<?xml version="1.0"?>
<div class="add">

<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">x1</font>,<font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c9">x3</font>,<font color="Maroon" title="c10">y1</font>,<font color="Maroon" title="c11">y2</font>,<font color="Maroon" title="c12">y3</font>);<br/>
<a NAME="E1:180_1_1"/><span class="lab"><font color="Green" title="E126">A1</font></span>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" onmouseover="rs('xreal_0/D1')" onmouseout="rh()">XREAL_0:def 1</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><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">x1</font>,<font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c9">x3</font>,<font color="Maroon" title="c10">y1</font>,<font color="Maroon" title="c11">y2</font>,<font color="Maroon" title="c12">y3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>
<a NAME="E2:180_1_1"/><span class="lab"><font color="Green" title="E127">A2</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="180_1_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:180_1_1_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">x1</font>,<font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c9">x3</font>,<font color="Maroon" title="c10">y1</font>,<font color="Maroon" title="c11">y2</font>,<font color="Maroon" title="c12">y3</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="E2:180_1_1_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="#E1:180_1_1"><span class="lab"><font color="Green" title="E126">A1</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" onmouseover="rs('zfmisc_1/T87')" onmouseout="rh()">ZFMISC_1:87</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="E3:180_1_1"/><span class="lab"><font color="Green" title="E128">A3</font></span>: 
 for <font color="Olive" title="b1">n</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">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</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="180_1_1_2"><span class="kw">proof </span></a><div class="add">

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

<span class="kw">assume </span><a NAME="E1:180_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">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"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<a NAME="E2:180_1_1_2"/><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">x1</font>,<font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c9">x3</font>,<font color="Maroon" title="c10">y1</font>,<font color="Maroon" title="c11">y2</font>,<font color="Maroon" title="c12">y3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</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">x1</font>,<font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c9">x3</font>,<font color="Maroon" title="c10">y1</font>,<font color="Maroon" title="c11">y2</font>,<font color="Maroon" title="c12">y3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T10" onmouseover="rs('mcart_1/T10')" onmouseout="rh()">MCART_1:10</a></span>;<br/>
<a NAME="E3:180_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E129">A4</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">x1</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">x1</font>,<font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c9">x3</font>,<font color="Maroon" title="c10">y1</font>,<font color="Maroon" title="c11">y2</font>,<font color="Maroon" title="c12">y3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</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">x2</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">x1</font>,<font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c9">x3</font>,<font color="Maroon" title="c10">y1</font>,<font color="Maroon" title="c11">y2</font>,<font color="Maroon" title="c12">y3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</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">x3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">y1</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">x1</font>,<font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c9">x3</font>,<font color="Maroon" title="c10">y1</font>,<font color="Maroon" title="c11">y2</font>,<font color="Maroon" title="c12">y3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</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">y2</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">x1</font>,<font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c9">x3</font>,<font color="Maroon" title="c10">y1</font>,<font color="Maroon" title="c11">y2</font>,<font color="Maroon" title="c12">y3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</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">y3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" onmouseover="rs('xreal_0/D1')" onmouseout="rh()">XREAL_0:def 1</a></span>;<br/>
<a NAME="E4:180_1_1_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">x1</font>,<font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c9">x3</font>,<font color="Maroon" title="c10">y1</font>,<font color="Maroon" title="c11">y2</font>,<font color="Maroon" title="c12">y3</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">n</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">x1</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">x1</font>,<font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c9">x3</font>,<font color="Maroon" title="c10">y1</font>,<font color="Maroon" title="c11">y2</font>,<font color="Maroon" title="c12">y3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</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">x2</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">x1</font>,<font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c9">x3</font>,<font color="Maroon" title="c10">y1</font>,<font color="Maroon" title="c11">y2</font>,<font color="Maroon" title="c12">y3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</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">x3</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">y1</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">x1</font>,<font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c9">x3</font>,<font color="Maroon" title="c10">y1</font>,<font color="Maroon" title="c11">y2</font>,<font color="Maroon" title="c12">y3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</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">y2</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">x1</font>,<font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c9">x3</font>,<font color="Maroon" title="c10">y1</font>,<font color="Maroon" title="c11">y2</font>,<font color="Maroon" title="c12">y3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</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">y3</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="E5:180_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">n</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="#E3:180_1_1_2"><span class="lab"><font color="Green" title="E129">A4</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" onmouseover="rs('zfmisc_1/T87')" onmouseout="rh()">ZFMISC_1:87</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="E4:180_1_1"/><span class="lab"><font color="Green" title="E129">A5</font></span>: 
 for <font color="Olive" title="b1">n</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">n</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="#E2:180_1_1"><span class="lab"><font color="Green" title="E127">A2</font></span></a>, <a class="txt" href="#E3:180_1_1"><span class="lab"><font color="Green" title="E128">A3</font></span></a>);<br/></span>
<span class="kw">thus </span><a NAME="E5:180_1_1"/>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <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">x1</font>,<font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c9">x3</font>,<font color="Maroon" title="c10">y1</font>,<font color="Maroon" title="c11">y2</font>,<font color="Maroon" title="c12">y3</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="relat_1.html#D19" onmouseover="rs('relat_1/D19')" onmouseout="rh()">RELAT_1:def 19</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><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="180_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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"> (  not <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <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">x1</font>,<font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c9">x3</font>,<font color="Maroon" title="c10">y1</font>,<font color="Maroon" title="c11">y2</font>,<font color="Maroon" title="c12">y3</font>)</span>)</span> or <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:180_1_1_3"/>
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <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">x1</font>,<font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c9">x3</font>,<font color="Maroon" title="c10">y1</font>,<font color="Maroon" title="c11">y2</font>,<font color="Maroon" title="c12">y3</font>)</span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span><br/>

<a NAME="E2:180_1_1_3"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <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">x1</font>,<font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c9">x3</font>,<font color="Maroon" title="c10">y1</font>,<font color="Maroon" title="c11">y2</font>,<font color="Maroon" title="c12">y3</font>)</span>)</span> &amp; <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">x1</font>,<font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c9">x3</font>,<font color="Maroon" title="c10">y1</font>,<font color="Maroon" title="c11">y2</font>,<font color="Maroon" title="c12">y3</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:180_1_1_3"/>
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:180_1_1"><span class="lab"><font color="Green" title="E129">A5</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>


</div>
