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<span class="kw">let </span><font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c3">D</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">Number</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">a</font>, <font color="Olive" title="b2">b</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">b</font> holds <br/>( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</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"><a href="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</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"><a href="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</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="c4">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">b</font> implies ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</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"><a href="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</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"><a href="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:138"/><span class="lab"><font color="Green" title="E88">A1</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">b</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="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</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"><a href="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</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"><a href="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">f</font> =  <a href="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</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>( $1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">a</font> implies ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> $1</span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`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="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> $1</span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`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="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  ) );<br/>
<a NAME="E2:138"/><span class="lab"><font color="Green" title="E89">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
;<br/>
<a NAME="E3:138"/><span class="lab"><font color="Green" title="E90">A3</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="138_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">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="c7">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">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="E1:138_1"/><span class="lab"><font color="Green" title="E91">A4</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">k</font>]
 <span class="kw">and </span><br/><a NAME="E2:138_1"/><span class="lab"><font color="Green" title="E92">A5</font></span>: 
<font color="Maroon" title="c7">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="c4">a</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="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`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="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`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="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  )</span><br/>

<a NAME="E3:138_1"/><span class="lab"><font color="Green" title="E93">A6</font></span>: 
<span class="p1">(<span class="default"><a href="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a></span>)</span> <a href="square_1.html#K2" title="SQUARE_1:func.2">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">D</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a></span>)</span> <a href="square_1.html#K2" title="SQUARE_1:func.2">^2</a></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a></span>)</span></span>)</span> <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="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a></span>)</span></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="number14.html#D19" target="_self" onmouseover="rs('number14/D19')" onmouseout="rh()">Def19</a></span>;<br/>
<a NAME="E4:138_1"/><span class="lab"><font color="Green" title="E94">A7</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`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> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E5:138_1"/><span class="kw">then </span>
2 <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="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/>
<a NAME="E6:138_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A8</font></span>: 
2 <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="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E7:138_1"/><span class="lab"><font color="Green" title="E96">A9</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a></span>)</span> <a href="square_1.html#K2" title="SQUARE_1:func.2">^2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 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="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:138_1"><span class="lab"><font color="Green" title="E94">A7</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="E8:138_1"/>
<font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:138_1"><span class="lab"><font color="Green" title="E92">A5</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="138_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="E1:138_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">a</font> or <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</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="138_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:138_1_1_1"/><span class="lab"><font color="Green" title="E97">A10</font></span>: 
<font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">a</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="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`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="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`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="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  )</span><br/><div class="add"><a NAME="E2:138_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a></span>)</span> <a href="square_1.html#K2" title="SQUARE_1:func.2">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">D</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"><span class="p5">(<span class="default"><a href="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a></span>)</span> <a href="square_1.html#K2" title="SQUARE_1:func.2">^2</a></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:138_1"><span class="lab"><font color="Green" title="E96">A9</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/><span class="kw">hence </span><a NAME="E3:138_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`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="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:138_1"><span class="lab"><font color="Green" title="E91">A4</font></span></a>, <a class="txt" href="#E3:138_1"><span class="lab"><font color="Green" title="E93">A6</font></span></a>, <a class="txt" href="#E1:138_1_1_1"><span class="lab"><font color="Green" title="E97">A10</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="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`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="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span><br/><a NAME="E4:138_1_1_1"/>
<span class="p1">(<span class="default">2 <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="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a></span>)</span></span>)</span> <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="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:138_1"><span class="lab"><font color="Green" title="E95">A8</font></span></a>, <a class="ref" href="xreal_1.html#T68" onmouseover="rs('xreal_1/T68')" onmouseout="rh()">XREAL_1:68</a></span>;<br/><span class="kw">hence </span><a NAME="E5:138_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`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="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:138_1"><span class="lab"><font color="Green" title="E91">A4</font></span></a>, <a class="txt" href="#E3:138_1"><span class="lab"><font color="Green" title="E93">A6</font></span></a>, <a class="txt" href="#E1:138_1_1_1"><span class="lab"><font color="Green" title="E97">A10</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="138_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:138_1_1_2"/><span class="lab"><font color="Green" title="E97">A11</font></span>: 
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</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="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`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="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`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="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  )</span><br/><div class="add"><a NAME="E2:138_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a></span>)</span> <a href="square_1.html#K2" title="SQUARE_1:func.2">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">D</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"><span class="p5">(<span class="default"><a href="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a></span>)</span> <a href="square_1.html#K2" title="SQUARE_1:func.2">^2</a></span>)</span></span>)</span> <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"><span class="p3">(<span class="default"><a href="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:138_1"><span class="lab"><font color="Green" title="E96">A9</font></span></a>, <a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E3:138_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`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="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:138_1"><span class="lab"><font color="Green" title="E93">A6</font></span></a>, <a class="txt" href="#E1:138_1_1_2"><span class="lab"><font color="Green" title="E97">A11</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"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`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="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span><br/><a NAME="E4:138_1_1_2"/>
<span class="p1">(<span class="default">2 <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="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a></span>)</span></span>)</span> <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="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 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="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:138_1"><span class="lab"><font color="Green" title="E95">A8</font></span></a>, <a class="ref" href="xreal_1.html#T68" onmouseover="rs('xreal_1/T68')" onmouseout="rh()">XREAL_1:68</a></span>;<br/><span class="kw">hence </span><a NAME="E5:138_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`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="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:138_1"><span class="lab"><font color="Green" title="E93">A6</font></span></a>, <a class="txt" href="#E1:138_1_1_2"><span class="lab"><font color="Green" title="E97">A11</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><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E4:138"/>
 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="#E2:138"><span class="lab"><font color="Green" title="E89">A2</font></span></a>, <a class="txt" href="#E3:138"><span class="lab"><font color="Green" title="E90">A3</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E5:138"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</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"><a href="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</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"><a href="number14.html#K10" title="NUMBER14:func.10">exampleSierpinski150</a> (<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">D</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:138"><span class="lab"><font color="Green" title="E88">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>
