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<b>reconsider </b><font color="Maroon">T</font> =  <a href="radix_2.html#K6" target="_self">Table1</a> <font color="Maroon">q</font>,<font color="Maroon">c</font>,<font color="Maroon">f</font>,<font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="mesfunc1.html#K2">INT</a>  <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">i1</font>, <font color="Olive">i2</font> being  <a href="int_1.html#NM1">Integer</a> st <br/>( <font color="Olive">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Olive">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>3</sub></font> &amp; <font color="Olive">i2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">i1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="radix_2.html#K6" target="_self">Table1</a> <font color="Maroon">q</font>,<font color="Maroon">c</font>,<font color="Maroon">f</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">f</font> );<br/>
<a NAME="E2:22_1_1"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> holds <br/>for <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="mesfunc1.html#K2">INT</a>   ex <font color="Olive">y</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="mesfunc1.html#K2">INT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:22_1_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 ;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="mesfunc1.html#K2">INT</a> ;<br/>

<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<b>consider </b><font color="Maroon">y</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E4:22_1_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="radix_2.html#K6" target="_self">Table1</a> <font color="Maroon">q</font>,<font color="Maroon">c</font>,<font color="Maroon">f</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">f</font>
 ;<br/>
<b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="mesfunc1.html#K2">INT</a>  <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<b>take </b>
<font color="Maroon">z</font>
;<br/>

<b>thus </b><a NAME="E6:22_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">r</font> being    <a href="trees_4.html#M1">FinSequence</a> of  <a href="mesfunc1.html#K2">INT</a> <b> such that </b><br/><a NAME="E4:22_1_1"/><i><font color="Green">E35</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; ( <font color="Maroon">r</font> <a href="wsierp_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 ) )
 <b>and </b><br/><a NAME="E5:22_1_1"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Maroon">r</font> <a href="wsierp_1.html#K1">.</a> <font color="Olive">i</font>,<font color="Maroon">r</font> <a href="wsierp_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 <b>from </b><i><a class="ref" href="recdef_1.html#S6">RECDEF_1:sch 6</a>();<br/></i>
<b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">r</font> as   <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">n</font>,<a href="mesfunc1.html#K2">INT</a>  <b>by </b><i>, <a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<b>take </b>
<font color="Maroon">r</font>
;<br/>

<b>thus </b><a NAME="E7:22_1_1"/>
<font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="radix_2.html#K6" target="_self">Table1</a> <font color="Maroon">q</font>,<font color="Maroon">c</font>,<font color="Maroon">f</font>,<font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/>

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E8:22_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1 )
 ;<br/>

<a NAME="E9:22_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="sppol_1.html#T5">SPPOL_1:5</a></i>;<br/>
<b>hence </b><a NAME="E10:22_1_1"/>
 ex <font color="Olive">I1</font>, <font color="Olive">I2</font> being  <a href="int_1.html#NM1">Integer</a> st <br/>( <font color="Olive">I1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Olive">I2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Olive">I2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">I1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="radix_2.html#K6" target="_self">Table1</a> <font color="Maroon">q</font>,<font color="Maroon">c</font>,<font color="Maroon">f</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">f</font> )
 <b>by </b><i/>;<br/>


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