</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="amistd_1.html#K5" target="_self">il.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="amistd_1.html#T26" target="_self">Th26</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E5:89"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="amistd_1.html#E1:89"><i><font color="Green">E48</font></i></a>, <a class="txt" href="amistd_1.html#E3:89"><i><font color="Green">E49</font></i></a>, <a class="ref" href="amistd_1.html#T50" target="_self">Th50</a></i>;<br/>
<a NAME="E6:89"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1
 
<b>by </b><i><a class="ref" href="jordan3.html#T12">JORDAN3:12</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:89_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1 or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1 )
 </span><b>by </b><i><a class="txt" href="amistd_1.html#E6:89"><i><font color="Green">E50</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_1"><b>suppose </b></a><a NAME="E1:89_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1
 ;<br/><div class="add"><a NAME="E2:89_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E3:89_1_1"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="card_2.html#T59">CARD_2:59</a></i>;<br/><a NAME="E4:89_1_1"/><b>then </b>
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><a NAME="E5:89_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="amistd_1.html#E2:89_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><a NAME="E6:89_1_1"/><b>then </b>
 <a href="amistd_1.html#K5" target="_self">il.</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="amistd_1.html#T50" target="_self">Th50</a></i>;<br/><a NAME="E7:89_1_1"/><b>then </b>
 <a href="amistd_1.html#K5" target="_self">il.</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="amistd_1.html#R1" target="_self">&lt;=</a>  <a href="amistd_1.html#K10" target="_self">LastLoc</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="amistd_1.html#T53" target="_self">Th53</a></i>;<br/><a NAME="E8:89_1_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="amistd_1.html#E3:89"><i><font color="Green">E49</font></i></a>, <a class="ref" href="amistd_1.html#T28" target="_self">Th28</a></i>;<br/><a NAME="E9:89_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E10:89_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="amistd_1.html#E8:89_1_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E11:89_1_1"/>
 <a href="amistd_1.html#K10" target="_self">LastLoc</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="amistd_1.html#K5" target="_self">il.</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2"><b>suppose </b></a><a NAME="E1:89_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:89_1_2"/>
 <a href="amistd_1.html#K10" target="_self">LastLoc</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="amistd_1.html#K5" target="_self">il.</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="amistd_1.html#E3:89"><i><font color="Green">E49</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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