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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span><font color="Maroon">F<sub>4</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>5</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/>
<span class="kw">assume </span><a NAME="E1:22_1"/>
<font color="Maroon">F<sub>4</sub></font>() <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon">F<sub>5</sub></font>()
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:22_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">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>4</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon">F<sub>5</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E4:22"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="funct_1.html#T2" title="FUNCT_1:th.2">FUNCT_1:2</a></span>;<br/>
<a NAME="E3:22_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A7</font></span>: 
 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>]
 
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c1">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E5:22_1"/><span class="lab"><font color="Green" title="E24">A8</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">m</font>]
 <span class="kw">and </span><br/><a NAME="E6:22_1"/><span class="lab"><font color="Green" title="E25">A9</font></span>: 
 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="c1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font>
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="#E3:22_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>);<br/></span>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c2">k</font> = <font color="Maroon" title="c1">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 3;<br/><a NAME="E1:22_1_1"/><span class="lab"><font color="Green" title="E26">A10</font></span>: 
( <font color="Maroon">F<sub>4</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="c1">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 3</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>(<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 3</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 3</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>4</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="c1">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 3</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>4</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="c1">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 3</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>) &amp; <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="c1">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 3</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>(<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 3</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="c1">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 3</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="c1">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 3</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="c1">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 3</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="txt" href="#E6:22"><span class="lab"><font color="Green" title="E22">A6</font></span></a></span>;<br/><span class="kw">assume </span><a NAME="E2:22_1_1"/>
( <font color="Maroon" title="c1">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="c1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Maroon" title="c1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 2 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E3:22_1_1"/><span class="kw">then </span>
3 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T27" title="NAT_1:th.27">NAT_1:27</a></span>;<br/><a NAME="E4:22_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A11</font></span>: 
<font color="Maroon" title="c1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 3</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/><a NAME="E5:22_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 3</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="c1">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E6:22_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A12</font></span>: 
<font color="Maroon">F<sub>4</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="c1">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 3</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>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="c1">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 3</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:22_1"><span class="lab"><font color="Green" title="E25">A9</font></span></a></span>;<br/><a NAME="E7:22_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 3</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="c1">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_1_1"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E8:22_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A13</font></span>: 
<font color="Maroon">F<sub>4</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="c1">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 3</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>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="c1">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 3</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:22_1"><span class="lab"><font color="Green" title="E25">A9</font></span></a></span>;<br/><a NAME="E9:22_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 3</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="c1">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_1_1"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><span class="kw">hence </span><a NAME="E10:22_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22_1"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="txt" href="#E6:22_1"><span class="lab"><font color="Green" title="E25">A9</font></span></a>, <a class="txt" href="#E4:22_1_1"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="txt" href="#E6:22_1_1"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="txt" href="#E8:22_1_1"><span class="lab"><font color="Green" title="E29">A13</font></span></a>, <a class="txt" href="#E1:22_1_1"><span class="lab"><font color="Green" title="E26">A10</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E8:22_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E5:22"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="txt" href="#E5:22_1"><span class="lab"><font color="Green" title="E24">A8</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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