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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="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="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>  &amp; ( ( <font color="Maroon" title="c7">a1</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="c8">a2</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>  ) or ( <font color="Maroon" title="c7">a1</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="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="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>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) or ( <font color="Maroon" title="c10">b1</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="c11">b2</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>  ) ) implies  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> 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">n</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>  &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">n</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><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:184"/><span class="lab"><font color="Green" title="E127">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="E2:184"/><span class="lab"><font color="Green" title="E128">B1</font></span>: 
( <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="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>  )
 <span class="kw">and </span><br/><a NAME="E3:184"/><span class="lab"><font color="Green" title="E129">A2</font></span>: 
( ( ( <font color="Maroon" title="c7">a1</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="c8">a2</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>  ) or ( <font color="Maroon" title="c7">a1</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="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="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>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) or ( <font color="Maroon" title="c10">b1</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="c11">b2</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>  ) ) )
 ; <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">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> 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">n</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>  &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">n</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><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">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>( <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>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  );<br/>
<a NAME="E4:184"/><span class="lab"><font color="Green" title="E130">A3</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="184_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:184_1"/>
<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> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <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"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></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/>
<span class="kw">hence </span><a NAME="E2:184_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:184"><span class="lab"><font color="Green" title="E127">A1</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><span class="kw">end;</span></div>
<a NAME="E5:184"/><span class="lab"><font color="Green" title="E131">A4</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="184_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">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="c14">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<a NAME="E1:184_2"/>
<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="c14">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="c14">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="c14">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="c14">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="c14">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/>
<span class="kw">hence </span><a NAME="E2:184_2"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:184"><span class="lab"><font color="Green" title="E128">B1</font></span></a>, <a class="txt" href="#E3:184"><span class="lab"><font color="Green" title="E129">A2</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><span class="kw">end;</span></div>
<a NAME="E6:184"/>
 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="#E4:184"><span class="lab"><font color="Green" title="E130">A3</font></span></a>, <a class="txt" href="#E5:184"><span class="lab"><font color="Green" title="E131">A4</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E7:184"/>
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> 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">n</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>  &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">n</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>  )
 ; <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>
