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<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:19"/><i><font color="Green" title="E16">A1</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">m</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">k1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:19"/><i><font color="Green" title="E17">A2</font></i>: 
<font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c3">k1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/>
<a NAME="E4:19"/>
<font color="Maroon" title="c3">k1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="clopban3.html#E1:19"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="clopban3.html#E3:19"><i><font color="Green" title="E17">A2</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">n1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E6:19"/><i><font color="Green" title="E18">A3</font></i>: 
<font color="Maroon" title="c3">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">n1</font> = <font color="Maroon" title="c4">n1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">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/>
<b>take </b><font color="Maroon" title="c6">k</font> = <font color="Maroon" title="c5">n1</font>;<br/>

<b>thus </b><a NAME="E8:19"/>
<font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">k</font>
 <b>by </b><i><a class="txt" href="clopban3.html#E3:19"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="clopban3.html#E6:19"><i><font color="Green" title="E18">A3</font></i></a></i>;<br/>


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