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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">k1</font>, <font color="Olive" title="b3">k2</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <span class="p1">(<span class="default">2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  &amp;  <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 &amp; <font color="Olive" title="b1">m</font> <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; <font color="Olive" title="b1">m</font> <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; <font color="Olive" title="b1">m</font> <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; <font color="Olive" title="b1">m</font> <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> &amp; <font color="Olive" title="b2">k1</font> <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; <font color="Olive" title="b2">k1</font> <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; <font color="Olive" title="b2">k1</font> <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; <font color="Olive" title="b2">k1</font> <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> &amp; <font color="Olive" title="b3">k2</font> <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#K10" title="IDEA_1:func.10">INV_MOD</a> (<span class="p1">(<span class="default"><font color="Olive" title="b2">k1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Olive" title="b3">k2</font> <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#K6" title="IDEA_1:func.6">NEG_MOD</a> (<span class="p1">(<span class="default"><font color="Olive" title="b2">k1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Olive" title="b3">k2</font> <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#K6" title="IDEA_1:func.6">NEG_MOD</a> (<span class="p1">(<span class="default"><font color="Olive" title="b2">k1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Olive" title="b3">k2</font> <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#K10" title="IDEA_1:func.10">INV_MOD</a> (<span class="p1">(<span class="default"><font color="Olive" title="b2">k1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4</span>)</span>,<font color="Maroon" title="c1">n</font>) holds <br/> <a href="idea_1.html#K11" title="IDEA_1:func.11">IDEAoperationA</a> (<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">k1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Olive" title="b3">k2</font>,<font color="Maroon" title="c1">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c3">k1</font>, <font color="Maroon" title="c4">k2</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default">2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  &amp;  <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 &amp; <font color="Maroon" title="c2">m</font> <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; <font color="Maroon" title="c2">m</font> <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; <font color="Maroon" title="c2">m</font> <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; <font color="Maroon" title="c2">m</font> <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> &amp; <font color="Maroon" title="c3">k1</font> <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; <font color="Maroon" title="c3">k1</font> <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; <font color="Maroon" title="c3">k1</font> <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; <font color="Maroon" title="c3">k1</font> <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> &amp; <font color="Maroon" title="c4">k2</font> <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#K10" title="IDEA_1:func.10">INV_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">k1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c4">k2</font> <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#K6" title="IDEA_1:func.6">NEG_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">k1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c4">k2</font> <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#K6" title="IDEA_1:func.6">NEG_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">k1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span>,<font color="Maroon" title="c1">n</font>) &amp; <font color="Maroon" title="c4">k2</font> <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#K10" title="IDEA_1:func.10">INV_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">k1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4</span>)</span>,<font color="Maroon" title="c1">n</font>) implies  <a href="idea_1.html#K11" title="IDEA_1:func.11">IDEAoperationA</a> (<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">k1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">k2</font>,<font color="Maroon" title="c1">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:42"/><span class="lab"><font color="Green" title="E36">A1</font></span>: 
<span class="p1">(<span class="default">2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> 
 <span class="kw">and </span><br/><a NAME="E2:42"/><span class="lab"><font color="Green" title="E37">A2</font></span>: 
 <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
 <span class="kw">and </span><br/><a NAME="E3:42"/><span class="lab"><font color="Green" title="E38">A3</font></span>: 
<font color="Maroon" title="c2">m</font> <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>
 <span class="kw">and </span><br/><a NAME="E4:42"/><span class="lab"><font color="Green" title="E39">A4</font></span>: 
<font color="Maroon" title="c2">m</font> <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>
 <span class="kw">and </span><br/><a NAME="E5:42"/><span class="lab"><font color="Green" title="E40">A5</font></span>: 
<font color="Maroon" title="c2">m</font> <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>
 <span class="kw">and </span><br/><a NAME="E6:42"/><span class="lab"><font color="Green" title="E41">A6</font></span>: 
<font color="Maroon" title="c2">m</font> <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 class="kw">and </span><br/><a NAME="E7:42"/><span class="lab"><font color="Green" title="E42">A7</font></span>: 
<font color="Maroon" title="c3">k1</font> <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>
 <span class="kw">and </span><br/><a NAME="E8:42"/><span class="lab"><font color="Green" title="E43">A8</font></span>: 
<font color="Maroon" title="c3">k1</font> <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>
 <span class="kw">and </span><br/><a NAME="E9:42"/><span class="lab"><font color="Green" title="E44">A9</font></span>: 
<font color="Maroon" title="c3">k1</font> <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>
 <span class="kw">and </span><br/><a NAME="E10:42"/><span class="lab"><font color="Green" title="E45">A10</font></span>: 
<font color="Maroon" title="c3">k1</font> <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 class="kw">and </span><br/><a NAME="E11:42"/><span class="lab"><font color="Green" title="E46">A11</font></span>: 
<font color="Maroon" title="c4">k2</font> <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#K10" title="IDEA_1:func.10">INV_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">k1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span>,<font color="Maroon" title="c1">n</font>)
 <span class="kw">and </span><br/><a NAME="E12:42"/><span class="lab"><font color="Green" title="E47">A12</font></span>: 
<font color="Maroon" title="c4">k2</font> <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#K6" title="IDEA_1:func.6">NEG_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">k1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span>)</span>,<font color="Maroon" title="c1">n</font>)
 <span class="kw">and </span><br/><a NAME="E13:42"/><span class="lab"><font color="Green" title="E48">A13</font></span>: 
<font color="Maroon" title="c4">k2</font> <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#K6" title="IDEA_1:func.6">NEG_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">k1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span>,<font color="Maroon" title="c1">n</font>)
 <span class="kw">and </span><br/><a NAME="E14:42"/><span class="lab"><font color="Green" title="E49">A14</font></span>: 
<font color="Maroon" title="c4">k2</font> <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#K10" title="IDEA_1:func.10">INV_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">k1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4</span>)</span>,<font color="Maroon" title="c1">n</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="idea_1.html#K11" title="IDEA_1:func.11">IDEAoperationA</a> (<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">k1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">k2</font>,<font color="Maroon" title="c1">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font></span><br/>

<a NAME="E15:42"/><span class="lab"><font color="Green" title="E50">A15</font></span>: 
<font color="Maroon" title="c4">k2</font> <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E10:42"><span class="lab"><font color="Green" title="E45">A10</font></span></a>, <a class="txt" href="#E14:42"><span class="lab"><font color="Green" title="E49">A14</font></span></a>, <a class="ref" href="idea_1.html#D10" target="_self" onmouseover="rs('idea_1/D10')" onmouseout="rh()">Def10</a></span>;<br/>
<a NAME="E16:42"/>
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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E17:42"/><span class="kw">then </span>
2 <a href="tarski.html#R2" title="TARSKI: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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/>
<a NAME="E18:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A16</font></span>: 
2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E19:42"/>
 <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E20:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A17</font></span>: 
4 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/>
<a NAME="E21:42"/>
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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E22:42"/><span class="kw">then </span>
1 <a href="tarski.html#R2" title="TARSKI: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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/>
<a NAME="E23:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A18</font></span>: 
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E24:42"/>
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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E25:42"/><span class="kw">then </span>
3 <a href="tarski.html#R2" title="TARSKI: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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/>
<a NAME="E26:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A19</font></span>: 
3 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">I1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E28:42"/><span class="lab"><font color="Green" title="E55">A20</font></span>: 
<font color="Maroon" title="c5">I1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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">k1</font>,<font color="Maroon" title="c1">n</font>)
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">I2</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E30:42"/><span class="lab"><font color="Green" title="E56">A21</font></span>: 
<font color="Maroon" title="c6">I2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K11" title="IDEA_1:func.11">IDEAoperationA</a> (<font color="Maroon" title="c5">I1</font>,<font color="Maroon" title="c4">k2</font>,<font color="Maroon" title="c1">n</font>)
 ;<br/>
<a NAME="E31:42"/><span class="lab"><font color="Green" title="E57">A22</font></span>: 
<font color="Maroon" title="c4">k2</font> <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E7:42"><span class="lab"><font color="Green" title="E42">A7</font></span></a>, <a class="txt" href="#E11:42"><span class="lab"><font color="Green" title="E46">A11</font></span></a>, <a class="ref" href="idea_1.html#D10" target="_self" onmouseover="rs('idea_1/D10')" onmouseout="rh()">Def10</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E58">A23</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="tarski.html#R2" title="TARSKI: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> holds <br/><font color="Maroon" title="c6">I2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">j</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">j</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">j</font> <a href="tarski.html#R2" title="TARSKI: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> implies <font color="Maroon" title="c6">I2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">j</font> )</span><br/><span class="kw">assume </span><a NAME="E1:42_1"/><span class="lab"><font color="Green" title="E59">A24</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="tarski.html#R2" title="TARSKI: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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">I2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">j</font></span><br/><a NAME="E2:42_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">j</font> <a href="tarski.html#R2" title="TARSKI: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="c5">I1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:42"><span class="lab"><font color="Green" title="E55">A20</font></span></a>, <a class="ref" href="idea_1.html#D11" target="_self" onmouseover="rs('idea_1/D11')" onmouseout="rh()">Def11</a></span>;<br/><a NAME="E3:42_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A25</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">I1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><a NAME="E4:42_1"/><span class="lab"><font color="Green" title="E61">A26</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_1"><span class="lab"><font color="Green" title="E59">A24</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">I2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">j</font></span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:42_1_1_1"/>
( <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 or ( <font color="Maroon" title="c7">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Maroon" title="c7">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 2 &amp; <font color="Maroon" title="c7">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 3 &amp; <font color="Maroon" title="c7">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 4 ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:42_1_1_1_1"/><span class="lab"><font color="Green" title="E62">A27</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">I2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">j</font></span><br/><div class="add"><span class="kw">hence </span><font color="Maroon" title="c6">I2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">j</font> = 
 <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">I1</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="c4">k2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:42"><span class="lab"><font color="Green" title="E56">A21</font></span></a>, <a class="txt" href="#E3:42_1"><span class="lab"><font color="Green" title="E60">A25</font></span></a>, <a class="ref" href="idea_1.html#D11" target="_self" onmouseover="rs('idea_1/D11')" onmouseout="rh()">Def11</a></span>
<br/>.= 
 <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<span class="p1">(<span class="default"><a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<span class="p2">(<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="p2">(<span class="default"><font color="Maroon" title="c3">k1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">k2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:42"><span class="lab"><font color="Green" title="E55">A20</font></span></a>, <a class="txt" href="#E23:42"><span class="lab"><font color="Green" title="E53">A18</font></span></a>, <a class="ref" href="idea_1.html#D11" target="_self" onmouseover="rs('idea_1/D11')" onmouseout="rh()">Def11</a></span>
<br/>.= 
 <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"><a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">k1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">k2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E3:42"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="txt" href="#E7:42"><span class="lab"><font color="Green" title="E42">A7</font></span></a>, <a class="txt" href="#E31:42"><span class="lab"><font color="Green" title="E57">A22</font></span></a>, <a class="ref" href="idea_1.html#T23" target="_self" onmouseover="rs('idea_1/T23')" onmouseout="rh()">Th23</a></span>
<br/>.= 
 <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (1,<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>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E7:42"><span class="lab"><font color="Green" title="E42">A7</font></span></a>, <a class="txt" href="#E11:42"><span class="lab"><font color="Green" title="E46">A11</font></span></a>, <a class="ref" href="idea_1.html#D10" target="_self" onmouseover="rs('idea_1/D10')" onmouseout="rh()">Def10</a></span>
<br/>.= 
<font color="Maroon" title="c2">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="txt" href="#E1:42_1_1_1_1"><span class="lab"><font color="Green" title="E62">A27</font></span></a>, <a class="ref" href="idea_1.html#T22" target="_self" onmouseover="rs('idea_1/T22')" onmouseout="rh()">Th22</a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:42_1_1_1_2"/><span class="lab"><font color="Green" title="E62">A28</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">I2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">j</font></span><br/><div class="add"><span class="kw">hence </span><font color="Maroon" title="c6">I2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">j</font> = 
 <a href="idea_1.html#K4" title="IDEA_1:func.4">ADD_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">I1</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="c4">k2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span>)</span>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:42"><span class="lab"><font color="Green" title="E56">A21</font></span></a>, <a class="txt" href="#E3:42_1"><span class="lab"><font color="Green" title="E60">A25</font></span></a>, <a class="ref" href="idea_1.html#D11" target="_self" onmouseover="rs('idea_1/D11')" onmouseout="rh()">Def11</a></span>
<br/>.= 
 <a href="idea_1.html#K4" title="IDEA_1:func.4">ADD_MOD</a> (<span class="p1">(<span class="default"><a href="idea_1.html#K4" title="IDEA_1:func.4">ADD_MOD</a> (<span class="p2">(<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="p2">(<span class="default"><font color="Maroon" title="c3">k1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span>)</span>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">k2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span>)</span>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:42"><span class="lab"><font color="Green" title="E55">A20</font></span></a>, <a class="txt" href="#E18:42"><span class="lab"><font color="Green" title="E51">A16</font></span></a>, <a class="ref" href="idea_1.html#D11" target="_self" onmouseover="rs('idea_1/D11')" onmouseout="rh()">Def11</a></span>
<br/>.= 
 <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"><a href="idea_1.html#K4" title="IDEA_1:func.4">ADD_MOD</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">k1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">k2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span>)</span>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="idea_1.html#T14" target="_self" onmouseover="rs('idea_1/T14')" onmouseout="rh()">Th14</a></span>
<br/>.= 
 <a href="idea_1.html#K4" title="IDEA_1:func.4">ADD_MOD</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</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>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:42"><span class="lab"><font color="Green" title="E43">A8</font></span></a>, <a class="txt" href="#E12:42"><span class="lab"><font color="Green" title="E47">A12</font></span></a>, <a class="ref" href="idea_1.html#T11" target="_self" onmouseover="rs('idea_1/T11')" onmouseout="rh()">Th11</a></span>
<br/>.= 
<font color="Maroon" title="c2">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="txt" href="#E1:42_1_1_1_2"><span class="lab"><font color="Green" title="E62">A28</font></span></a>, <a class="ref" href="idea_1.html#T13" target="_self" onmouseover="rs('idea_1/T13')" onmouseout="rh()">Th13</a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:42_1_1_1_3"/><span class="lab"><font color="Green" title="E62">A29</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">I2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">j</font></span><br/><div class="add"><span class="kw">hence </span><font color="Maroon" title="c6">I2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">j</font> = 
 <a href="idea_1.html#K4" title="IDEA_1:func.4">ADD_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">I1</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="c4">k2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:42"><span class="lab"><font color="Green" title="E56">A21</font></span></a>, <a class="txt" href="#E3:42_1"><span class="lab"><font color="Green" title="E60">A25</font></span></a>, <a class="ref" href="idea_1.html#D11" target="_self" onmouseover="rs('idea_1/D11')" onmouseout="rh()">Def11</a></span>
<br/>.= 
 <a href="idea_1.html#K4" title="IDEA_1:func.4">ADD_MOD</a> (<span class="p1">(<span class="default"><a href="idea_1.html#K4" title="IDEA_1:func.4">ADD_MOD</a> (<span class="p2">(<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="p2">(<span class="default"><font color="Maroon" title="c3">k1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">k2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:42"><span class="lab"><font color="Green" title="E55">A20</font></span></a>, <a class="txt" href="#E26:42"><span class="lab"><font color="Green" title="E54">A19</font></span></a>, <a class="ref" href="idea_1.html#D11" target="_self" onmouseover="rs('idea_1/D11')" onmouseout="rh()">Def11</a></span>
<br/>.= 
 <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"><a href="idea_1.html#K4" title="IDEA_1:func.4">ADD_MOD</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">k1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">k2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="idea_1.html#T14" target="_self" onmouseover="rs('idea_1/T14')" onmouseout="rh()">Th14</a></span>
<br/>.= 
 <a href="idea_1.html#K4" title="IDEA_1:func.4">ADD_MOD</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</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>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:42"><span class="lab"><font color="Green" title="E44">A9</font></span></a>, <a class="txt" href="#E13:42"><span class="lab"><font color="Green" title="E48">A13</font></span></a>, <a class="ref" href="idea_1.html#T11" target="_self" onmouseover="rs('idea_1/T11')" onmouseout="rh()">Th11</a></span>
<br/>.= 
<font color="Maroon" title="c2">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:42"><span class="lab"><font color="Green" title="E40">A5</font></span></a>, <a class="txt" href="#E1:42_1_1_1_3"><span class="lab"><font color="Green" title="E62">A29</font></span></a>, <a class="ref" href="idea_1.html#T13" target="_self" onmouseover="rs('idea_1/T13')" onmouseout="rh()">Th13</a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:42_1_1_1_4"/><span class="lab"><font color="Green" title="E62">A30</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">I2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">j</font></span><br/><div class="add"><span class="kw">hence </span><font color="Maroon" title="c6">I2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">j</font> = 
 <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">I1</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="c4">k2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4</span>)</span>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:42"><span class="lab"><font color="Green" title="E56">A21</font></span></a>, <a class="txt" href="#E3:42_1"><span class="lab"><font color="Green" title="E60">A25</font></span></a>, <a class="ref" href="idea_1.html#D11" target="_self" onmouseover="rs('idea_1/D11')" onmouseout="rh()">Def11</a></span>
<br/>.= 
 <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<span class="p1">(<span class="default"><a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<span class="p2">(<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="p2">(<span class="default"><font color="Maroon" title="c3">k1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4</span>)</span>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">k2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4</span>)</span>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:42"><span class="lab"><font color="Green" title="E55">A20</font></span></a>, <a class="txt" href="#E20:42"><span class="lab"><font color="Green" title="E52">A17</font></span></a>, <a class="ref" href="idea_1.html#D11" target="_self" onmouseover="rs('idea_1/D11')" onmouseout="rh()">Def11</a></span>
<br/>.= 
 <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"><a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">k1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">k2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4</span>)</span>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E6:42"><span class="lab"><font color="Green" title="E41">A6</font></span></a>, <a class="txt" href="#E10:42"><span class="lab"><font color="Green" title="E45">A10</font></span></a>, <a class="txt" href="#E15:42"><span class="lab"><font color="Green" title="E50">A15</font></span></a>, <a class="ref" href="idea_1.html#T23" target="_self" onmouseover="rs('idea_1/T23')" onmouseout="rh()">Th23</a></span>
<br/>.= 
 <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (1,<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>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E10:42"><span class="lab"><font color="Green" title="E45">A10</font></span></a>, <a class="txt" href="#E14:42"><span class="lab"><font color="Green" title="E49">A14</font></span></a>, <a class="ref" href="idea_1.html#D10" target="_self" onmouseover="rs('idea_1/D10')" onmouseout="rh()">Def10</a></span>
<br/>.= 
<font color="Maroon" title="c2">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:42"><span class="lab"><font color="Green" title="E41">A6</font></span></a>, <a class="txt" href="#E1:42_1_1_1_4"><span class="lab"><font color="Green" title="E62">A30</font></span></a>, <a class="ref" href="idea_1.html#T22" target="_self" onmouseover="rs('idea_1/T22')" onmouseout="rh()">Th22</a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_5"><span class="kw">suppose </span></a><a NAME="E1:42_1_1_1_5"/><span class="lab"><font color="Green" title="E62">A31</font></span>: 
( <font color="Maroon" title="c7">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Maroon" title="c7">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 2 &amp; <font color="Maroon" title="c7">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 3 &amp; <font color="Maroon" title="c7">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 4 )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">I2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">j</font></span><br/><div class="add"><span class="kw">hence </span><font color="Maroon" title="c6">I2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">j</font> = 
<font color="Maroon" title="c5">I1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:42"><span class="lab"><font color="Green" title="E56">A21</font></span></a>, <a class="txt" href="#E3:42_1"><span class="lab"><font color="Green" title="E60">A25</font></span></a>, <a class="ref" href="idea_1.html#D11" target="_self" onmouseover="rs('idea_1/D11')" onmouseout="rh()">Def11</a></span>
<br/>.= 
<font color="Maroon" title="c2">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:42"><span class="lab"><font color="Green" title="E55">A20</font></span></a>, <a class="txt" href="#E4:42_1"><span class="lab"><font color="Green" title="E61">A26</font></span></a>, <a class="txt" href="#E1:42_1_1_1_5"><span class="lab"><font color="Green" title="E62">A31</font></span></a>, <a class="ref" href="idea_1.html#D11" target="_self" onmouseover="rs('idea_1/D11')" onmouseout="rh()">Def11</a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E6:42_1"/>
<font color="Maroon" title="c6">I2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>
<a NAME="E33:42"/><span class="lab"><font color="Green" title="E59">A32</font></span>: 
 <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
 <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> = 
 <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="c5">I1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:42"><span class="lab"><font color="Green" title="E55">A20</font></span></a>, <a class="ref" href="idea_1.html#D11" target="_self" onmouseover="rs('idea_1/D11')" onmouseout="rh()">Def11</a></span>
<br/>.= 
 <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="c6">I2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:42"><span class="lab"><font color="Green" title="E56">A21</font></span></a>, <a class="ref" href="idea_1.html#D11" target="_self" onmouseover="rs('idea_1/D11')" onmouseout="rh()">Def11</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">I2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E35:42"/>
 <a href="idea_1.html#K11" title="IDEA_1:func.11">IDEAoperationA</a> (<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">k1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">k2</font>,<font color="Maroon" title="c1">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:42"><span class="lab"><font color="Green" title="E55">A20</font></span></a>, <a class="txt" href="#E30:42"><span class="lab"><font color="Green" title="E56">A21</font></span></a>, <a class="txt" href="#E33:42"><span class="lab"><font color="Green" title="E59">A32</font></span></a>, <a class="txt" href="#E32:42"><span class="lab"><font color="Green" title="E58">A23</font></span></a>, <a class="ref" href="finseq_1.html#T13" onmouseover="rs('finseq_1/T13')" onmouseout="rh()">FINSEQ_1:13</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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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