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<b>let </b><font color="Maroon" title="c1">A</font> be  <a href="freealg.html#V3" title="FREEALG:attr.3">free</a> <a href="unialg_1.html#NM2" title="UNIALG_1:NM.2">Universal_Algebra</a>;<br/><b>let </b><font color="Maroon" title="c2">o</font> be   <a href="pua2mss1.html#NM2" title="PUA2MSS1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">A</font>;<br/><b>let </b><font color="Maroon" title="c3">p</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>;<br/>





<b>assume </b><a NAME="E1:64"/><i><font color="Green" title="E38">A1</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c1">A</font></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E2:64"/><i><font color="Green" title="E39">A2</font></i>: 
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">p</font> &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">p</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">p</font> = <font color="Maroon" title="c3">p</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="txt" href="aofa_000.html#E1:64"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E4:64"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">p</font>
 
<b>by </b><i><a class="txt" href="aofa_000.html#E2:64"><i><font color="Green" title="E39">A2</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">a</font> = <font color="Maroon" title="c4">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font> ;<br/>
<b>set </b><font color="Maroon" title="c7">G</font> =  <a href="aofa_000.html#K11" title="AOFA_000:func.11">Generators</a> <font color="Maroon" title="c1">A</font>;<br/>
<b>consider </b><font color="Maroon" title="c8">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E7:64"/><i><font color="Green" title="E40">A3</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="aofa_000.html#K11" title="AOFA_000:func.11">Generators</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Maroon" title="c8">n</font>
 <b>by </b><i><a class="ref" href="aofa_000.html#T30" target="_self" title="AOFA_000:th.30">Th30</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">n</font> = <font color="Maroon" title="c8">n</font> as    <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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</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> ex <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font> ex <font color="Olive" title="b2">o</font> being  <a href="pua2mss1.html#NM2" title="PUA2MSS1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">A</font> ex <font color="Olive" title="b3">p</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">A</font> st <br/>( <font color="Olive" title="b3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Olive" title="b2">o</font>,<font color="Maroon" title="c1">A</font></span>)</span> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b3">p</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Olive" title="b2">o</font>,<font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">p</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="aofa_000.html#K11" title="AOFA_000:func.11">Generators</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> $1 );<br/>
<a NAME="E9:64"/>
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">p</font> &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="aofa_000.html#K11" title="AOFA_000:func.11">Generators</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Maroon" title="c9">n</font> )
 
<b>by </b><i><a class="txt" href="aofa_000.html#E2:64"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="aofa_000.html#E7:64"><i><font color="Green" title="E40">A3</font></i></a></i>;<br/>
<a NAME="E10:64"/><b>then </b><i><font color="Green" title="E41">A4</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="aofa_000.html#E1:64"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="aofa_000.html#E2:64"><i><font color="Green" title="E39">A2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E12:64"/><i><font color="Green" title="E42">A5</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">n</font>] &amp; ( for <font color="Olive" title="b1">m</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">m</font>] holds <br/><font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</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="aofa_000.html#E10:64"><i><font color="Green" title="E41">A4</font></i></a>);<br/></i>
<b>consider </b><font color="Maroon" title="c11">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c12">o</font> being   <a href="pua2mss1.html#NM2" title="PUA2MSS1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c13">p</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E14:64"/><i><font color="Green" title="E43">A6</font></i>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c12">o</font>,<font color="Maroon" title="c1">A</font></span>)</span> &amp; <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c13">p</font> &amp; <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c12">o</font>,<font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">p</font> &amp; <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="aofa_000.html#K11" title="AOFA_000:func.11">Generators</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Maroon" title="c10">n</font> )
 <b>by </b><i><a class="txt" href="aofa_000.html#E12:64"><i><font color="Green" title="E42">A5</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">op</font> =  <a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c12">o</font>,<font color="Maroon" title="c1">A</font> as    <a href="unialg_2.html#M2" title="UNIALG_2:mode.2">Element</a> of  <a href="unialg_2.html#K1" title="UNIALG_2:func.1">Operations</a> <font color="Maroon" title="c1">A</font> ;<br/>
<a NAME="E16:64"/>
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">op</font>
 
<b>by </b><i><a class="txt" href="aofa_000.html#E14:64"><i><font color="Green" title="E43">A6</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E17:64"/><b>then </b>
<font color="Maroon" title="c11">a</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a>  <a href="aofa_000.html#K11" title="AOFA_000:func.11">Generators</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="ref" href="aofa_000.html#T26" target="_self" title="AOFA_000:th.26">Th26</a></i>;<br/>
<a NAME="E18:64"/><b>then </b>
<font color="Maroon" title="c10">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="aofa_000.html#E14:64"><i><font color="Green" title="E43">A6</font></i></a>, <a class="ref" href="aofa_000.html#T18" target="_self" title="AOFA_000:th.18">Th18</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E20:64"/><i><font color="Green" title="E44">A7</font></i>: 
<font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">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="c16">k</font> = <font color="Maroon" title="c15">k</font> as    <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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E22:64"/><i><font color="Green" title="E45">A8</font></i>: 
<font color="Maroon" title="c16">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">n</font>
 
<b>by </b><i><a class="txt" href="aofa_000.html#E20:64"><i><font color="Green" title="E44">A7</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E23:64"/><b>then </b>
<font color="Maroon" title="c11">a</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <span class="p1">(<span class="default"><a href="aofa_000.html#K11" title="AOFA_000:func.11">Generators</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Maroon" title="c16">k</font>
 
<b>by </b><i><a class="txt" href="aofa_000.html#E12:64"><i><font color="Green" title="E42">A5</font></i></a>, <a class="txt" href="aofa_000.html#E14:64"><i><font color="Green" title="E43">A6</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c17">o'</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c18">p'</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> <b> such that </b><br/><a NAME="E25:64"/><i><font color="Green" title="E46">A9</font></i>: 
( <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c17">o'</font>,<font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">p'</font> &amp; <font color="Maroon" title="c18">p'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c17">o'</font>,<font color="Maroon" title="c1">A</font></span>)</span> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c18">p'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="aofa_000.html#K11" title="AOFA_000:func.11">Generators</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="aofa_000.html#K9" title="AOFA_000:func.9">|^</a> <font color="Maroon" title="c16">k</font> )
 <b>by </b><i><a class="txt" href="aofa_000.html#E14:64"><i><font color="Green" title="E43">A6</font></i></a>, <a class="txt" href="aofa_000.html#E20:64"><i><font color="Green" title="E44">A7</font></i></a>, <a class="ref" href="aofa_000.html#T20" target="_self" title="AOFA_000:th.20">Th20</a></i>;<br/>
<a NAME="E26:64"/>
<font color="Maroon" title="c18">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p</font>
 
<b>by </b><i><a class="txt" href="aofa_000.html#E14:64"><i><font color="Green" title="E43">A6</font></i></a>, <a class="txt" href="aofa_000.html#E25:64"><i><font color="Green" title="E46">A9</font></i></a>, <a class="ref" href="aofa_000.html#T36" target="_self" title="AOFA_000:th.36">Th36</a></i>;<br/>
<b>hence </b><a NAME="E27:64"/>
contradiction
 <b>by </b><i><a class="txt" href="aofa_000.html#E12:64"><i><font color="Green" title="E42">A5</font></i></a>, <a class="txt" href="aofa_000.html#E14:64"><i><font color="Green" title="E43">A6</font></i></a>, <a class="txt" href="aofa_000.html#E25:64"><i><font color="Green" title="E46">A9</font></i></a>, <a class="txt" href="aofa_000.html#E22:64"><i><font color="Green" title="E45">A8</font></i></a></i>;<br/>


</div>
