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<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c3">F</font>, <font color="Maroon" title="c4">G</font> be    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>





<b>reconsider </b><font color="Maroon" title="c5">FG</font> = <font color="Maroon" title="c3">F</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <font color="Maroon" title="c4">G</font> as    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<i><font color="Green" title="E14">A1</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <font color="Maroon" title="c4">G</font></span>)</span></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c5">FG</font></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c5">FG</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euler_2.html#D1" target="_self" title="EULER_2:def.1">Def1</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">FG</font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T6" title="NEWTON:th.6">NEWTON:6</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <font color="Maroon" title="c4">G</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/>
<i><font color="Green" title="E15">A2</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c3">F</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euler_2.html#D1" target="_self" title="EULER_2:def.1">Def1</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c3">F</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c4">G</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euler_2.html#D1" target="_self" title="EULER_2:def.1">Def1</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c4">G</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T6" title="NEWTON:th.6">NEWTON:6</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">G</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T6" title="NEWTON:th.6">NEWTON:6</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <font color="Maroon" title="c4">G</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <font color="Maroon" title="c4">G</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/>
<i><font color="Green" title="E16">A3</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <font color="Maroon" title="c4">G</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c5">FG</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">FG</font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T6" title="NEWTON:th.6">NEWTON:6</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <font color="Maroon" title="c4">G</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/>
<i><font color="Green" title="E17">A4</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c3">F</font></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c3">F</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">F</font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T6" title="NEWTON:th.6">NEWTON:6</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">F</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/>
<i><font color="Green" title="E18">A5</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c3">F</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euler_2.html#D1" target="_self" title="EULER_2:def.1">Def1</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">F</font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T6" title="NEWTON:th.6">NEWTON:6</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">F</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/>
<i><font color="Green" title="E19">A6</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c4">G</font></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c4">G</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">G</font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T6" title="NEWTON:th.6">NEWTON:6</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">G</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/>
<i><font color="Green" title="E20">A7</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c4">G</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euler_2.html#D1" target="_self" title="EULER_2:def.1">Def1</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">G</font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T6" title="NEWTON:th.6">NEWTON:6</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">G</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E9:20"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <font color="Maroon" title="c4">G</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <font color="Maroon" title="c4">G</font></span>)</span></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:20_8"/><i><font color="Green" title="E21">A8</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <font color="Maroon" title="c4">G</font></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c7">H</font> = <font color="Maroon" title="c3">F</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <font color="Maroon" title="c4">G</font> as    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c6">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="euler_2.html#E1:20_8"><i><font color="Green" title="E21">A8</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_8_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_8_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_8_1_1"/>
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">F</font> or  ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">G</font> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">n</font> ) )
 </span><b>by </b><i><a class="txt" href="euler_2.html#E1:20_8"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="finseq_1.html#T38" title="FINSEQ_1:th.38">FINSEQ_1:38</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_8_1_1_1"><b>suppose </b></a><a NAME="E1:20_8_1_1_1"/><i><font color="Green" title="E22">A9</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">F</font>
 ;<br/><div class="add"><i><font color="Green" title="E23">A10</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <font color="Maroon" title="c4">G</font></span>)</span></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <font color="Maroon" title="c4">G</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">m</font>
<b>by </b><i><a class="txt" href="euler_2.html#E4:20"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="euler_2.html#E1:20_8"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="euler_2.html#D1" target="_self" title="EULER_2:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">m</font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T66" title="RVSUM_1:th.66">RVSUM_1:66</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">m</font>
<b>by </b><i><a class="txt" href="euler_2.html#E1:20_8_1_1_1"><i><font color="Green" title="E22">A9</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">x</font>
<b>by </b><i><a class="txt" href="euler_2.html#E6:20"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="euler_2.html#E1:20_8_1_1_1"><i><font color="Green" title="E22">A9</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">m</font>
<b>by </b><i><a class="txt" href="euler_2.html#E5:20"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="euler_2.html#E1:20_8_1_1_1"><i><font color="Green" title="E22">A9</font></i></a>, <a class="ref" href="euler_2.html#D1" target="_self" title="EULER_2:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">m</font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T66" title="RVSUM_1:th.66">RVSUM_1:66</a></i>
;<br/><b>hence </b><a NAME="E4:20_8_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <font color="Maroon" title="c4">G</font></span>)</span></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="txt" href="euler_2.html#E2:20_8_1_1_1"><i><font color="Green" title="E23">A10</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_8_1_1_2"><b>suppose </b></a><a NAME="E1:20_8_1_1_2"/>
 ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">G</font> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">n</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c9">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:20_8_1_1_2"/><i><font color="Green" title="E22">A11</font></i>: 
( <font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">G</font> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">n</font> )
 ;<br/><i><font color="Green" title="E23">A12</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c3">F</font></span>)</span>
<b>by </b><i><a class="ref" href="euler_2.html#D1" target="_self" title="EULER_2:def.1">Def1</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">F</font>
<b>by </b><i><a class="ref" href="newton.html#T6" title="NEWTON:th.6">NEWTON:6</a></i>
;<br/><i><font color="Green" title="E24">A13</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <font color="Maroon" title="c4">G</font></span>)</span></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <font color="Maroon" title="c4">G</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">m</font>
<b>by </b><i><a class="txt" href="euler_2.html#E4:20"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="euler_2.html#E1:20_8"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="euler_2.html#D1" target="_self" title="EULER_2:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">H</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">m</font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T66" title="RVSUM_1:th.66">RVSUM_1:66</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">m</font>
<b>by </b><i><a class="txt" href="euler_2.html#E3:20_8_1_1_2"><i><font color="Green" title="E22">A11</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">n</font>
<b>by </b><i><a class="txt" href="euler_2.html#E8:20"><i><font color="Green" title="E20">A7</font></i></a>, <a class="txt" href="euler_2.html#E3:20_8_1_1_2"><i><font color="Green" title="E22">A11</font></i></a>, <a class="txt" href="euler_2.html#E4:20_8_1_1_2"><i><font color="Green" title="E23">A12</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">m</font>
<b>by </b><i><a class="txt" href="euler_2.html#E7:20"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="euler_2.html#E3:20_8_1_1_2"><i><font color="Green" title="E22">A11</font></i></a>, <a class="ref" href="euler_2.html#D1" target="_self" title="EULER_2:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">m</font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T66" title="RVSUM_1:th.66">RVSUM_1:66</a></i>
;<br/><b>hence </b><a NAME="E7:20_8_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <font color="Maroon" title="c4">G</font></span>)</span></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="txt" href="euler_2.html#E5:20_8_1_1_2"><i><font color="Green" title="E24">A13</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:20_8"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <font color="Maroon" title="c4">G</font></span>)</span></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:20"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <font color="Maroon" title="c4">G</font></span>)</span></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="euler_2.html#E2:20"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="euler_2.html#E3:20"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


</div>
