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<b>let </b><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>

<a NAME="E1:1"/><i><font color="Green" title="E2">A1</font></i>: 
(  <a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> )
 
<b>by </b><i><a class="ref" href="pre_circ.html#D4" title="PRE_CIRC:def.4">PRE_CIRC:def 4</a></i>;<br/>
<a NAME="E2:1"/>
(  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c1">A</font> or  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c2">B</font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></i>;<br/>
<a NAME="E3:1"/><b>then </b><i><font color="Green" title="E3">A2</font></i>: 
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">B</font>
 
<b>by </b><i><a class="txt" href="jordan_a.html#E1:1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E4:1"/>
for <font color="Olive" title="b1">k</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">B</font> holds <br/> <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">k</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E1:1_1"/><i><font color="Green" title="E4">A3</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">B</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1"/>
( <font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> or <font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> )
 </span><b>by </b><i><a class="txt" href="jordan_a.html#E1:1_1"><i><font color="Green" title="E4">A3</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_1"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 ;<br/><div class="add"><a NAME="E2:1_1_1_1"/><b>then </b><i><font color="Green" title="E5">A4</font></i>: 
 <a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="ref" href="pre_circ.html#D4" title="PRE_CIRC:def.4">PRE_CIRC:def 4</a></i>;<br/><a NAME="E3:1_1_1_1"/>
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></i>;<br/><b>hence </b><a NAME="E4:1_1_1_1"/>
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="jordan_a.html#E2:1_1_1_1"><i><font color="Green" title="E5">A4</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_2"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font>
 ;<br/><div class="add"><a NAME="E2:1_1_1_2"/><b>then </b><i><font color="Green" title="E5">A5</font></i>: 
 <a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c2">B</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="ref" href="pre_circ.html#D4" title="PRE_CIRC:def.4">PRE_CIRC:def 4</a></i>;<br/><a NAME="E3:1_1_1_2"/>
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></i>;<br/><b>hence </b><a NAME="E4:1_1_1_2"/>
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="jordan_a.html#E2:1_1_1_2"><i><font color="Green" title="E5">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E5:1"/>
 <a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <font color="Maroon" title="c2">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan_a.html#E3:1"><i><font color="Green" title="E3">A2</font></i></a>, <a class="ref" href="pre_circ.html#D4" title="PRE_CIRC:def.4">PRE_CIRC:def 4</a></i>;<br/>


</div>
