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<b>let </b><font color="Maroon" title="c1">n</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">k</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4 holds <br/>( <span class="p1">(<span class="default"><a href="idea_1.html#K11" title="IDEA_1:func.11">IDEAoperationA</a> <font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">k</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <span class="p1">(<span class="default"><a href="idea_1.html#K11" title="IDEA_1:func.11">IDEAoperationA</a> <font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">k</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <span class="p1">(<span class="default"><a href="idea_1.html#K11" title="IDEA_1:func.11">IDEAoperationA</a> <font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">k</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <span class="p1">(<span class="default"><a href="idea_1.html#K11" title="IDEA_1:func.11">IDEAoperationA</a> <font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">k</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> )</span><br/><b>let </b><font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c3">k</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4 implies ( <span class="p1">(<span class="default"><a href="idea_1.html#K11" title="IDEA_1:func.11">IDEAoperationA</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <span class="p1">(<span class="default"><a href="idea_1.html#K11" title="IDEA_1:func.11">IDEAoperationA</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <span class="p1">(<span class="default"><a href="idea_1.html#K11" title="IDEA_1:func.11">IDEAoperationA</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <span class="p1">(<span class="default"><a href="idea_1.html#K11" title="IDEA_1:func.11">IDEAoperationA</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> ) )</span><br/>





<b>assume </b><a NAME="E1:40"/><i><font color="Green" title="E32">A1</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="idea_1.html#K11" title="IDEA_1:func.11">IDEAoperationA</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <span class="p1">(<span class="default"><a href="idea_1.html#K11" title="IDEA_1:func.11">IDEAoperationA</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <span class="p1">(<span class="default"><a href="idea_1.html#K11" title="IDEA_1:func.11">IDEAoperationA</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <span class="p1">(<span class="default"><a href="idea_1.html#K11" title="IDEA_1:func.11">IDEAoperationA</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> )</span><br/>

<a NAME="E2:40"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">m</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E3:40"/><b>then </b>
1 <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" title="c2">m</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:40"/><b>then </b>
1 <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" title="c2">m</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/>
<a NAME="E5:40"/><b>then </b><i><font color="Green" title="E33">A2</font></i>: 
<span class="p1">(<span class="default"><a href="idea_1.html#K11" title="IDEA_1:func.11">IDEAoperationA</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span>,<font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="idea_1.html#D11" target="_self" title="IDEA_1:def.11">Def11</a></i>;<br/>
<a NAME="E6:40"/>
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">m</font>
 
<b>by </b><i><a class="txt" href="#E1:40"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:40"/><b>then </b>
2 <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" title="c2">m</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E8:40"/><b>then </b>
2 <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" title="c2">m</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/>
<a NAME="E9:40"/><b>then </b><i><font color="Green" title="E34">A3</font></i>: 
<span class="p1">(<span class="default"><a href="idea_1.html#K11" title="IDEA_1:func.11">IDEAoperationA</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K4" title="IDEA_1:func.4">ADD_MOD</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span>)</span>,<font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="idea_1.html#D11" target="_self" title="IDEA_1:def.11">Def11</a></i>;<br/>
<a NAME="E10:40"/>
3 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">m</font>
 
<b>by </b><i><a class="txt" href="#E1:40"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E11:40"/><b>then </b>
3 <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" title="c2">m</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E12:40"/><b>then </b>
3 <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" title="c2">m</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/>
<a NAME="E13:40"/><b>then </b><i><font color="Green" title="E35">A4</font></i>: 
<span class="p1">(<span class="default"><a href="idea_1.html#K11" title="IDEA_1:func.11">IDEAoperationA</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K4" title="IDEA_1:func.4">ADD_MOD</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span>,<font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="idea_1.html#D11" target="_self" title="IDEA_1:def.11">Def11</a></i>;<br/>
<a NAME="E14:40"/>
4 <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" title="c2">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:40"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E15:40"/><b>then </b>
4 <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" title="c2">m</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/>
<a NAME="E16:40"/><b>then </b>
<span class="p1">(<span class="default"><a href="idea_1.html#K11" title="IDEA_1:func.11">IDEAoperationA</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4</span>)</span>,<font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="idea_1.html#D11" target="_self" title="IDEA_1:def.11">Def11</a></i>;<br/>
<b>hence </b><a NAME="E17:40"/>
( <span class="p1">(<span class="default"><a href="idea_1.html#K11" title="IDEA_1:func.11">IDEAoperationA</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <span class="p1">(<span class="default"><a href="idea_1.html#K11" title="IDEA_1:func.11">IDEAoperationA</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <span class="p1">(<span class="default"><a href="idea_1.html#K11" title="IDEA_1:func.11">IDEAoperationA</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <span class="p1">(<span class="default"><a href="idea_1.html#K11" title="IDEA_1:func.11">IDEAoperationA</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="#E5:40"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="#E9:40"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="#E13:40"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="idea_1.html#T16" target="_self" title="IDEA_1:th.16">Th16</a>, <a class="ref" href="idea_1.html#T25" target="_self" title="IDEA_1:th.25">Th25</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
