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<span class="kw">set </span><font color="Maroon" title="c4">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">set </span><font color="Maroon" title="c5">A</font> =  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>
<span class="kw">thus </span><a NAME="E1:133_1_1"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <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="relset_1.html#R1" title="RELSET_1: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="133_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">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="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <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> or <font color="Maroon" title="c6">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="E1:133_1_1_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <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 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="c6">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">then </span><span class="kw">consider </span><font color="Maroon" title="c7">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:133_1_1_1"/><span class="lab"><font color="Green" title="E87">A1</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> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T113" onmouseover="rs('funct_2/T113')" onmouseout="rh()">FUNCT_2:113</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="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="c1">m</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="E4:133_1_1_1"/><span class="lab"><font color="Green" title="E88">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="133_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:133_1_1_1_1"/>
<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> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>)</span>)</span>,<span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">m</font></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>)</span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<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="ref" href="number14.html#D19" target="_self" onmouseover="rs('number14/D19')" onmouseout="rh()">Def19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:133_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <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><span class="kw">end;</span></div>
<a NAME="E5:133_1_1_1"/><span class="lab"><font color="Green" title="E89">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="133_1_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">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="c8">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:133_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">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="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">fn</font> = <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="c8">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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="E3:133_1_1_1_2"/>
<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="c8">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>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">fn</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">D</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"><font color="Maroon" title="c9">fn</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>)</span>)</span>,<span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">fn</font> <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="p4">(<span class="default"><font color="Maroon" title="c9">fn</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a></span>)</span></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>)</span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<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="ref" href="number14.html#D19" target="_self" onmouseover="rs('number14/D19')" onmouseout="rh()">Def19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:133_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:133_1_1_1"/>
 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="#E4:133_1_1_1"><span class="lab"><font color="Green" title="E88">A2</font></span></a>, <a class="txt" href="#E5:133_1_1_1"><span class="lab"><font color="Green" title="E89">A3</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E7:133_1_1_1"/>
<font color="Maroon" title="c6">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="#E3:133_1_1_1"><span class="lab"><font color="Green" title="E87">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>


</div>
