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<b>assume </b><a NAME="E1:26_1"/>
<font color="Maroon">F<sub>5</sub></font>() <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon">F<sub>6</sub></font>()
 ;<br/>

<b>then consider </b><font color="Maroon" title="c1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:26_1"/><i><font color="Green" title="E22">A7</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon">F<sub>5</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon">F<sub>6</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x</font> )
 <b>by </b><i><a class="txt" href="recdef_2.html#E1:26"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="recdef_2.html#E4:26"><i><font color="Green" title="E19">A4</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b><font color="Maroon">F<sub>5</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon">F<sub>6</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/>
<a NAME="E4:26_1"/>
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>by </b><i><a class="txt" href="recdef_2.html#E3:26_1"><i><font color="Green" title="E22">A7</font></i></a></i>;<br/>
<a NAME="E5:26_1"/><b>then </b><i><font color="Green" title="E23">A8</font></i>: 
 ex <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>by </b><i><a class="txt" href="recdef_2.html#E3:26_1"><i><font color="Green" title="E22">A7</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c2">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E7:26_1"/><i><font color="Green" title="E24">A9</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">m</font>]
 <b>and </b><br/><a NAME="E8:26_1"/><i><font color="Green" title="E25">A10</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font>
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="recdef_2.html#E5:26_1"><i><font color="Green" title="E23">A8</font></i></a>);<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1"><b>now </b></a><div class="add"><b>set </b><font color="Maroon" title="c3">k</font> = <font color="Maroon" title="c2">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 4;<br/><b>assume </b><a NAME="E1:26_1_1"/>
( <font color="Maroon" title="c2">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Maroon" title="c2">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 2 &amp; <font color="Maroon" title="c2">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 3 )
 ;<br/><a NAME="E2:26_1_1"/><b>then </b>
4 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T29" title="NAT_1:th.29">NAT_1:29</a></i>;<br/><a NAME="E3:26_1_1"/><b>then </b><i><font color="Green" title="E26">A11</font></i>: 
<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="c2">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 4</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4
 <b>by </b><i><a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/><a NAME="E4:26_1_1"/><b>then </b><i><font color="Green" title="E27">A12</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 4</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">m</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E5:26_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 4</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">m</font>
 <b>by </b><i><a class="txt" href="recdef_2.html#E3:26_1_1"><i><font color="Green" title="E26">A11</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E6:26_1_1"/><b>then </b><i><font color="Green" title="E28">A13</font></i>: 
<font color="Maroon">F<sub>5</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 4</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 4</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="recdef_2.html#E8:26_1"><i><font color="Green" title="E25">A10</font></i></a></i>;<br/><a NAME="E7:26_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 4</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">m</font>
 <b>by </b><i><a class="txt" href="recdef_2.html#E3:26_1_1"><i><font color="Green" title="E26">A11</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E8:26_1_1"/><b>then </b><i><font color="Green" title="E29">A14</font></i>: 
<font color="Maroon">F<sub>5</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 4</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 4</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="recdef_2.html#E8:26_1"><i><font color="Green" title="E25">A10</font></i></a></i>;<br/><a NAME="E9:26_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 4</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">m</font>
 <b>by </b><i><a class="txt" href="recdef_2.html#E3:26_1_1"><i><font color="Green" title="E26">A11</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E10:26_1_1"/><b>then </b><i><font color="Green" title="E30">A15</font></i>: 
<font color="Maroon">F<sub>5</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 4</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 4</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3</span>)</span>
 <b>by </b><i><a class="txt" href="recdef_2.html#E8:26_1"><i><font color="Green" title="E25">A10</font></i></a></i>;<br/><a NAME="E11:26_1_1"/>
( <font color="Maroon">F<sub>5</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 4</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 4</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>5</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 4</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>5</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 4</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>5</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 4</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>5</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 4</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3</span>)</span></span>)</span>) &amp; <font color="Maroon">F<sub>6</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 4</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 4</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 4</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 4</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 4</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 4</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3</span>)</span></span>)</span>) )
 <b>by </b><i><a class="txt" href="recdef_2.html#E3:26"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="recdef_2.html#E6:26"><i><font color="Green" title="E21">A6</font></i></a></i>;<br/><b>hence </b><a NAME="E12:26_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="recdef_2.html#E7:26_1"><i><font color="Green" title="E24">A9</font></i></a>, <a class="txt" href="recdef_2.html#E8:26_1"><i><font color="Green" title="E25">A10</font></i></a>, <a class="txt" href="recdef_2.html#E3:26_1_1"><i><font color="Green" title="E26">A11</font></i></a>, <a class="txt" href="recdef_2.html#E4:26_1_1"><i><font color="Green" title="E27">A12</font></i></a>, <a class="txt" href="recdef_2.html#E6:26_1_1"><i><font color="Green" title="E28">A13</font></i></a>, <a class="txt" href="recdef_2.html#E8:26_1_1"><i><font color="Green" title="E29">A14</font></i></a>, <a class="txt" href="recdef_2.html#E10:26_1_1"><i><font color="Green" title="E30">A15</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:26_1"/>
contradiction
 <b>by </b><i><a class="txt" href="recdef_2.html#E2:26"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="recdef_2.html#E5:26"><i><font color="Green" title="E20">A5</font></i></a>, <a class="txt" href="recdef_2.html#E7:26_1"><i><font color="Green" title="E24">A9</font></i></a></i>;<br/>


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