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<b>let </b><font color="Maroon" title="c1">b</font>, <font color="Maroon" title="c2">a</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="E1:58"/><i><font color="Green" title="E27">A1</font></i>: 
<font color="Maroon" title="c1">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c3">b</font> = <font color="Maroon" title="c1">b</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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>reconsider </b><font color="Maroon" title="c4">a</font> = <font color="Maroon" title="c2">a</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:58_1"/>
<font color="Maroon" title="c3">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c4">a</font>
 ;<br/><a NAME="E2:58_1"/><b>then </b><i><font color="Green" title="E28">A2</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c4">a</font>
 <b>by </b><i><a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></i>;<br/><a NAME="E3:58_1"/>
<font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">a</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E4:58_1"/><b>then </b>
<font color="Maroon" title="c3">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c4">a</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="radix_1.html#T7" title="RADIX_1:th.7">RADIX_1:7</a></i>;<br/><a NAME="E5:58_1"/><b>then </b>
<font color="Maroon" title="c3">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></i>;<br/><a NAME="E6:58_1"/><b>then </b>
<font color="Maroon" title="c3">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="nat_3.html#E2:58_1"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E7:58_1"/><b>then </b>
<font color="Maroon" title="c4">a</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <b>by </b><i><a class="txt" href="nat_3.html#E1:58"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="pepin.html#T31" title="PEPIN:th.31">PEPIN:31</a></i>;<br/><b>hence </b><a NAME="E8:58_1"/>
contradiction
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:58"/>
<font color="Maroon" title="c1">b</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E1:58"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="nat_3.html#D7" target="_self" title="NAT_3:def.7">Def7</a></i>;<br/>


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