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<i><font color="Green" title="E5">A4</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon">F<sub>3</sub></font>() = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span>
<b>by </b><i><a class="txt" href="#E2:9"><i><font color="Green" title="E3">A2</font></i></a>, <a class="txt" href="#E3:9"><i><font color="Green" title="E4">A3</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon">F<sub>4</sub></font>()
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/>
<b>assume </b><a NAME="E2:9_1"/>
<font color="Maroon">F<sub>3</sub></font>() <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon">F<sub>4</sub></font>()
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><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="E4:9_1"/><i><font color="Green" title="E6">A5</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Maroon">F<sub>3</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>4</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="#E1:9_1"><i><font color="Green" title="E5">A4</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<a NAME="E5:9_1"/><i><font color="Green" title="E7">A6</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span>
 
<b>by </b><i><a class="txt" href="#E4:9_1"><i><font color="Green" title="E6">A5</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c2">x</font> = <font color="Maroon" title="c1">x</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="txt" href="#E4:9_1"><i><font color="Green" title="E6">A5</font></i></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>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 &amp; $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Maroon">F<sub>3</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>4</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 );<br/>
<a NAME="E7:9_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">F<sub>2</sub></font>() )
 
<b>by </b><i><a class="txt" href="#E2:9"><i><font color="Green" title="E3">A2</font></i></a>, <a class="txt" href="#E5:9_1"><i><font color="Green" title="E7">A6</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E8:9_1"/><b>then </b><i><font color="Green" title="E8">A7</font></i>: 
 ex <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>]
 
<b>by </b><i><a class="txt" href="#E4:9_1"><i><font color="Green" title="E6">A5</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c3">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E10:9_1"/><i><font color="Green" title="E9">A8</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font>] &amp; ( for <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>] holds <br/><font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</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="#E8:9_1"><i><font color="Green" title="E8">A7</font></i></a>);<br/></i>
<a NAME="E11:9_1"/>
<font color="Maroon" title="c3">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 
<b>by </b><i><a class="txt" href="#E2:9"><i><font color="Green" title="E3">A2</font></i></a>, <a class="txt" href="#E3:9"><i><font color="Green" title="E4">A3</font></i></a>, <a class="txt" href="#E10:9_1"><i><font color="Green" title="E9">A8</font></i></a></i>;<br/>
<a NAME="E12:9_1"/><b>then </b><i><font color="Green" title="E10">A9</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</font>
 
<b>by </b><i><a class="txt" href="#E10:9_1"><i><font color="Green" title="E9">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<a NAME="E13:9_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">n</font>
 
<b>by </b><i><a class="txt" href="#E10:9_1"><i><font color="Green" title="E9">A8</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E15:9_1"/><i><font color="Green" title="E11">A10</font></i>: 
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</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">k</font> = <font color="Maroon" title="c4">k</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/>
<a NAME="E17:9_1"/><i><font color="Green" title="E12">A11</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font>
 
<b>by </b><i><a class="txt" href="#E12:9_1"><i><font color="Green" title="E10">A9</font></i></a>, <a class="txt" href="#E15:9_1"><i><font color="Green" title="E11">A10</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E18:9_1"/>
<font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</font>
 
<b>by </b><i><a class="txt" href="#E15:9_1"><i><font color="Green" title="E11">A10</font></i></a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>
<a NAME="E19:9_1"/><b>then </b><i><font color="Green" title="E13">A12</font></i>: 
<font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon">F<sub>2</sub></font>()
 
<b>by </b><i><a class="txt" href="#E10:9_1"><i><font color="Green" title="E9">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E20:9_1"/>
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">k</font>
 
<b>by </b><i><a class="txt" href="#E15:9_1"><i><font color="Green" title="E11">A10</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E21:9_1"/><b>then </b>
<font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font>
 
<b>by </b><i><a class="txt" href="#E10:9_1"><i><font color="Green" title="E9">A8</font></i></a>, <a class="txt" href="#E17:9_1"><i><font color="Green" title="E12">A11</font></i></a>, <a class="txt" href="#E19:9_1"><i><font color="Green" title="E13">A12</font></i></a></i>;<br/>
<a NAME="E22:9_1"/><b>then </b>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>,<font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font>,<font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>,<font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font>,<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"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>] )
 
<b>by </b><i><a class="txt" href="#E2:9"><i><font color="Green" title="E3">A2</font></i></a>, <a class="txt" href="#E3:9"><i><font color="Green" title="E4">A3</font></i></a>, <a class="txt" href="#E17:9_1"><i><font color="Green" title="E12">A11</font></i></a>, <a class="txt" href="#E19:9_1"><i><font color="Green" title="E13">A12</font></i></a></i>;<br/>
<b>hence </b><a NAME="E23:9_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:9"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="#E10:9_1"><i><font color="Green" title="E9">A8</font></i></a>, <a class="txt" href="#E15:9_1"><i><font color="Green" title="E11">A10</font></i></a>, <a class="txt" href="#E17:9_1"><i><font color="Green" title="E12">A11</font></i></a>, <a class="txt" href="#E19:9_1"><i><font color="Green" title="E13">A12</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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