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<span class="kw">let </span><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c7">a1</font>, <font color="Maroon" title="c8">a2</font>, <font color="Maroon" title="c9">a3</font>, <font color="Maroon" title="c10">b1</font>, <font color="Maroon" title="c11">b2</font>, <font color="Maroon" title="c12">b3</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</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="c5">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c6">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c7">a1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c8">a2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c9">a3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c10">b1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c11">b2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c12">b3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies  for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b2">n</font> holds <br/>( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:186"/><span class="lab"><font color="Green" title="E129">A1</font></span>: 
( <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c6">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 <span class="kw">and </span><br/><a NAME="E2:186"/><span class="lab"><font color="Green" title="E130">A2</font></span>: 
( <font color="Maroon" title="c7">a1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c8">a2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c9">a3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 <span class="kw">and </span><br/><a NAME="E3:186"/><span class="lab"><font color="Green" title="E131">A3</font></span>: 
( <font color="Maroon" title="c10">b1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c11">b2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c12">b3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b2">n</font> holds <br/>( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c13">f</font> =  <a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>);<br/>
<span class="kw">let </span><font color="Maroon" title="c14">m</font>, <font color="Maroon" title="c15">n</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="c14">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c15">n</font> implies ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  ) )</span><br/>



<span class="kw">assume </span><a NAME="E4:186"/><span class="lab"><font color="Green" title="E132">A4</font></span>: 
<font color="Maroon" title="c14">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c15">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"> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span>( $1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c15">n</font> implies ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  ) );<br/>
<a NAME="E5:186"/><span class="lab"><font color="Green" title="E133">A5</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
;<br/>
<a NAME="E6:186"/><span class="lab"><font color="Green" title="E134">A6</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="186_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">k</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">S<sub>1</sub></font>[<font color="Maroon" title="c16">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:186_1"/><span class="lab"><font color="Green" title="E135">A7</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c16">k</font>]
 <span class="kw">and </span><br/><a NAME="E2:186_1"/><span class="lab"><font color="Green" title="E136">A8</font></span>: 
<font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c15">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"> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  )</span><br/>

<a NAME="E3:186_1"/><span class="lab"><font color="Green" title="E137">A9</font></span>: 
<span class="p1">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">a2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">a3</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">b1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">b2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">b3</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="number08.html#D10" target="_self" onmouseover="rs('number08/D10')" onmouseout="rh()">Def10</a></span>;<br/>
<a NAME="E4:186_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:186"><span class="lab"><font color="Green" title="E129">A1</font></span></a>, <a class="txt" href="#E2:186"><span class="lab"><font color="Green" title="E130">A2</font></span></a>, <a class="txt" href="#E3:186"><span class="lab"><font color="Green" title="E131">A3</font></span></a>, <a class="ref" href="number08.html#T90" target="_self" onmouseover="rs('number08/T90')" onmouseout="rh()">Th90</a></span>;<br/>
<a NAME="E5:186_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E138">A10</font></span>: 
1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:186"><span class="lab"><font color="Green" title="E130">A2</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/>
<a NAME="E6:186_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:186"><span class="lab"><font color="Green" title="E129">A1</font></span></a>, <a class="txt" href="#E2:186"><span class="lab"><font color="Green" title="E130">A2</font></span></a>, <a class="txt" href="#E3:186"><span class="lab"><font color="Green" title="E131">A3</font></span></a>, <a class="ref" href="number08.html#T89" target="_self" onmouseover="rs('number08/T89')" onmouseout="rh()">Th89</a></span>;<br/>
<a NAME="E7:186_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E139">B11</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:186"><span class="lab"><font color="Green" title="E130">A2</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/>
<a NAME="E8:186_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">a2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">a2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">a3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:186"><span class="lab"><font color="Green" title="E130">A2</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/>
<a NAME="E9:186_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E140">A11</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">a2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">a3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:186_1"><span class="lab"><font color="Green" title="E139">B11</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="E10:186_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:186"><span class="lab"><font color="Green" title="E129">A1</font></span></a>, <a class="txt" href="#E2:186"><span class="lab"><font color="Green" title="E130">A2</font></span></a>, <a class="txt" href="#E3:186"><span class="lab"><font color="Green" title="E131">A3</font></span></a>, <a class="ref" href="number08.html#T90" target="_self" onmouseover="rs('number08/T90')" onmouseout="rh()">Th90</a></span>;<br/>
<a NAME="E11:186_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E141">A12</font></span>: 
1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">b2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:186"><span class="lab"><font color="Green" title="E131">A3</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/>
<a NAME="E12:186_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:186"><span class="lab"><font color="Green" title="E129">A1</font></span></a>, <a class="txt" href="#E2:186"><span class="lab"><font color="Green" title="E130">A2</font></span></a>, <a class="txt" href="#E3:186"><span class="lab"><font color="Green" title="E131">A3</font></span></a>, <a class="ref" href="number08.html#T89" target="_self" onmouseover="rs('number08/T89')" onmouseout="rh()">Th89</a></span>;<br/>
<a NAME="E13:186_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E142">B13</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">b2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">b1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">b2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:186"><span class="lab"><font color="Green" title="E131">A3</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/>
<a NAME="E14:186_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">b1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">b2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">b1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">b2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">b3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:186"><span class="lab"><font color="Green" title="E131">A3</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/>
<a NAME="E15:186_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E143">A13</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">b2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">b1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">b2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">b3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:186_1"><span class="lab"><font color="Green" title="E142">B13</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="E16:186_1"/>
<font color="Maroon" title="c16">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c15">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:186_1"><span class="lab"><font color="Green" title="E136">A8</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="186_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="E1:186_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c16">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c15">n</font> or <font color="Maroon" title="c16">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="186_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:186_1_1_1"/><span class="lab"><font color="Green" title="E144">A14</font></span>: 
<font color="Maroon" title="c16">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c15">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"> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  )</span><br/><div class="add"><a NAME="E2:186_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:186_1"><span class="lab"><font color="Green" title="E135">A7</font></span></a>, <a class="txt" href="#E5:186_1"><span class="lab"><font color="Green" title="E138">A10</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E3:186_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:186_1"><span class="lab"><font color="Green" title="E137">A9</font></span></a>, <a class="txt" href="#E9:186_1"><span class="lab"><font color="Green" title="E140">A11</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</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"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a> </span><br/><a NAME="E4:186_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">b2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:186_1"><span class="lab"><font color="Green" title="E135">A7</font></span></a>, <a class="txt" href="#E11:186_1"><span class="lab"><font color="Green" title="E141">A12</font></span></a>, <a class="txt" href="#E1:186_1_1_1"><span class="lab"><font color="Green" title="E144">A14</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E5:186_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:186_1"><span class="lab"><font color="Green" title="E137">A9</font></span></a>, <a class="txt" href="#E15:186_1"><span class="lab"><font color="Green" title="E143">A13</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="186_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:186_1_1_2"/>
<font color="Maroon" title="c16">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">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"> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:186_1_1_2"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:186_1"><span class="lab"><font color="Green" title="E137">A9</font></span></a>, <a class="txt" href="#E5:186_1"><span class="lab"><font color="Green" title="E138">A10</font></span></a>, <a class="txt" href="#E9:186_1"><span class="lab"><font color="Green" title="E140">A11</font></span></a>, <a class="txt" href="#E11:186_1"><span class="lab"><font color="Green" title="E141">A12</font></span></a>, <a class="txt" href="#E15:186_1"><span class="lab"><font color="Green" title="E143">A13</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E7:186"/>
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E5:186"><span class="lab"><font color="Green" title="E133">A5</font></span></a>, <a class="txt" href="#E6:186"><span class="lab"><font color="Green" title="E134">A6</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E8:186"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number08.html#K5" title="NUMBER08:func.5">recSeqCart</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">a3</font>,<font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:186"><span class="lab"><font color="Green" title="E132">A4</font></span></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>
