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<b>let </b><font color="Maroon">f</font> be   <a href="seqm_3.html#NM1">Seq_of_Nat</a>, <font color="Maroon">g</font> be   <a href="seqm_3.html#NM1">Seq_of_Nat</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:25_1_2"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span> )
 <b>and </b><br/><a NAME="E2:25_1_2"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E3:25_1_2"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font> &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span> )
 <b>and </b><br/><a NAME="E4:25_1_2"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font>, <font color="Maroon">g</font> = <font color="Maroon">g</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E6:25_1_2"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E7:25_1_2"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font> &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E8:25_1_2"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>)
 
<b>by </b><i/>;<br/>
<a NAME="E9:25_1_2"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>,<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font>,<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>)
 
<b>by </b><i/>;<br/>
<a NAME="E10:25_1_2"/>
<font color="Maroon">f</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">g</font>
 
<b>from </b><i><a class="ref" href="recdef_2.html#S7">RECDEF_2:sch 7</a>(, , , );<br/></i>
<b>hence </b><a NAME="E11:25_1_2"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 ;<br/>


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