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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> 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> 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> 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> 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#K13" title="IDEA_1:func.13">IDEAoperationC</a> <span class="p2">(<span class="default"><a href="idea_1.html#K11" title="IDEA_1:func.11">IDEAoperationA</a> (<span class="p3">(<span class="default"><a href="idea_1.html#K13" title="IDEA_1:func.13">IDEAoperationC</a> <font color="Olive" title="b1">m</font></span>)</span>,<font color="Olive" title="b2">k1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></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> 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> 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> 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> 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#K13" title="IDEA_1:func.13">IDEAoperationC</a> <span class="p2">(<span class="default"><a href="idea_1.html#K11" title="IDEA_1:func.11">IDEAoperationA</a> (<span class="p3">(<span class="default"><a href="idea_1.html#K13" title="IDEA_1:func.13">IDEAoperationC</a> <font color="Maroon" title="c2">m</font></span>)</span>,<font color="Maroon" title="c3">k1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></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="E1:43"/><span class="lab"><font color="Green" title="E37">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="E2:43"/><span class="lab"><font color="Green" title="E38">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="E3:43"/><span class="lab"><font color="Green" title="E39">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="E4:43"/><span class="lab"><font color="Green" title="E40">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="E5:43"/><span class="lab"><font color="Green" title="E41">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="E6:43"/><span class="lab"><font color="Green" title="E42">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="E7:43"/><span class="lab"><font color="Green" title="E43">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="E8:43"/><span class="lab"><font color="Green" title="E44">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="E9:43"/><span class="lab"><font color="Green" title="E45">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="E10:43"/><span class="lab"><font color="Green" title="E46">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="E11:43"/><span class="lab"><font color="Green" title="E47">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="E12:43"/><span class="lab"><font color="Green" title="E48">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> 3</span>)</span>,<font color="Maroon" title="c1">n</font>)
 <span class="kw">and </span><br/><a NAME="E13:43"/><span class="lab"><font color="Green" title="E49">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> 2</span>)</span>,<font color="Maroon" title="c1">n</font>)
 <span class="kw">and </span><br/><a NAME="E14:43"/><span class="lab"><font color="Green" title="E50">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#K13" title="IDEA_1:func.13">IDEAoperationC</a> <span class="p2">(<span class="default"><a href="idea_1.html#K11" title="IDEA_1:func.11">IDEAoperationA</a> (<span class="p3">(<span class="default"><a href="idea_1.html#K13" title="IDEA_1:func.13">IDEAoperationC</a> <font color="Maroon" title="c2">m</font></span>)</span>,<font color="Maroon" title="c3">k1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></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="E15:43"/><span class="lab"><font color="Green" title="E51">A15</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="#E1:43"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="txt" href="#E7:43"><span class="lab"><font color="Green" title="E43">A7</font></span></a>, <a class="txt" href="#E11:43"><span class="lab"><font color="Green" title="E47">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/>
<a NAME="E16:43"/>
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="#E2:43"><span class="lab"><font color="Green" title="E38">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="E17:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A16</font></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="E18:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A17</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="E19:43"/><span class="lab"><font color="Green" title="E54">A18</font></span>: 
4 <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="txt" href="#E2:43"><span class="lab"><font color="Green" title="E38">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="E20:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A19</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="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E21:43"/>
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="#E2:43"><span class="lab"><font color="Green" title="E38">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="E22:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A20</font></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="E23:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A21</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/>
<a NAME="E24:43"/><span class="lab"><font color="Green" title="E58">A22</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="#E1:43"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="txt" href="#E10:43"><span class="lab"><font color="Green" title="E46">A10</font></span></a>, <a class="txt" href="#E14:43"><span class="lab"><font color="Green" title="E50">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/>
<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="E26:43"/><span class="lab"><font color="Green" title="E59">A23</font></span>: 
<font color="Maroon" title="c5">I1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K13" title="IDEA_1:func.13">IDEAoperationC</a> <font color="Maroon" title="c2">m</font>
 ;<br/>
<a NAME="E27:43"/>
4 <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="#E26:43"><span class="lab"><font color="Green" title="E59">A23</font></span></a>, <a class="txt" href="#E19:43"><span class="lab"><font color="Green" title="E54">A18</font></span></a>, <a class="ref" href="idea_1.html#D13" target="_self" onmouseover="rs('idea_1/D13')" onmouseout="rh()">Def13</a></span>;<br/>
<a NAME="E28:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A24</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="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="E29:43"/>
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="c5">I1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:43"><span class="lab"><font color="Green" title="E59">A23</font></span></a>, <a class="txt" href="#E17:43"><span class="lab"><font color="Green" title="E52">A16</font></span></a>, <a class="ref" href="idea_1.html#D13" target="_self" onmouseover="rs('idea_1/D13')" onmouseout="rh()">Def13</a></span>;<br/>
<a NAME="E30:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A25</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="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="E31:43"/>
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="#E2:43"><span class="lab"><font color="Green" title="E38">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="E32:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A26</font></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="E33:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A27</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="E34:43"/>
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="c5">I1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:43"><span class="lab"><font color="Green" title="E59">A23</font></span></a>, <a class="txt" href="#E22:43"><span class="lab"><font color="Green" title="E56">A20</font></span></a>, <a class="ref" href="idea_1.html#D13" target="_self" onmouseover="rs('idea_1/D13')" onmouseout="rh()">Def13</a></span>;<br/>
<a NAME="E35:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A28</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="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="E36:43"/>
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="c5">I1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:43"><span class="lab"><font color="Green" title="E59">A23</font></span></a>, <a class="txt" href="#E32:43"><span class="lab"><font color="Green" title="E62">A26</font></span></a>, <a class="ref" href="idea_1.html#D13" target="_self" onmouseover="rs('idea_1/D13')" onmouseout="rh()">Def13</a></span>;<br/>
<a NAME="E37:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A29</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="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/>
<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="E39:43"/><span class="lab"><font color="Green" title="E66">A30</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="c3">k1</font>,<font color="Maroon" title="c1">n</font>)
 ;<br/>
<a NAME="E40:43"/><span class="lab"><font color="Green" title="E67">A31</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">I2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">I1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:43"><span class="lab"><font color="Green" title="E66">A30</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="E41:43"/><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="c6">I2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:43"><span class="lab"><font color="Green" title="E59">A23</font></span></a>, <a class="txt" href="#E32:43"><span class="lab"><font color="Green" title="E62">A26</font></span></a>, <a class="ref" href="idea_1.html#D13" target="_self" onmouseover="rs('idea_1/D13')" onmouseout="rh()">Def13</a></span>;<br/>
<a NAME="E42:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A32</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="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/>
<a NAME="E43:43"/>
4 <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="c6">I2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:43"><span class="lab"><font color="Green" title="E59">A23</font></span></a>, <a class="txt" href="#E40:43"><span class="lab"><font color="Green" title="E67">A31</font></span></a>, <a class="txt" href="#E19:43"><span class="lab"><font color="Green" title="E54">A18</font></span></a>, <a class="ref" href="idea_1.html#D13" target="_self" onmouseover="rs('idea_1/D13')" onmouseout="rh()">Def13</a></span>;<br/>
<a NAME="E44:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A33</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="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/>
<a NAME="E45:43"/>
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="c6">I2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:43"><span class="lab"><font color="Green" title="E59">A23</font></span></a>, <a class="txt" href="#E40:43"><span class="lab"><font color="Green" title="E67">A31</font></span></a>, <a class="txt" href="#E17:43"><span class="lab"><font color="Green" title="E52">A16</font></span></a>, <a class="ref" href="idea_1.html#D13" target="_self" onmouseover="rs('idea_1/D13')" onmouseout="rh()">Def13</a></span>;<br/>
<a NAME="E46:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A34</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="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/>
<a NAME="E47:43"/>
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="c6">I2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:43"><span class="lab"><font color="Green" title="E59">A23</font></span></a>, <a class="txt" href="#E40:43"><span class="lab"><font color="Green" title="E67">A31</font></span></a>, <a class="txt" href="#E22:43"><span class="lab"><font color="Green" title="E56">A20</font></span></a>, <a class="ref" href="idea_1.html#D13" target="_self" onmouseover="rs('idea_1/D13')" onmouseout="rh()">Def13</a></span>;<br/>
<a NAME="E48:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A35</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="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">consider </span><font color="Maroon" title="c7">I3</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="E50:43"/><span class="lab"><font color="Green" title="E72">A36</font></span>: 
<font color="Maroon" title="c7">I3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K13" title="IDEA_1:func.13">IDEAoperationC</a> <font color="Maroon" title="c6">I2</font>
 ;<br/>
<a NAME="E51:43"/><span class="lab"><font color="Green" title="E73">A37</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">I3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">I2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:43"><span class="lab"><font color="Green" title="E72">A36</font></span></a>, <a class="ref" href="idea_1.html#D13" target="_self" onmouseover="rs('idea_1/D13')" onmouseout="rh()">Def13</a></span>;<br/>
<a NAME="E52:43"/><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="c7">I3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:43"><span class="lab"><font color="Green" title="E59">A23</font></span></a>, <a class="txt" href="#E40:43"><span class="lab"><font color="Green" title="E67">A31</font></span></a>, <a class="txt" href="#E32:43"><span class="lab"><font color="Green" title="E62">A26</font></span></a>, <a class="ref" href="idea_1.html#D13" target="_self" onmouseover="rs('idea_1/D13')" onmouseout="rh()">Def13</a></span>;<br/>
<a NAME="E53:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A38</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="c7">I3</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="E54:43"/>
4 <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="c7">I3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:43"><span class="lab"><font color="Green" title="E59">A23</font></span></a>, <a class="txt" href="#E40:43"><span class="lab"><font color="Green" title="E67">A31</font></span></a>, <a class="txt" href="#E51:43"><span class="lab"><font color="Green" title="E73">A37</font></span></a>, <a class="txt" href="#E19:43"><span class="lab"><font color="Green" title="E54">A18</font></span></a>, <a class="ref" href="idea_1.html#D13" target="_self" onmouseover="rs('idea_1/D13')" onmouseout="rh()">Def13</a></span>;<br/>
<a NAME="E55:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A39</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="c7">I3</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="E56:43"/>
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="c7">I3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:43"><span class="lab"><font color="Green" title="E59">A23</font></span></a>, <a class="txt" href="#E40:43"><span class="lab"><font color="Green" title="E67">A31</font></span></a>, <a class="txt" href="#E51:43"><span class="lab"><font color="Green" title="E73">A37</font></span></a>, <a class="txt" href="#E22:43"><span class="lab"><font color="Green" title="E56">A20</font></span></a>, <a class="ref" href="idea_1.html#D13" target="_self" onmouseover="rs('idea_1/D13')" onmouseout="rh()">Def13</a></span>;<br/>
<a NAME="E57:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A40</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="c7">I3</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="c8">I4</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="E59:43"/><span class="lab"><font color="Green" title="E77">A41</font></span>: 
<font color="Maroon" title="c8">I4</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="c7">I3</font>,<font color="Maroon" title="c4">k2</font>,<font color="Maroon" title="c1">n</font>)
 ;<br/>
<a NAME="E60:43"/>
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="c7">I3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:43"><span class="lab"><font color="Green" title="E59">A23</font></span></a>, <a class="txt" href="#E40:43"><span class="lab"><font color="Green" title="E67">A31</font></span></a>, <a class="txt" href="#E51:43"><span class="lab"><font color="Green" title="E73">A37</font></span></a>, <a class="txt" href="#E17:43"><span class="lab"><font color="Green" title="E52">A16</font></span></a>, <a class="ref" href="idea_1.html#D13" target="_self" onmouseover="rs('idea_1/D13')" onmouseout="rh()">Def13</a></span>;<br/>
<a NAME="E61:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A42</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="c7">I3</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/>
<div><span class="lab"><font color="Green" title="E79">A43</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="43_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="c8">I4</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="c9">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="c9">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="c8">I4</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">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="c9">j</font> )</span><br/><span class="kw">assume </span><a NAME="E1:43_1"/><span class="lab"><font color="Green" title="E80">A44</font></span>: 
<font color="Maroon" title="c9">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="c8">I4</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">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="c9">j</font></span><br/><a NAME="E2:43_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A45</font></span>: 
<font color="Maroon" title="c9">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="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><a NAME="E3:43_1"/><span class="lab"><font color="Green" title="E82">A46</font></span>: 
<font color="Maroon" title="c9">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="#E26:43"><span class="lab"><font color="Green" title="E59">A23</font></span></a>, <a class="txt" href="#E1:43_1"><span class="lab"><font color="Green" title="E80">A44</font></span></a>, <a class="ref" href="idea_1.html#D13" target="_self" onmouseover="rs('idea_1/D13')" onmouseout="rh()">Def13</a></span>;<br/><a NAME="E4:43_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A47</font></span>: 
<font color="Maroon" title="c9">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="c6">I2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:43"><span class="lab"><font color="Green" title="E66">A30</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="E5:43_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A48</font></span>: 
<font color="Maroon" title="c9">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="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/><a NAME="E6:43_1"/>
<font color="Maroon" title="c9">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="c7">I3</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:43"><span class="lab"><font color="Green" title="E72">A36</font></span></a>, <a class="txt" href="#E4:43_1"><span class="lab"><font color="Green" title="E83">A47</font></span></a>, <a class="ref" href="idea_1.html#D13" target="_self" onmouseover="rs('idea_1/D13')" onmouseout="rh()">Def13</a></span>;<br/><a NAME="E7:43_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A49</font></span>: 
<font color="Maroon" title="c9">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="c7">I3</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="E8:43_1"/><span class="lab"><font color="Green" title="E86">A50</font></span>: 
<font color="Maroon" title="c9">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="txt" href="#E3:43_1"><span class="lab"><font color="Green" title="E82">A46</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="43_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="c8">I4</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">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="c9">j</font></span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_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="E1:43_1_1_1"/>
( <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 or ( <font color="Maroon" title="c9">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Maroon" title="c9">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 2 &amp; <font color="Maroon" title="c9">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 3 &amp; <font color="Maroon" title="c9">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="43_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:43_1_1_1_1"/><span class="lab"><font color="Green" title="E87">A51</font></span>: 
<font color="Maroon" title="c9">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="c8">I4</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">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="c9">j</font></span><br/><div class="add"><span class="kw">hence </span><font color="Maroon" title="c8">I4</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">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="c7">I3</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="#E59:43"><span class="lab"><font color="Green" title="E77">A41</font></span></a>, <a class="txt" href="#E61:43"><span class="lab"><font color="Green" title="E78">A42</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="c6">I2</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="#E50:43"><span class="lab"><font color="Green" title="E72">A36</font></span></a>, <a class="txt" href="#E46:43"><span class="lab"><font color="Green" title="E70">A34</font></span></a>, <a class="txt" href="idea_1.html#E45"><span class="lab"><font color="Green" title="E34">Lm5</font></span></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="c5">I1</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="#E39:43"><span class="lab"><font color="Green" title="E66">A30</font></span></a>, <a class="txt" href="#E30:43"><span class="lab"><font color="Green" title="E61">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="#E26:43"><span class="lab"><font color="Green" title="E59">A23</font></span></a>, <a class="txt" href="#E18:43"><span class="lab"><font color="Green" title="E53">A17</font></span></a>, <a class="txt" href="idea_1.html#E45"><span class="lab"><font color="Green" title="E34">Lm5</font></span></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="#E1:43"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="txt" href="#E7:43"><span class="lab"><font color="Green" title="E43">A7</font></span></a>, <a class="txt" href="#E15:43"><span class="lab"><font color="Green" title="E51">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> 1</span>)</span>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="txt" href="#E7:43"><span class="lab"><font color="Green" title="E43">A7</font></span></a>, <a class="txt" href="#E11:43"><span class="lab"><font color="Green" title="E47">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="c9">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="txt" href="#E1:43_1_1_1_1"><span class="lab"><font color="Green" title="E87">A51</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="43_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:43_1_1_1_2"/><span class="lab"><font color="Green" title="E87">A52</font></span>: 
<font color="Maroon" title="c9">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="c8">I4</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">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="c9">j</font></span><br/><div class="add"><span class="kw">hence </span><font color="Maroon" title="c8">I4</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">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="c7">I3</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="#E59:43"><span class="lab"><font color="Green" title="E77">A41</font></span></a>, <a class="txt" href="#E53:43"><span class="lab"><font color="Green" title="E74">A38</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="c6">I2</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> 2</span>)</span>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:43"><span class="lab"><font color="Green" title="E72">A36</font></span></a>, <a class="txt" href="#E42:43"><span class="lab"><font color="Green" title="E68">A32</font></span></a>, <a class="txt" href="idea_1.html#E43"><span class="lab"><font color="Green" title="E32">Lm3</font></span></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="c5">I1</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> 2</span>)</span>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:43"><span class="lab"><font color="Green" title="E66">A30</font></span></a>, <a class="txt" href="#E35:43"><span class="lab"><font color="Green" title="E64">A28</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> 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> 2</span>)</span>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:43"><span class="lab"><font color="Green" title="E59">A23</font></span></a>, <a class="txt" href="#E23:43"><span class="lab"><font color="Green" title="E57">A21</font></span></a>, <a class="txt" href="idea_1.html#E44"><span class="lab"><font color="Green" title="E33">Lm4</font></span></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> 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> 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="#E9:43"><span class="lab"><font color="Green" title="E45">A9</font></span></a>, <a class="txt" href="#E12:43"><span class="lab"><font color="Green" title="E48">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="c9">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="txt" href="#E1:43_1_1_1_2"><span class="lab"><font color="Green" title="E87">A52</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="43_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:43_1_1_1_3"/><span class="lab"><font color="Green" title="E87">A53</font></span>: 
<font color="Maroon" title="c9">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="c8">I4</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">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="c9">j</font></span><br/><div class="add"><span class="kw">hence </span><font color="Maroon" title="c8">I4</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">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="c7">I3</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="#E59:43"><span class="lab"><font color="Green" title="E77">A41</font></span></a>, <a class="txt" href="#E57:43"><span class="lab"><font color="Green" title="E76">A40</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="c6">I2</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> 3</span>)</span>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:43"><span class="lab"><font color="Green" title="E72">A36</font></span></a>, <a class="txt" href="#E48:43"><span class="lab"><font color="Green" title="E71">A35</font></span></a>, <a class="txt" href="idea_1.html#E44"><span class="lab"><font color="Green" title="E33">Lm4</font></span></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="c5">I1</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> 3</span>)</span>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:43"><span class="lab"><font color="Green" title="E66">A30</font></span></a>, <a class="txt" href="#E37:43"><span class="lab"><font color="Green" title="E65">A29</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> 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> 3</span>)</span>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:43"><span class="lab"><font color="Green" title="E59">A23</font></span></a>, <a class="txt" href="#E33:43"><span class="lab"><font color="Green" title="E63">A27</font></span></a>, <a class="txt" href="idea_1.html#E43"><span class="lab"><font color="Green" title="E32">Lm3</font></span></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> 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> 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="#E8:43"><span class="lab"><font color="Green" title="E44">A8</font></span></a>, <a class="txt" href="#E13:43"><span class="lab"><font color="Green" title="E49">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="c9">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43"><span class="lab"><font color="Green" title="E41">A5</font></span></a>, <a class="txt" href="#E1:43_1_1_1_3"><span class="lab"><font color="Green" title="E87">A53</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="43_1_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:43_1_1_1_4"/><span class="lab"><font color="Green" title="E87">A54</font></span>: 
<font color="Maroon" title="c9">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="c8">I4</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">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="c9">j</font></span><br/><div class="add"><span class="kw">hence </span><font color="Maroon" title="c8">I4</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">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="c7">I3</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="#E59:43"><span class="lab"><font color="Green" title="E77">A41</font></span></a>, <a class="txt" href="#E55:43"><span class="lab"><font color="Green" title="E75">A39</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="c6">I2</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="#E50:43"><span class="lab"><font color="Green" title="E72">A36</font></span></a>, <a class="txt" href="#E44:43"><span class="lab"><font color="Green" title="E69">A33</font></span></a>, <a class="txt" href="idea_1.html#E45"><span class="lab"><font color="Green" title="E34">Lm5</font></span></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="c5">I1</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="#E39:43"><span class="lab"><font color="Green" title="E66">A30</font></span></a>, <a class="txt" href="#E28:43"><span class="lab"><font color="Green" title="E60">A24</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="#E26:43"><span class="lab"><font color="Green" title="E59">A23</font></span></a>, <a class="txt" href="#E20:43"><span class="lab"><font color="Green" title="E55">A19</font></span></a>, <a class="txt" href="idea_1.html#E45"><span class="lab"><font color="Green" title="E34">Lm5</font></span></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="#E1:43"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="txt" href="#E6:43"><span class="lab"><font color="Green" title="E42">A6</font></span></a>, <a class="txt" href="#E10:43"><span class="lab"><font color="Green" title="E46">A10</font></span></a>, <a class="txt" href="#E24:43"><span class="lab"><font color="Green" title="E58">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> 4</span>)</span>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="txt" href="#E10:43"><span class="lab"><font color="Green" title="E46">A10</font></span></a>, <a class="txt" href="#E14:43"><span class="lab"><font color="Green" title="E50">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="c9">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43"><span class="lab"><font color="Green" title="E42">A6</font></span></a>, <a class="txt" href="#E1:43_1_1_1_4"><span class="lab"><font color="Green" title="E87">A54</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="43_1_1_1_5"><span class="kw">suppose </span></a><a NAME="E1:43_1_1_1_5"/><span class="lab"><font color="Green" title="E87">A55</font></span>: 
( <font color="Maroon" title="c9">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Maroon" title="c9">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 2 &amp; <font color="Maroon" title="c9">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 3 &amp; <font color="Maroon" title="c9">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="c8">I4</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">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="c9">j</font></span><br/><div class="add"><span class="kw">hence </span><font color="Maroon" title="c8">I4</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">j</font> = 
<font color="Maroon" title="c7">I3</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E59:43"><span class="lab"><font color="Green" title="E77">A41</font></span></a>, <a class="txt" href="#E7:43_1"><span class="lab"><font color="Green" title="E85">A49</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="c6">I2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:43"><span class="lab"><font color="Green" title="E72">A36</font></span></a>, <a class="txt" href="#E5:43_1"><span class="lab"><font color="Green" title="E84">A48</font></span></a>, <a class="txt" href="#E1:43_1_1_1_5"><span class="lab"><font color="Green" title="E87">A55</font></span></a>, <a class="txt" href="idea_1.html#E45"><span class="lab"><font color="Green" title="E34">Lm5</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c5">I1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:43"><span class="lab"><font color="Green" title="E66">A30</font></span></a>, <a class="txt" href="#E8:43_1"><span class="lab"><font color="Green" title="E86">A50</font></span></a>, <a class="txt" href="#E1:43_1_1_1_5"><span class="lab"><font color="Green" title="E87">A55</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="c9">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:43"><span class="lab"><font color="Green" title="E59">A23</font></span></a>, <a class="txt" href="#E2:43_1"><span class="lab"><font color="Green" title="E81">A45</font></span></a>, <a class="txt" href="#E1:43_1_1_1_5"><span class="lab"><font color="Green" title="E87">A55</font></span></a>, <a class="txt" href="idea_1.html#E45"><span class="lab"><font color="Green" title="E34">Lm5</font></span></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="E10:43_1"/>
<font color="Maroon" title="c8">I4</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">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="c9">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="E63:43"/><span class="lab"><font color="Green" title="E80">A56</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="#E26:43"><span class="lab"><font color="Green" title="E59">A23</font></span></a>, <a class="ref" href="idea_1.html#D13" target="_self" onmouseover="rs('idea_1/D13')" onmouseout="rh()">Def13</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="#E39:43"><span class="lab"><font color="Green" title="E66">A30</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="c7">I3</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:43"><span class="lab"><font color="Green" title="E72">A36</font></span></a>, <a class="ref" href="idea_1.html#D13" target="_self" onmouseover="rs('idea_1/D13')" onmouseout="rh()">Def13</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="c8">I4</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E59:43"><span class="lab"><font color="Green" title="E77">A41</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="c8">I4</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="E65:43"/>
 <a href="idea_1.html#K11" title="IDEA_1:func.11">IDEAoperationA</a> (<span class="p1">(<span class="default"><a href="idea_1.html#K13" title="IDEA_1:func.13">IDEAoperationC</a> <span class="p2">(<span class="default"><a href="idea_1.html#K11" title="IDEA_1:func.11">IDEAoperationA</a> (<span class="p3">(<span class="default"><a href="idea_1.html#K13" title="IDEA_1:func.13">IDEAoperationC</a> <font color="Maroon" title="c2">m</font></span>)</span>,<font color="Maroon" title="c3">k1</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></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="#E26:43"><span class="lab"><font color="Green" title="E59">A23</font></span></a>, <a class="txt" href="#E39:43"><span class="lab"><font color="Green" title="E66">A30</font></span></a>, <a class="txt" href="#E50:43"><span class="lab"><font color="Green" title="E72">A36</font></span></a>, <a class="txt" href="#E59:43"><span class="lab"><font color="Green" title="E77">A41</font></span></a>, <a class="txt" href="#E63:43"><span class="lab"><font color="Green" title="E80">A56</font></span></a>, <a class="txt" href="#E62:43"><span class="lab"><font color="Green" title="E79">A43</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/>


</div>
