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<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c3">k</font>, <font color="Maroon" title="c4">t</font>, <font color="Maroon" title="c5">k1</font>, <font color="Maroon" title="c6">t1</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>











<b>assume </b><a NAME="E1:31"/><i><font color="Green" title="E12">A1</font></i>: 
( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">t</font> &amp; <font color="Maroon" title="c4">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">m</font> &amp; <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">k1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">t1</font> &amp; <font color="Maroon" title="c6">t1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">m</font> )
 ;<br/>

<div><i><font color="Green" title="E13">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:31_1"/>
<font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k1</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c7">u</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:31_1"/><i><font color="Green" title="E14">A3</font></i>: 
<font color="Maroon" title="c5">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">u</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T10" target="_self" title="NAT_1:th.10">Th28</a></i>;<br/><a NAME="E4:31_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">u</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">u</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">t1</font></span>)</span>
 ;<br/><a NAME="E5:31_1"/><b>then </b><i><font color="Green" title="E15">A4</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">u</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t</font>
 <b>by </b><i><a class="txt" href="nat_1.html#E1:31"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="nat_1.html#E3:31_1"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="xcmplx_1.html#T2" title="XCMPLX_1:th.2">XCMPLX_1:2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c8">w</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><a NAME="E1:31_1_1"/><i><font color="Green" title="E16">A5</font></i>: 
<font color="Maroon" title="c7">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">w</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 ;<br/><a NAME="E2:31_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">u</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">w</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">t1</font></span>)</span>
 <b>by </b><i><a class="txt" href="nat_1.html#E1:31_1_1"><i><font color="Green" title="E16">A5</font></i></a></i>;<br/><b>hence </b><a NAME="E3:31_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_1.html#E1:31"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="nat_1.html#E5:31_1"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="nat_1.html#T11" target="_self" title="NAT_1:th.11">Th29</a></i>;<br/></div><b>end;</b></div><a NAME="E7:31_1"/><b>then </b><i><font color="Green" title="E16">A6</font></i>: 
<font color="Maroon" title="c7">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T6" target="_self" title="NAT_1:th.6">Th22</a></i>;<br/><b>hence </b><a NAME="E8:31_1"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k1</font>
 <b>by </b><i><a class="txt" href="nat_1.html#E3:31_1"><i><font color="Green" title="E14">A3</font></i></a></i>;<br/><b>thus </b><a NAME="E9:31_1"/>
<font color="Maroon" title="c4">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">t1</font>
 <b>by </b><i><a class="txt" href="nat_1.html#E1:31"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="nat_1.html#E3:31_1"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="nat_1.html#E7:31_1"><i><font color="Green" title="E16">A6</font></i></a>, <a class="ref" href="xcmplx_1.html#T2" title="XCMPLX_1:th.2">XCMPLX_1:2</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:31_2"/>
<font color="Maroon" title="c5">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c7">u</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:31_2"/><i><font color="Green" title="E14">A7</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">u</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T10" target="_self" title="NAT_1:th.10">Th28</a></i>;<br/><a NAME="E4:31_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">u</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">k1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">u</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">t</font></span>)</span>
 ;<br/><a NAME="E5:31_2"/><b>then </b><i><font color="Green" title="E15">A8</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">u</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">t1</font>
 <b>by </b><i><a class="txt" href="nat_1.html#E1:31"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="nat_1.html#E3:31_2"><i><font color="Green" title="E14">A7</font></i></a>, <a class="ref" href="xcmplx_1.html#T2" title="XCMPLX_1:th.2">XCMPLX_1:2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c8">w</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><a NAME="E1:31_2_1"/><i><font color="Green" title="E16">A9</font></i>: 
<font color="Maroon" title="c7">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">w</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 ;<br/><a NAME="E2:31_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">u</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">w</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">t</font></span>)</span>
 <b>by </b><i><a class="txt" href="nat_1.html#E1:31_2_1"><i><font color="Green" title="E16">A9</font></i></a></i>;<br/><b>hence </b><a NAME="E3:31_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_1.html#E1:31"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="nat_1.html#E5:31_2"><i><font color="Green" title="E15">A8</font></i></a>, <a class="ref" href="nat_1.html#T11" target="_self" title="NAT_1:th.11">Th29</a></i>;<br/></div><b>end;</b></div><a NAME="E7:31_2"/><b>then </b><i><font color="Green" title="E16">A10</font></i>: 
<font color="Maroon" title="c7">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T6" target="_self" title="NAT_1:th.6">Th22</a></i>;<br/><b>hence </b><a NAME="E8:31_2"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k1</font>
 <b>by </b><i><a class="txt" href="nat_1.html#E3:31_2"><i><font color="Green" title="E14">A7</font></i></a></i>;<br/><b>thus </b><a NAME="E9:31_2"/>
<font color="Maroon" title="c4">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">t1</font>
 <b>by </b><i><a class="txt" href="nat_1.html#E1:31"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="nat_1.html#E3:31_2"><i><font color="Green" title="E14">A7</font></i></a>, <a class="txt" href="nat_1.html#E7:31_2"><i><font color="Green" title="E16">A10</font></i></a>, <a class="ref" href="xcmplx_1.html#T2" title="XCMPLX_1:th.2">XCMPLX_1:2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:31"/>
( <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k1</font> &amp; <font color="Maroon" title="c4">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">t1</font> )
 <b>by </b><i><a class="txt" href="nat_1.html#E2:31"><i><font color="Green" title="E13">A2</font></i></a></i>;<br/>


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