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<b>let </b><font color="Maroon" title="c1">p</font> be   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>;<br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c4">f</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>







<a NAME="E1:50"/><i><font color="Green" title="E36">A1</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>assume </b><a NAME="E2:50"/><i><font color="Green" title="E37">A2</font></i>: 
( <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 &amp; <font color="Maroon" title="c2">a</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p1">(<span class="default"><a href="newton.html#K4" title="NEWTON:func.4">idseq</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span> &amp; <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">p</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 ) </span>}</span>  )
 ;<br/>

<a NAME="E3:50"/><b>then </b><i><font color="Green" title="E38">A3</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="int_5.html#E3"><i><font color="Green" title="E3">Lm1</font></i></a></i>;<br/>
<a NAME="E4:50"/><b>then </b>
 <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E2:50"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="int_5.html#T28" target="_self" title="INT_5:th.28">Th28</a></i>;<br/>
<a NAME="E5:50"/><b>then </b><i><font color="Green" title="E39">A4</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span>, <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="ref" href="int_1.html#T35" title="INT_1:th.35">INT_1:35</a></i>;<br/>
<a NAME="E6:50"/><i><font color="Green" title="E40">A5</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">p'</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2;<br/>
<b>set </b><font color="Maroon" title="c6">f1</font> = <font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font>;<br/>
<b>set </b><font color="Maroon" title="c7">f2</font> =  <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c8">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">p</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 ) </span>}</span> ;<br/>
<a NAME="E7:50"/><i><font color="Green" title="E41">A6</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="newton.html#K4" title="NEWTON:func.4">idseq</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">I</font> =  <a href="newton.html#K4" title="NEWTON:func.4">idseq</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E9:50"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">I</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E2:50"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>;<br/>
<b>then </b><i><font color="Green" title="E42">A7</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">I</font>
<b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
<b>by </b><i><a class="ref" href="finseq_2.html#T55" title="FINSEQ_2:th.55">FINSEQ_2:55</a></i>
;<br/>
<a NAME="E11:50"/>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Product</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E6:50"><i><font color="Green" title="E40">A5</font></i></a>, <a class="ref" href="euler_2.html#T26" title="EULER_2:th.26">EULER_2:26</a></i>;<br/>
<a NAME="E12:50"/><b>then </b><i><font color="Green" title="E43">A8</font></i>: 
 <a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span>, <a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Product</a> <font color="Maroon" title="c4">f</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="ref" href="int_3.html#T12" title="INT_3:th.12">INT_3:12</a></i>;<br/>
<a NAME="E13:50"/>
<font color="Maroon" title="c9">I</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="finseq_2.html#T131" title="FINSEQ_2:th.131">FINSEQ_2:131</a></i>;<br/>
<a NAME="E14:50"/><b>then </b>
<font color="Maroon" title="c9">I</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="finseq_2.html#T129" title="FINSEQ_2:th.129">FINSEQ_2:129</a></i>;<br/>
<b>then </b><i><font color="Green" title="E44">A9</font></i>:  <a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Product</a> <font color="Maroon" title="c4">f</font> = 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Product</a> <font color="Maroon" title="c9">I</font></span>)</span>
<b>by </b><i><a class="txt" href="int_5.html#E2:50"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="rvsum_1.html#T138" title="RVSUM_1:th.138">RVSUM_1:138</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Product</a> <font color="Maroon" title="c9">I</font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#D1" title="NEWTON:def.1">NEWTON:def 1</a></i>
;<br/>
<a NAME="E16:50"/><i><font color="Green" title="E45">A10</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<b>by </b><i><a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/>
<a NAME="E17:50"/><b>then </b><i><font color="Green" title="E46">A11</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/>
<a NAME="E18:50"/>
<font color="Maroon" title="c1">p</font> <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
 
<b>by </b><i><a class="txt" href="int_5.html#E2:50"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E19:50"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E20:50"/><b>then </b><i><font color="Green" title="E47">A12</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="txt" href="int_5.html#E17:50"><i><font color="Green" title="E46">A11</font></i></a>, <a class="ref" href="nat_2.html#T15" title="NAT_2:th.15">NAT_2:15</a></i>;<br/>
<a NAME="E21:50"/>
not <font color="Maroon" title="c1">p</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 
<b>by </b><i><a class="txt" href="int_5.html#E2:50"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="pepin.html#T17" title="PEPIN:th.17">PEPIN:17</a></i>;<br/>
<a NAME="E22:50"/><b>then </b><i><font color="Green" title="E48">A13</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 is <a href="abian.html#V1" title="ABIAN:attr.1">even</a>
 
<b>by </b><i><a class="txt" href="int_5.html#E17:50"><i><font color="Green" title="E46">A11</font></i></a>, <a class="ref" href="hilbert3.html#T2" title="HILBERT3:th.2">HILBERT3:2</a></i>;<br/>
<a NAME="E23:50"/><b>then </b>
2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 
<b>by </b><i><a class="ref" href="pepin.html#T22" title="PEPIN:th.22">PEPIN:22</a></i>;<br/>
<a NAME="E24:50"/><b>then </b><i><font color="Green" title="E49">A14</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="nat_d.html#T3" title="NAT_D:th.3">NAT_D:3</a></i>;<br/>
<a NAME="E25:50"/><b>then </b><i><font color="Green" title="E50">A15</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 
<b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/>
<a NAME="E26:50"/>
<font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="int_5.html#E16:50"><i><font color="Green" title="E45">A10</font></i></a>, <a class="txt" href="int_5.html#E17:50"><i><font color="Green" title="E46">A11</font></i></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/>
<a NAME="E27:50"/><b>then </b><i><font color="Green" title="E51">A16</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 
<b>by </b><i><a class="txt" href="int_5.html#E25:50"><i><font color="Green" title="E50">A15</font></i></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></i>;<br/>
<a NAME="E28:50"/><i><font color="Green" title="E52">A17</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T148" title="XREAL_1:th.148">XREAL_1:148</a></i>;<br/>
<a NAME="E29:50"/><b>then </b><i><font color="Green" title="E53">A18</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E17:50"><i><font color="Green" title="E46">A11</font></i></a>, <a class="txt" href="int_5.html#E27:50"><i><font color="Green" title="E51">A16</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<i><font color="Green" title="E54">A19</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
<b>by </b><i><a class="txt" href="int_5.html#E22:50"><i><font color="Green" title="E48">A13</font></i></a>, <a class="ref" href="nat_2.html#T28" title="NAT_2:th.28">NAT_2:28</a></i>
<br/>.= 
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
<b>by </b><i><a class="txt" href="int_5.html#E16:50"><i><font color="Green" title="E45">A10</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>
;<br/>
<a NAME="E31:50"/>
for <font color="Olive" title="b1">d</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>   st <font color="Olive" title="b1">d</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="c9">I</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Olive" title="b1">d</font></span>)</span> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">d</font> be    <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> ;<br/>

<b>assume </b><a NAME="E1:50_4"/>
<font color="Maroon" title="c10">d</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="c9">I</font>
 ;<br/>

<a NAME="E2:50_4"/><b>then </b>
<font color="Maroon" title="c10">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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="c9">I</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/>
<a NAME="E3:50_4"/><b>then </b><i><font color="Green" title="E55">A20</font></i>: 
<font color="Maroon" title="c10">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T55" title="FINSEQ_2:th.55">FINSEQ_2:55</a></i>;<br/>
<a NAME="E4:50_4"/><b>then </b><i><font color="Green" title="E56">A21</font></i>: 
<font color="Maroon" title="c9">I</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c10">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">d</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E20:50"><i><font color="Green" title="E47">A12</font></i></a>, <a class="ref" href="finseq_2.html#T57" title="FINSEQ_2:th.57">FINSEQ_2:57</a></i>;<br/>
<a NAME="E5:50_4"/>
( <font color="Maroon" title="c10">d</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c10">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 )
 
<b>by </b><i><a class="txt" href="int_5.html#E3:50_4"><i><font color="Green" title="E55">A20</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E6:50_4"/><b>then </b>
( <font color="Maroon" title="c10">d</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c10">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">p</font> )
 
<b>by </b><i><a class="txt" href="int_5.html#E29:50"><i><font color="Green" title="E53">A18</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:50_4"/><b>then </b>
<font color="Maroon" title="c10">d</font>,<font color="Maroon" title="c1">p</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 
<b>by </b><i><a class="txt" href="int_5.html#E6:50"><i><font color="Green" title="E40">A5</font></i></a>, <a class="ref" href="euler_1.html#T3" title="EULER_1:th.3">EULER_1:3</a></i>;<br/>
<b>hence </b><a NAME="E8:50_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c10">d</font></span>)</span> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="int_5.html#E4:50_4"><i><font color="Green" title="E56">A21</font></i></a>, <a class="ref" href="int_2.html#D6" title="INT_2:def.6">INT_2:def 6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E32:50"/><b>then </b>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Product</a> <font color="Maroon" title="c9">I</font></span>)</span> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="int_5.html#E6:50"><i><font color="Green" title="E40">A5</font></i></a>, <a class="ref" href="wsierp_1.html#T43" title="WSIERP_1:th.43">WSIERP_1:43</a></i>;<br/>
<a NAME="E33:50"/><b>then </b><i><font color="Green" title="E55">A22</font></i>: 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Product</a> <font color="Maroon" title="c9">I</font></span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="int_5.html#E3"><i><font color="Green" title="E3">Lm1</font></i></a></i>;<br/>
<a NAME="E34:50"/><i><font color="Green" title="E56">A23</font></i>: 
for <font color="Olive" title="b1">d</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">d</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">f</font> holds <br/><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:50_5"/><i><font color="Green" title="E57">A24</font></i>: 
<font color="Maroon" title="c10">d</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">f</font>
 ;<br/>

<a NAME="E2:50_5"/><b>then </b>
<font color="Maroon" title="c10">d</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="c9">I</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E2:50"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>;<br/>
<a NAME="E3:50_5"/><b>then </b>
<font color="Maroon" title="c10">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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="c9">I</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/>
<a NAME="E4:50_5"/><b>then </b><i><font color="Green" title="E58">A25</font></i>: 
<font color="Maroon" title="c10">d</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <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"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T55" title="FINSEQ_2:th.55">FINSEQ_2:55</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">d</font> = 
<font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">d</font></span>)</span>
<b>by </b><i><a class="txt" href="int_5.html#E2:50"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="int_5.html#E1:50_5"><i><font color="Green" title="E57">A24</font></i></a>, <a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>
<br/>.= 
<font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">d</font>
<b>by </b><i><a class="txt" href="int_5.html#E20:50"><i><font color="Green" title="E47">A12</font></i></a>, <a class="txt" href="int_5.html#E4:50_5"><i><font color="Green" title="E58">A25</font></i></a>, <a class="ref" href="finseq_2.html#T57" title="FINSEQ_2:th.57">FINSEQ_2:57</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E35:50"/><i><font color="Green" title="E57">A26</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="ref" href="euler_2.html#D1" title="EULER_2:def.1">EULER_2:def 1</a></i>;<br/>
<a NAME="E36:50"/><b>then </b><i><font color="Green" title="E58">A27</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="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E37:50"/>
<font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="int_5.html#E10:50"><i><font color="Green" title="E42">A7</font></i></a>, <a class="txt" href="int_5.html#E20:50"><i><font color="Green" title="E47">A12</font></i></a>, <a class="txt" href="int_5.html#E35:50"><i><font color="Green" title="E57">A26</font></i></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E38:50"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T27" title="FINSEQ_1:th.27">FINSEQ_1:27</a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">n1</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="E40:50"/><i><font color="Green" title="E59">A28</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c10">n1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="heyting3.html#T3" title="HEYTING3:th.3">HEYTING3:3</a></i>;<br/>
<a NAME="E41:50"/>
not  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:50_6"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:50_6"/><i><font color="Green" title="E60">A29</font></i>: 
( <font color="Maroon" title="c11">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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">a</font> = <font color="Maroon" title="c2">a</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="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">p</font>
<b>by </b><i><a class="txt" href="int_5.html#E36:50"><i><font color="Green" title="E58">A27</font></i></a>, <a class="txt" href="int_5.html#E3:50_6"><i><font color="Green" title="E60">A29</font></i></a>, <a class="ref" href="euler_2.html#D1" title="EULER_2:def.1">EULER_2:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">p</font>
<b>by </b><i><a class="txt" href="int_5.html#E34:50"><i><font color="Green" title="E56">A23</font></i></a>, <a class="txt" href="int_5.html#E36:50"><i><font color="Green" title="E58">A27</font></i></a>, <a class="txt" href="int_5.html#E3:50_6"><i><font color="Green" title="E60">A29</font></i></a></i>
;<br/>
<a NAME="E6:50_6"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c12">a</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c11">n</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E6:50"><i><font color="Green" title="E40">A5</font></i></a>, <a class="ref" href="pepin.html#T6" title="PEPIN:th.6">PEPIN:6</a></i>;<br/>
<a NAME="E7:50_6"/><b>then </b><i><font color="Green" title="E61">A30</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c11">n</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E2:50"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="int_5.html#E6:50"><i><font color="Green" title="E40">A5</font></i></a>, <a class="ref" href="wsierp_1.html#T37" title="WSIERP_1:th.37">WSIERP_1:37</a></i>;<br/>
<a NAME="E8:50_6"/><i><font color="Green" title="E62">A31</font></i>: 
( <font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 )
 
<b>by </b><i><a class="txt" href="int_5.html#E10:50"><i><font color="Green" title="E42">A7</font></i></a>, <a class="txt" href="int_5.html#E35:50"><i><font color="Green" title="E57">A26</font></i></a>, <a class="txt" href="int_5.html#E3:50_6"><i><font color="Green" title="E60">A29</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E9:50_6"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E7:50_6"><i><font color="Green" title="E61">A30</font></i></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></i>;<br/>
<b>hence </b><a NAME="E10:50_6"/>
contradiction
 <b>by </b><i><a class="txt" href="int_5.html#E29:50"><i><font color="Green" title="E53">A18</font></i></a>, <a class="txt" href="int_5.html#E8:50_6"><i><font color="Green" title="E62">A31</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E42:50"/><b>then </b><i><font color="Green" title="E60">A32</font></i>: 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T56" title="ZFMISC_1:th.56">ZFMISC_1:56</a></i>;<br/>
<a NAME="E43:50"/><b>then </b><i><font color="Green" title="E61">A33</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c10">n1</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E40:50"><i><font color="Green" title="E59">A28</font></i></a>, <a class="ref" href="xboole_1.html#T73" title="XBOOLE_1:th.73">XBOOLE_1:73</a></i>;<br/>
<a NAME="E44:50"/><b>then </b><i><font color="Green" title="E62">A34</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></i>;<br/>
<a NAME="E45:50"/>
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> <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <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>  : ( <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> &amp; <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">p</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 ) </span>}</span>  holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:50_7"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">p</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">k</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="E3:50_7"/><i><font color="Green" title="E63">A35</font></i>: 
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">k</font> &amp; <font color="Maroon" title="c12">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> &amp; <font color="Maroon" title="c12">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">p</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 )
 ;<br/>
<b>thus </b><a NAME="E4:50_7"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_5.html#E3:50_7"><i><font color="Green" title="E63">A35</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E46:50"/><b>then </b><i><font color="Green" title="E63">A36</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">p</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">p</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 ) </span>}</span>  as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/>
<a NAME="E48:50"/><i><font color="Green" title="E64">A37</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">X</font> = <font color="Maroon" title="c11">X</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="int_5.html#E46:50"><i><font color="Green" title="E63">A36</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E50:50"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">X</font> is    <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> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">m</font> = <font color="Maroon" title="c3">m</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="int_5.html#E2:50"><i><font color="Green" title="E37">A2</font></i></a></i>;<br/>
<a NAME="E52:50"/><i><font color="Green" title="E65">A38</font></i>: 
<span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c12">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">Y</font> = <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c12">X</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="int_5.html#E48:50"><i><font color="Green" title="E64">A37</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E54:50"/><i><font color="Green" title="E66">A39</font></i>: 
for <font color="Olive" title="b1">d</font>, <font color="Olive" title="b2">e</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>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">d</font> &amp; <font color="Olive" title="b1">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">e</font> &amp; <font color="Olive" title="b2">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Olive" title="b2">e</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">d</font>, <font color="Maroon" title="c16">e</font> be    <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> ;<br/>

<b>assume </b><a NAME="E1:50_8"/><i><font color="Green" title="E67">A40</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">d</font> &amp; <font color="Maroon" title="c15">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">e</font> &amp; <font color="Maroon" title="c16">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> )
 ;<br/>

<a NAME="E2:50_8"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">d</font> &amp; <font color="Maroon" title="c15">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">e</font> &amp; <font color="Maroon" title="c16">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> )
 
<b>by </b><i><a class="ref" href="euler_2.html#D1" title="EULER_2:def.1">EULER_2:def 1</a></i>;<br/>
<a NAME="E3:50_8"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">d</font> &amp; <font color="Maroon" title="c15">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">e</font> &amp; <font color="Maroon" title="c16">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E4:50_8"/><b>then </b><i><font color="Green" title="E68">A41</font></i>: 
( <font color="Maroon" title="c15">d</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">f</font> &amp; <font color="Maroon" title="c16">e</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">f</font> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E5:50_8"/><b>then </b><i><font color="Green" title="E69">A42</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c15">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c15">d</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">p</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c16">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c16">e</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">p</font> )
 
<b>by </b><i><a class="ref" href="euler_2.html#D1" title="EULER_2:def.1">EULER_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_8_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_8_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c15">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c16">e</font>
 ;<br/><a NAME="E2:50_8_1"/><b>then </b>
<font color="Maroon" title="c4">f</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c16">e</font>,<font color="Maroon" title="c4">f</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c15">d</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c1">p</font>
 <b>by </b><i><a class="txt" href="int_5.html#E5:50_8"><i><font color="Green" title="E69">A42</font></i></a>, <a class="ref" href="int_3.html#T12" title="INT_3:th.12">INT_3:12</a></i>;<br/><a NAME="E3:50_8_1"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c16">e</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c15">d</font></span>)</span>
 <b>by </b><i><a class="ref" href="int_2.html#T19" title="INT_2:th.19">INT_2:19</a></i>;<br/><a NAME="E4:50_8_1"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c16">e</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c15">d</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_5.html#E34:50"><i><font color="Green" title="E56">A23</font></i></a>, <a class="txt" href="int_5.html#E4:50_8"><i><font color="Green" title="E68">A41</font></i></a></i>;<br/><a NAME="E5:50_8_1"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c16">e</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c15">d</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_5.html#E34:50"><i><font color="Green" title="E56">A23</font></i></a>, <a class="txt" href="int_5.html#E4:50_8"><i><font color="Green" title="E68">A41</font></i></a></i>;<br/><a NAME="E6:50_8_1"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">e</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">d</font></span>)</span>
 ;<br/><a NAME="E7:50_8_1"/><b>then </b><i><font color="Green" title="E70">A43</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c16">e</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">d</font>
 <b>by </b><i><a class="txt" href="int_5.html#E3:50"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="wsierp_1.html#T36" title="WSIERP_1:th.36">WSIERP_1:36</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c17">dd</font> = <font color="Maroon" title="c16">e</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">d</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="int_5.html#E1:50_8"><i><font color="Green" title="E67">A40</font></i></a>, <a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a></i>;<br/><a NAME="E9:50_8_1"/><i><font color="Green" title="E71">A44</font></i>: 
<font color="Maroon" title="c17">dd</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="int_5.html#E1:50_8"><i><font color="Green" title="E67">A40</font></i></a></i>;<br/><a NAME="E10:50_8_1"/>
(  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> &amp;  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c17">dd</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">dd</font> )
 <b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><a NAME="E11:50_8_1"/><b>then </b><i><font color="Green" title="E72">A45</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">dd</font>
 <b>by </b><i><a class="txt" href="int_5.html#E7:50_8_1"><i><font color="Green" title="E70">A43</font></i></a>, <a class="txt" href="int_5.html#E9:50_8_1"><i><font color="Green" title="E71">A44</font></i></a>, <a class="ref" href="int_4.html#T6" title="INT_4:th.6">INT_4:6</a></i>;<br/><a NAME="E12:50_8_1"/><i><font color="Green" title="E73">A46</font></i>: 
<font color="Maroon" title="c17">dd</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="txt" href="int_5.html#E10:50"><i><font color="Green" title="E42">A7</font></i></a>, <a class="txt" href="int_5.html#E35:50"><i><font color="Green" title="E57">A26</font></i></a>, <a class="txt" href="int_5.html#E1:50_8"><i><font color="Green" title="E67">A40</font></i></a>, <a class="ref" href="xreal_1.html#T15" title="XREAL_1:th.15">XREAL_1:15</a></i>;<br/><a NAME="E13:50_8_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">p</font>
 <b>by </b><i><a class="txt" href="int_5.html#E29:50"><i><font color="Green" title="E53">A18</font></i></a>, <a class="ref" href="xreal_1.html#T149" title="XREAL_1:th.149">XREAL_1:149</a></i>;<br/><b>hence </b><a NAME="E14:50_8_1"/>
contradiction
 <b>by </b><i><a class="txt" href="int_5.html#E11:50_8_1"><i><font color="Green" title="E72">A45</font></i></a>, <a class="txt" href="int_5.html#E12:50_8_1"><i><font color="Green" title="E73">A46</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:50_8"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c15">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c16">e</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E55:50"/><b>then </b><i><font color="Green" title="E67">A47</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E35:50"><i><font color="Green" title="E57">A26</font></i></a>, <a class="ref" href="graph_5.html#T10" title="GRAPH_5:th.10">GRAPH_5:10</a></i>;<br/>
<a NAME="E56:50"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">X</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E46:50"><i><font color="Green" title="E63">A36</font></i></a>, <a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c15">n</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c13">m</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="int_5.html#E2:50"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="int_5.html#E10:50"><i><font color="Green" title="E42">A7</font></i></a>, <a class="txt" href="int_5.html#E55:50"><i><font color="Green" title="E67">A47</font></i></a>, <a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a></i>;<br/>
<a NAME="E58:50"/><i><font color="Green" title="E68">A48</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E54:50"><i><font color="Green" title="E66">A39</font></i></a>, <a class="ref" href="graph_5.html#T10" title="GRAPH_5:th.10">GRAPH_5:10</a></i>;<br/>
<a NAME="E59:50"/><b>then </b><i><font color="Green" title="E69">A49</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="finseq_4.html#T77" title="FINSEQ_4:th.77">FINSEQ_4:77</a></i>;<br/>
<a NAME="E60:50"/><i><font color="Green" title="E70">A50</font></i>: 
 <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="int_5.html#E43:50"><i><font color="Green" title="E61">A33</font></i></a>, <a class="ref" href="finseq_3.html#T99" title="FINSEQ_3:th.99">FINSEQ_3:99</a></i>;<br/>
<a NAME="E61:50"/><b>then </b>
<font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font>, <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="rfinseq.html#R1" title="RFINSEQ:pred.1">are_fiberwise_equipotent</a> 
 
<b>by </b><i><a class="txt" href="int_5.html#E44:50"><i><font color="Green" title="E62">A34</font></i></a>, <a class="txt" href="int_5.html#E59:50"><i><font color="Green" title="E69">A49</font></i></a>, <a class="ref" href="rfinseq.html#T39" title="RFINSEQ:th.39">RFINSEQ:39</a></i>;<br/>
<a NAME="E62:50"/><b>then </b><i><font color="Green" title="E71">A51</font></i>: 
 <a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="euler_2.html#T25" title="EULER_2:th.25">EULER_2:25</a></i>;<br/>
<a NAME="E63:50"/><i><font color="Green" title="E72">A52</font></i>: 
( <font color="Maroon" title="c12">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c10">n1</font> &amp; <font color="Maroon" title="c14">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c10">n1</font> )
 
<b>by </b><i><a class="txt" href="int_5.html#E43:50"><i><font color="Green" title="E61">A33</font></i></a>, <a class="txt" href="int_5.html#E46:50"><i><font color="Green" title="E63">A36</font></i></a>, <a class="txt" href="int_5.html#E52:50"><i><font color="Green" title="E65">A38</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_9"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:50_9"/>
( <font color="Maroon" title="c12">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  not <font color="Maroon" title="c12">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_9_1"><b>suppose </b></a><a NAME="E1:50_9_1"/><i><font color="Green" title="E73">A53</font></i>: 
<font color="Maroon" title="c12">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><a NAME="E2:50_9_1"/>
for <font color="Olive" title="b1">d</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">d</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="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> holds <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_9_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:50_9_1_1"/><i><font color="Green" title="E74">A54</font></i>: 
<font color="Maroon" title="c16">d</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="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c17">d</font> = <font color="Maroon" title="c16">d</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="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E3:50_9_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c17">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> &amp; not <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c17">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">X</font> )
 
<b>by </b><i><a class="txt" href="int_5.html#E1:50_9_1"><i><font color="Green" title="E73">A53</font></i></a>, <a class="txt" href="int_5.html#E1:50_9_1_1"><i><font color="Green" title="E74">A54</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E4:50_9_1_1"/><b>then </b><i><font color="Green" title="E75">A55</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c17">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">p</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 
;<br/>
<a NAME="E5:50_9_1_1"/><i><font color="Green" title="E76">A56</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c17">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E1:50_9_1_1"><i><font color="Green" title="E74">A54</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E6:50_9_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c17">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E7:50_9_1_1"/><b>then </b>
not <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c17">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="int_5.html#E42:50"><i><font color="Green" title="E60">A32</font></i></a>, <a class="txt" href="int_5.html#E5:50_9_1_1"><i><font color="Green" title="E76">A56</font></i></a>, <a class="ref" href="xboole_0.html#T5" title="XBOOLE_0:th.5">XBOOLE_0:5</a></i>;<br/>
<a NAME="E8:50_9_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c17">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E9:50_9_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">d</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 )
 <b>by </b><i><a class="txt" href="int_5.html#E30:50"><i><font color="Green" title="E54">A19</font></i></a>, <a class="txt" href="int_5.html#E4:50_9_1_1"><i><font color="Green" title="E75">A55</font></i></a>, <a class="ref" href="int_1.html#T81" title="INT_1:th.81">INT_1:81</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:50_9_1"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">I</font>
 <b>by </b><i><a class="txt" href="int_5.html#E7:50"><i><font color="Green" title="E41">A6</font></i></a>, <a class="txt" href="int_5.html#E10:50"><i><font color="Green" title="E42">A7</font></i></a>, <a class="txt" href="int_5.html#E35:50"><i><font color="Green" title="E57">A26</font></i></a>, <a class="txt" href="int_5.html#E59:50"><i><font color="Green" title="E69">A49</font></i></a>, <a class="ref" href="int_5.html#T40" target="_self" title="INT_5:th.40">Th40</a></i>;<br/><a NAME="E4:50_9_1"/><b>then </b>
<font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c9">I</font> <a href="rfinseq.html#R1" title="RFINSEQ:pred.1">are_fiberwise_equipotent</a> 
 <b>by </b><i><a class="txt" href="int_5.html#E59:50"><i><font color="Green" title="E69">A49</font></i></a>, <a class="ref" href="rfinseq.html#T39" title="RFINSEQ:th.39">RFINSEQ:39</a></i>;<br/><a NAME="E5:50_9_1"/><b>then </b>
 <a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Product</a> <font color="Maroon" title="c9">I</font>
 <b>by </b><i><a class="ref" href="euler_2.html#T25" title="EULER_2:th.25">EULER_2:25</a></i>;<br/><a NAME="E6:50_9_1"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Product</a> <font color="Maroon" title="c9">I</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Product</a> <font color="Maroon" title="c9">I</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="int_5.html#E12:50"><i><font color="Green" title="E43">A8</font></i></a>, <a class="txt" href="int_5.html#E15:50"><i><font color="Green" title="E44">A9</font></i></a>, <a class="ref" href="int_2.html#T19" title="INT_2:th.19">INT_2:19</a></i>;<br/><a NAME="E7:50_9_1"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Product</a> <font color="Maroon" title="c9">I</font></span>)</span>
 ;<br/><a NAME="E8:50_9_1"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="int_5.html#E33:50"><i><font color="Green" title="E55">A22</font></i></a>, <a class="ref" href="wsierp_1.html#T36" title="WSIERP_1:th.36">WSIERP_1:36</a></i>;<br/><a NAME="E9:50_9_1"/><b>then </b>
1,<font color="Maroon" title="c2">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c1">p</font>
 <b>by </b><i><a class="ref" href="int_2.html#T19" title="INT_2:th.19">INT_2:19</a></i>;<br/><a NAME="E10:50_9_1"/><b>then </b><i><font color="Green" title="E74">A57</font></i>: 
1, <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c1">p</font>
 <b>by </b><i><a class="txt" href="int_5.html#E5:50"><i><font color="Green" title="E39">A4</font></i></a>, <a class="ref" href="int_1.html#T36" title="INT_1:th.36">INT_1:36</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_9_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_9_1_2"/>
 <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1
 ;<br/><a NAME="E2:50_9_1_2"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="int_5.html#E10:50_9_1"><i><font color="Green" title="E74">A57</font></i></a>, <a class="ref" href="int_2.html#T19" title="INT_2:th.19">INT_2:19</a></i>;<br/><b>hence </b><a NAME="E3:50_9_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="int_5.html#E2:50"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></i>;<br/></div><b>end;</b></div><a NAME="E12:50_9_1"/><b>then </b>
 <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="int_5.html#T25" target="_self" title="INT_5:th.25">Th25</a></i>;<br/><b>hence </b><a NAME="E13:50_9_1"/>
 <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">m</font>
 <b>by </b><i><a class="txt" href="int_5.html#E2:50"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="int_5.html#E1:50_9_1"><i><font color="Green" title="E73">A53</font></i></a>, <a class="ref" href="card_1.html#T78" title="CARD_1:th.78">CARD_1:78</a>, <a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_9_2"><b>suppose </b></a><a NAME="E1:50_9_2"/><i><font color="Green" title="E73">A58</font></i>: 
not <font color="Maroon" title="c12">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><a NAME="E2:50_9_2"/><i><font color="Green" title="E74">A59</font></i>: 
 <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c14">Y</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">^</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c12">X</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_9_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:50_9_2_1"/>
for <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">l</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>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Y</font> &amp; <font color="Olive" title="b2">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">X</font> holds <br/><font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">l</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_9_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">k</font>, <font color="Maroon" title="c17">l</font> be    <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> ;<br/>

<b>assume </b><a NAME="E1:50_9_2_1_1"/><i><font color="Green" title="E75">A60</font></i>: 
( <font color="Maroon" title="c16">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Y</font> &amp; <font color="Maroon" title="c17">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">X</font> )
 ;<br/>

<a NAME="E2:50_9_2_1_1"/><b>then </b>
( <font color="Maroon" title="c16">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> &amp; not <font color="Maroon" title="c16">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">X</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E3:50_9_2_1_1"/><b>then </b><i><font color="Green" title="E76">A61</font></i>: 
<font color="Maroon" title="c16">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">p</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 
;<br/>
<a NAME="E4:50_9_2_1_1"/>
 ex <font color="Olive" title="b1">l1</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>  st <br/>( <font color="Olive" title="b1">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">l</font> &amp; <font color="Olive" title="b1">l1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> &amp; <font color="Olive" title="b1">l1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">p</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 )
 
<b>by </b><i><a class="txt" href="int_5.html#E1:50_9_2_1_1"><i><font color="Green" title="E75">A60</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:50_9_2_1_1"/>
<font color="Maroon" title="c16">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">l</font>
 <b>by </b><i><a class="txt" href="int_5.html#E3:50_9_2_1_1"><i><font color="Green" title="E76">A61</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:50_9_2_1"/><b>then </b>
 <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">Y</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c12">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c14">Y</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">^</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c12">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E63:50"><i><font color="Green" title="E72">A52</font></i></a>, <a class="ref" href="finseq_3.html#T48" title="FINSEQ_3:th.48">FINSEQ_3:48</a></i>;<br/>
<a NAME="E3:50_9_2_1"/><b>then </b>
 <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c14">Y</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">^</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c12">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T39" title="XBOOLE_1:th.39">XBOOLE_1:39</a></i>;<br/>
<b>hence </b><a NAME="E4:50_9_2_1"/>
 <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c14">Y</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">^</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c12">X</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_5.html#E46:50"><i><font color="Green" title="E63">A36</font></i></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:50_9_2"/><i><font color="Green" title="E75">A62</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c14">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_9_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:50_9_2_2"/>
<font color="Maroon" title="c14">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c10">n1</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E43:50"><i><font color="Green" title="E61">A33</font></i></a>, <a class="txt" href="int_5.html#E52:50"><i><font color="Green" title="E65">A38</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>then </b> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c14">Y</font></span>)</span> = 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c14">Y</font>
<b>by </b><i><a class="ref" href="finseq_3.html#T44" title="FINSEQ_3:th.44">FINSEQ_3:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c13">m</font>
<b>by </b><i><a class="txt" href="int_5.html#E2:50"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="int_5.html#E10:50"><i><font color="Green" title="E42">A7</font></i></a>, <a class="txt" href="int_5.html#E35:50"><i><font color="Green" title="E57">A26</font></i></a>, <a class="txt" href="int_5.html#E46:50"><i><font color="Green" title="E63">A36</font></i></a>, <a class="txt" href="int_5.html#E58:50"><i><font color="Green" title="E68">A48</font></i></a>, <a class="ref" href="card_2.html#T63" title="CARD_2:th.63">CARD_2:63</a></i>
;<br/>
<b>hence </b><a NAME="E3:50_9_2_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c14">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E4:50_9_2"/><b>then </b><i><font color="Green" title="E76">A63</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c12">X</font>
 <b>by </b><i><a class="txt" href="int_5.html#E2:50_9_2"><i><font color="Green" title="E74">A59</font></i></a>, <a class="ref" href="finseq_5.html#T40" title="FINSEQ_5:th.40">FINSEQ_5:40</a></i>;<br/><a NAME="E5:50_9_2"/>
 <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="rfinseq.html#T21" title="RFINSEQ:th.21">RFINSEQ:21</a></i>;<br/><a NAME="E6:50_9_2"/><b>then </b><i><font color="Green" title="E77">A64</font></i>: 
( <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c15">n</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 <b>by </b><i><a class="txt" href="int_5.html#E60:50"><i><font color="Green" title="E70">A50</font></i></a>, <a class="ref" href="finseq_3.html#T98" title="FINSEQ_3:th.98">FINSEQ_3:98</a></i>;<br/><a NAME="E7:50_9_2"/>
 <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c14">Y</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="int_5.html#E2:50_9_2"><i><font color="Green" title="E74">A59</font></i></a>, <a class="ref" href="finseq_6.html#T12" title="FINSEQ_6:th.12">FINSEQ_6:12</a></i>;<br/><a NAME="E8:50_9_2"/><b>then </b><i><font color="Green" title="E78">A65</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c15">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c14">Y</font>
 <b>by </b><i><a class="txt" href="int_5.html#E3:50_9_2"><i><font color="Green" title="E75">A62</font></i></a>, <a class="ref" href="finseq_3.html#T122" title="FINSEQ_3:th.122">FINSEQ_3:122</a></i>;<br/><a NAME="E9:50_9_2"/><i><font color="Green" title="E79">A66</font></i>: 
<font color="Maroon" title="c13">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="int_5.html#E2:50"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="int_5.html#E1:50_9_2"><i><font color="Green" title="E73">A58</font></i></a>, <a class="ref" href="card_4.html#T17" title="CARD_4:th.17">CARD_4:17</a></i>;<br/><a NAME="E10:50_9_2"/><i><font color="Green" title="E80">A67</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 <b>by </b><i><a class="txt" href="int_5.html#E10:50"><i><font color="Green" title="E42">A7</font></i></a>, <a class="txt" href="int_5.html#E35:50"><i><font color="Green" title="E57">A26</font></i></a>, <a class="txt" href="int_5.html#E43:50"><i><font color="Green" title="E61">A33</font></i></a>, <a class="txt" href="int_5.html#E58:50"><i><font color="Green" title="E68">A48</font></i></a>, <a class="ref" href="finseq_3.html#T44" title="FINSEQ_3:th.44">FINSEQ_3:44</a></i>;<br/><a NAME="E11:50_9_2"/><b>then </b><i><font color="Green" title="E81">A68</font></i>: 
<font color="Maroon" title="c15">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T45" title="XREAL_1:th.45">XREAL_1:45</a></i>;<br/><a NAME="E12:50_9_2"/><b>then </b><i><font color="Green" title="E82">A69</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"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/><b>set </b><font color="Maroon" title="c16">f3</font> = <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"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p5">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c17">f4</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span>;<br/><i><font color="Green" title="E83">A70</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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D4" title="SEQ_1:def.4">SEQ_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</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/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p5">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_2.html#T69" title="FINSEQ_2:th.69">FINSEQ_2:69</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</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#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span>

;<br/><a NAME="E14:50_9_2"/><b>then </b><i><font color="Green" title="E84">A71</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"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><i><font color="Green" title="E85">A72</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"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span> = 
<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"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">n</font>
<b>by </b><i><a class="ref" href="binarith.html#T76" title="BINARITH:th.76">BINARITH:76</a></i>
<br/>.= 
<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"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">n</font>
<b>by </b><i><a class="txt" href="int_5.html#E11:50_9_2"><i><font color="Green" title="E81">A68</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon" title="c13">m</font>
<b>by </b><i><a class="txt" href="int_5.html#E10:50_9_2"><i><font color="Green" title="E80">A67</font></i></a></i>
;<br/><a NAME="E16:50_9_2"/><i><font color="Green" title="E86">A73</font></i>: 
for <font color="Olive" title="b1">d</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">d</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_9_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:50_9_2_5"/><i><font color="Green" title="E87">A74</font></i>: 
<font color="Maroon" title="c18">d</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:50_9_2_5"/><b>then </b><i><font color="Green" title="E88">A75</font></i>: 
<font color="Maroon" title="c18">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E13:50_9_2"><i><font color="Green" title="E83">A70</font></i></a>, <a class="txt" href="int_5.html#E1:50_9_2_5"><i><font color="Green" title="E87">A74</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:50_9_2_5"/>
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E4:50_9_2_5"/><b>then </b>
<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"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p5">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E9:50_9_2"><i><font color="Green" title="E79">A66</font></i></a>, <a class="txt" href="int_5.html#E15:50_9_2"><i><font color="Green" title="E85">A72</font></i></a>, <a class="txt" href="int_5.html#E2:50_9_2_5"><i><font color="Green" title="E88">A75</font></i></a>, <a class="ref" href="finseq_2.html#T71" title="FINSEQ_2:th.71">FINSEQ_2:71</a></i>;<br/>
<b>hence </b><a NAME="E5:50_9_2_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">d</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_5.html#E1:50_9_2_5"><i><font color="Green" title="E87">A74</font></i></a>, <a class="ref" href="seq_1.html#D4" title="SEQ_1:def.4">SEQ_1:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E17:50_9_2"/><i><font color="Green" title="E87">A76</font></i>: 
for <font color="Olive" title="b1">d</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">d</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> holds <br/>( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_9_2_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:50_9_2_6"/><i><font color="Green" title="E88">A77</font></i>: 
<font color="Maroon" title="c18">d</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:50_9_2_6"/><b>then </b><i><font color="Green" title="E89">A78</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">d</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E16:50_9_2"><i><font color="Green" title="E86">A73</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c19">w</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">d</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  <b>by </b><i><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>;<br/>
<a NAME="E4:50_9_2_6"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c12">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c12">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E4:50_9_2"><i><font color="Green" title="E76">A63</font></i></a>, <a class="txt" href="int_5.html#E13:50_9_2"><i><font color="Green" title="E83">A70</font></i></a>, <a class="txt" href="int_5.html#E1:50_9_2_6"><i><font color="Green" title="E88">A77</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E5:50_9_2_6"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c12">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">X</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E63:50"><i><font color="Green" title="E72">A52</font></i></a>, <a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c20">ll</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="E7:50_9_2_6"/><i><font color="Green" title="E90">A79</font></i>: 
( <font color="Maroon" title="c20">ll</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c12">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">d</font> &amp; <font color="Maroon" title="c20">ll</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> &amp; <font color="Maroon" title="c20">ll</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">p</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 )
 ;<br/>
<a NAME="E8:50_9_2_6"/><i><font color="Green" title="E91">A80</font></i>: 
<font color="Maroon" title="c19">w</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">p</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E4:50_9_2"><i><font color="Green" title="E76">A63</font></i></a>, <a class="txt" href="int_5.html#E2:50_9_2_6"><i><font color="Green" title="E89">A78</font></i></a>, <a class="txt" href="int_5.html#E7:50_9_2_6"><i><font color="Green" title="E90">A79</font></i></a>, <a class="ref" href="xreal_1.html#T12" title="XREAL_1:th.12">XREAL_1:12</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c21">e</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E10:50_9_2_6"/><i><font color="Green" title="E92">A81</font></i>: 
( <font color="Maroon" title="c21">e</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="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">d</font> )
 <b>by </b><i><a class="txt" href="int_5.html#E4:50_9_2"><i><font color="Green" title="E76">A63</font></i></a>, <a class="txt" href="int_5.html#E7:50_9_2_6"><i><font color="Green" title="E90">A79</font></i></a>, <a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E11:50_9_2_6"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">e</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E36:50"><i><font color="Green" title="E58">A27</font></i></a>, <a class="txt" href="int_5.html#E10:50_9_2_6"><i><font color="Green" title="E92">A81</font></i></a>, <a class="ref" href="euler_2.html#D1" title="EULER_2:def.1">EULER_2:def 1</a></i>;<br/>
<a NAME="E12:50_9_2_6"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="ref" href="nat_d.html#T1" title="NAT_D:th.1">NAT_D:1</a></i>;<br/>
<b>hence </b><a NAME="E13:50_9_2_6"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">d</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 )
 <b>by </b><i><a class="txt" href="int_5.html#E30:50"><i><font color="Green" title="E54">A19</font></i></a>, <a class="txt" href="int_5.html#E2:50_9_2_6"><i><font color="Green" title="E89">A78</font></i></a>, <a class="txt" href="int_5.html#E8:50_9_2_6"><i><font color="Green" title="E91">A80</font></i></a>, <a class="ref" href="int_1.html#T81" title="INT_1:th.81">INT_1:81</a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/>


</div><b>end;</b></div><a NAME="E18:50_9_2"/>
for <font color="Olive" title="b1">d</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">d</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_9_2_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:50_9_2_7"/>
<font color="Maroon" title="c18">d</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:50_9_2_7"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">d</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">d</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="int_5.html#E16:50_9_2"><i><font color="Green" title="E86">A73</font></i></a>, <a class="txt" href="int_5.html#E17:50_9_2"><i><font color="Green" title="E87">A76</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:50_9_2_7"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c18">f3</font> = <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"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p5">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T14" title="FINSEQ_2:th.14">FINSEQ_2:14</a></i>;<br/><a NAME="E20:50_9_2"/>
for <font color="Olive" title="b1">d</font>, <font color="Olive" title="b2">e</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>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">d</font> &amp; <font color="Olive" title="b1">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">e</font> &amp; <font color="Olive" title="b2">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">f3</font> holds <br/><font color="Maroon" title="c18">f3</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c18">f3</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Olive" title="b2">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_9_2_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">d</font>, <font color="Maroon" title="c20">e</font> be    <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> ;<br/>

<b>assume </b><a NAME="E1:50_9_2_8"/><i><font color="Green" title="E88">A82</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">d</font> &amp; <font color="Maroon" title="c19">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">e</font> &amp; <font color="Maroon" title="c20">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">f3</font> )
 ;<br/>

<a NAME="E2:50_9_2_8"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">d</font> &amp; <font color="Maroon" title="c19">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">f3</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">e</font> &amp; <font color="Maroon" title="c20">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">f3</font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E3:50_9_2_8"/><b>then </b><i><font color="Green" title="E89">A83</font></i>: 
( <font color="Maroon" title="c19">d</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="c18">f3</font> &amp; <font color="Maroon" title="c20">e</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="c18">f3</font> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E4:50_9_2_8"/><b>then </b>
( <font color="Maroon" title="c18">f3</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c19">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c19">d</font></span>)</span> &amp; <font color="Maroon" title="c18">f3</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c20">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c20">e</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="int_5.html#E16:50_9_2"><i><font color="Green" title="E86">A73</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:50_9_2_8"/>
<font color="Maroon" title="c18">f3</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c19">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c18">f3</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c20">e</font>
 <b>by </b><i><a class="txt" href="int_5.html#E6:50_9_2"><i><font color="Green" title="E77">A64</font></i></a>, <a class="txt" href="int_5.html#E13:50_9_2"><i><font color="Green" title="E83">A70</font></i></a>, <a class="txt" href="int_5.html#E1:50_9_2_8"><i><font color="Green" title="E88">A82</font></i></a>, <a class="txt" href="int_5.html#E3:50_9_2_8"><i><font color="Green" title="E89">A83</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>


</div><b>end;</b></div><a NAME="E21:50_9_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">f3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c18">f3</font></span>)</span>
 <b>by </b><i><a class="ref" href="graph_5.html#T10" title="GRAPH_5:th.10">GRAPH_5:10</a></i>;<br/><a NAME="E22:50_9_2"/><b>then </b><i><font color="Green" title="E88">A84</font></i>: 
<font color="Maroon" title="c18">f3</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="finseq_4.html#T77" title="FINSEQ_4:th.77">FINSEQ_4:77</a></i>;<br/><a NAME="E23:50_9_2"/>
( <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c15">n</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c18">f3</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 ;<br/><b>then reconsider </b><font color="Maroon" title="c19">f4</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_1.html#T96" title="FINSEQ_1:th.96">FINSEQ_1:96</a></i>;<br/><i><font color="Green" title="E89">A85</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">f4</font> = 
<font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">m</font>
<b>by </b><i><a class="txt" href="int_5.html#E12:50_9_2"><i><font color="Green" title="E82">A69</font></i></a>, <a class="txt" href="int_5.html#E14:50_9_2"><i><font color="Green" title="E84">A71</font></i></a>, <a class="txt" href="int_5.html#E15:50_9_2"><i><font color="Green" title="E85">A72</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font>
<b>by </b><i><a class="txt" href="int_5.html#E10:50"><i><font color="Green" title="E42">A7</font></i></a></i>
;<br/><a NAME="E26:50_9_2"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c18">f3</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_9_2_10"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:50_9_2_10"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c18">f3</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c20">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:50_9_2_10"/><i><font color="Green" title="E90">A86</font></i>: 
( <font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c15">n</font></span>)</span> &amp; <font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c18">f3</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E4:50_9_2_10"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T21" title="FINSEQ_5:th.21">FINSEQ_5:21</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c21">d1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E6:50_9_2_10"/><i><font color="Green" title="E91">A87</font></i>: 
( <font color="Maroon" title="c21">d1</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="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x</font> )
 <b>by </b><i><a class="txt" href="int_5.html#E44:50"><i><font color="Green" title="E62">A34</font></i></a>, <a class="txt" href="int_5.html#E3:50_9_2_10"><i><font color="Green" title="E90">A86</font></i></a>, <a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E7:50_9_2_10"/><i><font color="Green" title="E92">A88</font></i>: 
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">d1</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E36:50"><i><font color="Green" title="E58">A27</font></i></a>, <a class="txt" href="int_5.html#E6:50_9_2_10"><i><font color="Green" title="E91">A87</font></i></a>, <a class="ref" href="euler_2.html#D1" title="EULER_2:def.1">EULER_2:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c22">e</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E9:50_9_2_10"/><i><font color="Green" title="E93">A89</font></i>: 
( <font color="Maroon" title="c22">e</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="c18">f3</font> &amp; <font color="Maroon" title="c18">f3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x</font> )
 <b>by </b><i><a class="txt" href="int_5.html#E3:50_9_2_10"><i><font color="Green" title="E90">A86</font></i></a>, <a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E10:50_9_2_10"/>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">e</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E16:50_9_2"><i><font color="Green" title="E86">A73</font></i></a>, <a class="txt" href="int_5.html#E9:50_9_2_10"><i><font color="Green" title="E93">A89</font></i></a></i>;<br/>
<a NAME="E11:50_9_2_10"/><b>then </b>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">e</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E11:50_9_2"><i><font color="Green" title="E81">A68</font></i></a>, <a class="txt" href="int_5.html#E13:50_9_2"><i><font color="Green" title="E83">A70</font></i></a>, <a class="txt" href="int_5.html#E9:50_9_2_10"><i><font color="Green" title="E93">A89</font></i></a>, <a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></i>;<br/>
<a NAME="E12:50_9_2_10"/><b>then </b><i><font color="Green" title="E94">A90</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">d1</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">e</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E7:50_9_2_10"><i><font color="Green" title="E92">A88</font></i></a></i>;<br/>
<a NAME="E13:50_9_2_10"/>
<font color="Maroon" title="c22">e</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">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> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E13:50_9_2"><i><font color="Green" title="E83">A70</font></i></a>, <a class="txt" href="int_5.html#E9:50_9_2_10"><i><font color="Green" title="E93">A89</font></i></a>, <a class="ref" href="finseq_5.html#T29" title="FINSEQ_5:th.29">FINSEQ_5:29</a></i>;<br/>
<a NAME="E14:50_9_2_10"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">e</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E44:50"><i><font color="Green" title="E62">A34</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c23">e1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E16:50_9_2_10"/><i><font color="Green" title="E95">A91</font></i>: 
( <font color="Maroon" title="c23">e1</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="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">e</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">n</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E17:50_9_2_10"/><i><font color="Green" title="E96">A92</font></i>: 
<font color="Maroon" title="c23">e1</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">f</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E35:50"><i><font color="Green" title="E57">A26</font></i></a>, <a class="txt" href="int_5.html#E16:50_9_2_10"><i><font color="Green" title="E95">A91</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E18:50_9_2_10"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">d1</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">e1</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E12:50_9_2_10"><i><font color="Green" title="E94">A90</font></i></a>, <a class="txt" href="int_5.html#E16:50_9_2_10"><i><font color="Green" title="E95">A91</font></i></a>, <a class="ref" href="euler_2.html#D1" title="EULER_2:def.1">EULER_2:def 1</a></i>;<br/>
<a NAME="E19:50_9_2_10"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">d1</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">e1</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="nat_d.html#T25" title="NAT_D:th.25">NAT_D:25</a></i>;<br/>
<a NAME="E20:50_9_2_10"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">d1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">e1</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="euler_2.html#T8" title="EULER_2:th.8">EULER_2:8</a></i>;<br/>
<a NAME="E21:50_9_2_10"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">d1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">e1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pepin.html#T6" title="PEPIN:th.6">PEPIN:6</a></i>;<br/>
<a NAME="E22:50_9_2_10"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">d1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">e1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E34:50"><i><font color="Green" title="E56">A23</font></i></a>, <a class="txt" href="int_5.html#E36:50"><i><font color="Green" title="E58">A27</font></i></a>, <a class="txt" href="int_5.html#E6:50_9_2_10"><i><font color="Green" title="E91">A87</font></i></a></i>;<br/>
<a NAME="E23:50_9_2_10"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">d1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">e1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E34:50"><i><font color="Green" title="E56">A23</font></i></a>, <a class="txt" href="int_5.html#E17:50_9_2_10"><i><font color="Green" title="E96">A92</font></i></a></i>;<br/>
<a NAME="E24:50_9_2_10"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">d1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c23">e1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">a</font>
 
;<br/>
<a NAME="E25:50_9_2_10"/><b>then </b><i><font color="Green" title="E97">A93</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c21">d1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c23">e1</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E1:50"><i><font color="Green" title="E36">A1</font></i></a>, <a class="txt" href="int_5.html#E2:50"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="wsierp_1.html#T37" title="WSIERP_1:th.37">WSIERP_1:37</a></i>;<br/>
<a NAME="E26:50_9_2_10"/>
( <font color="Maroon" title="c21">d1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c21">d1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 &amp; <font color="Maroon" title="c23">e1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c23">e1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 )
 
<b>by </b><i><a class="txt" href="int_5.html#E10:50"><i><font color="Green" title="E42">A7</font></i></a>, <a class="txt" href="int_5.html#E35:50"><i><font color="Green" title="E57">A26</font></i></a>, <a class="txt" href="int_5.html#E6:50_9_2_10"><i><font color="Green" title="E91">A87</font></i></a>, <a class="txt" href="int_5.html#E16:50_9_2_10"><i><font color="Green" title="E95">A91</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E27:50_9_2_10"/><b>then </b>
( <font color="Maroon" title="c21">d1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c23">e1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c21">d1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c23">e1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E28:50_9_2_10"/><b>then </b>
( <font color="Maroon" title="c21">d1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c23">e1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c21">d1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c23">e1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">p</font> )
 
<b>by </b><i><a class="txt" href="int_5.html#E17:50"><i><font color="Green" title="E46">A11</font></i></a>, <a class="txt" href="int_5.html#E24:50"><i><font color="Green" title="E49">A14</font></i></a>, <a class="txt" href="int_5.html#E28:50"><i><font color="Green" title="E52">A17</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E29:50_9_2_10"/>
contradiction
 <b>by </b><i><a class="txt" href="int_5.html#E25:50_9_2_10"><i><font color="Green" title="E97">A93</font></i></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></i>;<br/>


</div><b>end;</b></div><a NAME="E27:50_9_2"/><b>then </b><i><font color="Green" title="E90">A94</font></i>: 
<font color="Maroon" title="c19">f4</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="int_5.html#E6:50_9_2"><i><font color="Green" title="E77">A64</font></i></a>, <a class="txt" href="int_5.html#E22:50_9_2"><i><font color="Green" title="E88">A84</font></i></a>, <a class="ref" href="finseq_3.html#T98" title="FINSEQ_3:th.98">FINSEQ_3:98</a></i>;<br/><a NAME="E28:50_9_2"/>
for <font color="Olive" title="b1">d</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">d</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="c19">f4</font> holds <br/>( <font color="Maroon" title="c19">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c19">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_9_2_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:50_9_2_11"/><i><font color="Green" title="E91">A95</font></i>: 
<font color="Maroon" title="c20">d</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="c19">f4</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_9_2_11_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:50_9_2_11_1"/>
( <font color="Maroon" title="c20">d</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"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c15">n</font></span>)</span> or  ex <font color="Olive" title="b1">l</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">l</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="c18">f3</font> &amp; <font color="Maroon" title="c20">d</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">l</font> ) )
 </span><b>by </b><i><a class="txt" href="int_5.html#E1:50_9_2_11"><i><font color="Green" title="E91">A95</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="50_9_2_11_1_1"><b>suppose </b></a><a NAME="E1:50_9_2_11_1_1"/><i><font color="Green" title="E92">A96</font></i>: 
<font color="Maroon" title="c20">d</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"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c15">n</font></span>)</span>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c21">d</font> = <font color="Maroon" title="c20">d</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="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E3:50_9_2_11_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c21">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c14">Y</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_5.html#E8:50_9_2"><i><font color="Green" title="E78">A65</font></i></a>, <a class="txt" href="int_5.html#E1:50_9_2_11_1_1"><i><font color="Green" title="E92">A96</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><a NAME="E4:50_9_2_11_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c21">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Y</font>
 <b>by </b><i><a class="txt" href="int_5.html#E63:50"><i><font color="Green" title="E72">A52</font></i></a>, <a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></i>;<br/><a NAME="E5:50_9_2_11_1_1"/><b>then </b><i><font color="Green" title="E93">A97</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c21">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> &amp; not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c21">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">X</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E6:50_9_2_11_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c21">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">p</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 ;<br/><a NAME="E7:50_9_2_11_1_1"/><b>then </b><i><font color="Green" title="E94">A98</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c21">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 <b>by </b><i><a class="txt" href="int_5.html#E30:50"><i><font color="Green" title="E54">A19</font></i></a>, <a class="ref" href="int_1.html#T81" title="INT_1:th.81">INT_1:81</a></i>;<br/><a NAME="E8:50_9_2_11_1_1"/>
not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c21">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="int_5.html#E42:50"><i><font color="Green" title="E60">A32</font></i></a>, <a class="txt" href="int_5.html#E5:50_9_2_11_1_1"><i><font color="Green" title="E93">A97</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><a NAME="E9:50_9_2_11_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c21">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E10:50_9_2_11_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c21">d</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><b>hence </b><a NAME="E11:50_9_2_11_1_1"/>
( <font color="Maroon" title="c19">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">d</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c19">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 )
 <b>by </b><i><a class="txt" href="int_5.html#E1:50_9_2_11_1_1"><i><font color="Green" title="E92">A96</font></i></a>, <a class="txt" href="int_5.html#E7:50_9_2_11_1_1"><i><font color="Green" title="E94">A98</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_9_2_11_1_2"><b>suppose </b></a><a NAME="E1:50_9_2_11_1_2"/>
 ex <font color="Olive" title="b1">l</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">l</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="c18">f3</font> &amp; <font color="Maroon" title="c20">d</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">l</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c21">l</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="E3:50_9_2_11_1_2"/><i><font color="Green" title="E92">A99</font></i>: 
( <font color="Maroon" title="c21">l</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="c18">f3</font> &amp; <font color="Maroon" title="c20">d</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c21">l</font> )
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E4:50_9_2_11_1_2"/>
<font color="Maroon" title="c19">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">f3</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c21">l</font>
 <b>by </b><i><a class="txt" href="int_5.html#E3:50_9_2_11_1_2"><i><font color="Green" title="E92">A99</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/><b>hence </b><a NAME="E5:50_9_2_11_1_2"/>
( <font color="Maroon" title="c19">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">d</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c19">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 )
 <b>by </b><i><a class="txt" href="int_5.html#E17:50_9_2"><i><font color="Green" title="E87">A76</font></i></a>, <a class="txt" href="int_5.html#E3:50_9_2_11_1_2"><i><font color="Green" title="E92">A99</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E29:50_9_2"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c19">f4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">I</font>
 <b>by </b><i><a class="txt" href="int_5.html#E7:50"><i><font color="Green" title="E41">A6</font></i></a>, <a class="txt" href="int_5.html#E10:50"><i><font color="Green" title="E42">A7</font></i></a>, <a class="txt" href="int_5.html#E25:50_9_2"><i><font color="Green" title="E89">A85</font></i></a>, <a class="txt" href="int_5.html#E27:50_9_2"><i><font color="Green" title="E90">A94</font></i></a>, <a class="ref" href="int_5.html#T40" target="_self" title="INT_5:th.40">Th40</a></i>;<br/><a NAME="E30:50_9_2"/><b>then </b>
<font color="Maroon" title="c19">f4</font>,<font color="Maroon" title="c9">I</font> <a href="rfinseq.html#R1" title="RFINSEQ:pred.1">are_fiberwise_equipotent</a> 
 <b>by </b><i><a class="txt" href="int_5.html#E27:50_9_2"><i><font color="Green" title="E90">A94</font></i></a>, <a class="ref" href="rfinseq.html#T39" title="RFINSEQ:th.39">RFINSEQ:39</a></i>;<br/><a NAME="E31:50_9_2"/><b>then </b>
 <a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Product</a> <font color="Maroon" title="c19">f4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Product</a> <font color="Maroon" title="c9">I</font>
 <b>by </b><i><a class="ref" href="euler_2.html#T25" title="EULER_2:th.25">EULER_2:25</a></i>;<br/><a NAME="E32:50_9_2"/><b>then </b><i><font color="Green" title="E91">A100</font></i>: 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Product</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Product</a> <font color="Maroon" title="c18">f3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Product</a> <font color="Maroon" title="c9">I</font>
 <b>by </b><i><a class="ref" href="rvsum_1.html#T127" title="RVSUM_1:th.127">RVSUM_1:127</a></i>;<br/><a NAME="E33:50_9_2"/><i><font color="Green" title="E92">A101</font></i>: 
( <font color="Maroon" title="c18">f3</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  )
 <b>by </b><i><a class="ref" href="finseq_2.html#T27" title="FINSEQ_2:th.27">FINSEQ_2:27</a>, <a class="ref" href="numbers.html#T17" title="NUMBERS:th.17">NUMBERS:17</a></i>;<br/><a NAME="E34:50_9_2"/>
for <font color="Olive" title="b1">d</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">d</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="c18">f3</font> holds <br/><font color="Maroon" title="c18">f3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font>, <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c1">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_9_2_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:50_9_2_12"/>
<font color="Maroon" title="c20">d</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="c18">f3</font>
 ;<br/>

<b>then </b><span class="p1">(<span class="default"><font color="Maroon" title="c18">f3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">d</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">p</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p5">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">d</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">p</font>
<b>by </b><i><a class="txt" href="int_5.html#E16:50_9_2"><i><font color="Green" title="E86">A73</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">d</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">p</font>

<br/>.= 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p5">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">d</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">p</font>
<b>by </b><i><a class="ref" href="euler_1.html#T13" title="EULER_1:th.13">EULER_1:13</a></i>
;<br/>
<b>hence </b><a NAME="E3:50_9_2_12"/>
<font color="Maroon" title="c18">f3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">d</font>, <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">d</font></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c1">p</font>
 <b>by </b><i><a class="ref" href="int_3.html#T12" title="INT_3:th.12">INT_3:12</a></i>;<br/>


</div><b>end;</b></div><a NAME="E35:50_9_2"/><b>then </b>
 <a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Product</a> <font color="Maroon" title="c18">f3</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Product</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c1">p</font>
 <b>by </b><i><a class="txt" href="int_5.html#E14:50_9_2"><i><font color="Green" title="E84">A71</font></i></a>, <a class="txt" href="int_5.html#E15:50_9_2"><i><font color="Green" title="E85">A72</font></i></a>, <a class="txt" href="int_5.html#E33:50_9_2"><i><font color="Green" title="E92">A101</font></i></a>, <a class="ref" href="int_5.html#T33" target="_self" title="INT_5:th.33">Th33</a></i>;<br/><a NAME="E36:50_9_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Product</a> <font color="Maroon" title="c18">f3</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Product</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Product</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p5">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Product</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c1">p</font>
 <b>by </b><i><a class="ref" href="int_4.html#T11" title="INT_4:th.11">INT_4:11</a></i>;<br/><a NAME="E37:50_9_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Product</a> <font color="Maroon" title="c18">f3</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Product</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">m</font></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"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Product</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p5">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Product</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p5">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c1">p</font>
 ;<br/><a NAME="E38:50_9_2"/><b>then </b>
 <a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Product</a> <font color="Maroon" title="c9">I</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Product</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p5">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p5">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c1">p</font>
 <b>by </b><i><a class="txt" href="int_5.html#E32:50_9_2"><i><font color="Green" title="E91">A100</font></i></a>, <a class="ref" href="rvsum_1.html#T127" title="RVSUM_1:th.127">RVSUM_1:127</a></i>;<br/><a NAME="E39:50_9_2"/><b>then </b><i><font color="Green" title="E93">A102</font></i>: 
 <a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Product</a> <font color="Maroon" title="c9">I</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c1">p</font>
 <b>by </b><i><a class="txt" href="int_5.html#E62:50"><i><font color="Green" title="E71">A51</font></i></a>, <a class="ref" href="rfinseq.html#T21" title="RFINSEQ:th.21">RFINSEQ:21</a></i>;<br/><a NAME="E40:50_9_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Product</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c1">p</font>
 <b>by </b><i><a class="txt" href="int_5.html#E12:50"><i><font color="Green" title="E43">A8</font></i></a>, <a class="ref" href="int_4.html#T11" title="INT_4:th.11">INT_4:11</a></i>;<br/><a NAME="E41:50_9_2"/><b>then </b>
 <a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Product</a> <font color="Maroon" title="c9">I</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Product</a> <font color="Maroon" title="c9">I</font></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c1">p</font>
 <b>by </b><i><a class="txt" href="int_5.html#E15:50"><i><font color="Green" title="E44">A9</font></i></a>, <a class="txt" href="int_5.html#E39:50_9_2"><i><font color="Green" title="E93">A102</font></i></a>, <a class="ref" href="int_1.html#T36" title="INT_1:th.36">INT_1:36</a></i>;<br/><a NAME="E42:50_9_2"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Product</a> <font color="Maroon" title="c9">I</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <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="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Product</a> <font color="Maroon" title="c9">I</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="int_2.html#T19" title="INT_2:th.19">INT_2:19</a></i>;<br/><a NAME="E43:50_9_2"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Product</a> <font color="Maroon" title="c9">I</font></span>)</span>
 ;<br/><a NAME="E44:50_9_2"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="int_5.html#E33:50"><i><font color="Green" title="E55">A22</font></i></a>, <a class="ref" href="wsierp_1.html#T36" title="WSIERP_1:th.36">WSIERP_1:36</a></i>;<br/><a NAME="E45:50_9_2"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="int_2.html#T12" title="INT_2:th.12">INT_2:12</a></i>;<br/><a NAME="E46:50_9_2"/><b>then </b><i><font color="Green" title="E94">A103</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <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="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">m</font></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"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span></span>)</span>
 ;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">m</font></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"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">m</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">m</font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c13">m</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 2</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">m</font>
<b>by </b><i><a class="ref" href="newton.html#T14" title="NEWTON:th.14">NEWTON:14</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> 2</span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c13">m</font>
<b>by </b><i><a class="ref" href="wsierp_1.html#T2" title="WSIERP_1:th.2">WSIERP_1:2</a></i>
<br/>.= 
1 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c13">m</font>
<b>by </b><i><a class="ref" href="newton.html#T100" title="NEWTON:th.100">NEWTON:100</a>, <a class="ref" href="square_1.html#T59" title="SQUARE_1:th.59">SQUARE_1:59</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="newton.html#T15" title="NEWTON:th.15">NEWTON:15</a></i>
;<br/><a NAME="E48:50_9_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">m</font>,<font color="Maroon" title="c2">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c1">p</font>
 <b>by </b><i><a class="txt" href="int_5.html#E46:50_9_2"><i><font color="Green" title="E94">A103</font></i></a>, <a class="ref" href="int_2.html#T19" title="INT_2:th.19">INT_2:19</a></i>;<br/><a NAME="E49:50_9_2"/><b>then </b><i><font color="Green" title="E95">A104</font></i>: 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">m</font>, <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c1">p</font>
 <b>by </b><i><a class="txt" href="int_5.html#E5:50"><i><font color="Green" title="E39">A4</font></i></a>, <a class="ref" href="int_1.html#T36" title="INT_1:th.36">INT_1:36</a></i>;<br/><a NAME="E50:50_9_2"/>
 <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="series_2.html#T1" title="SERIES_2:th.1">SERIES_2:1</a></i>;<br/><a NAME="E51:50_9_2"/><b>then </b><i><font color="Green" title="E96">A105</font></i>: 
( <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="ref" href="absvalue.html#T1" title="ABSVALUE:th.1">ABSVALUE:1</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_9_2_14"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:50_9_2_14"/>
( <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 )
 </span><b>by </b><i><a class="txt" href="int_5.html#E51:50_9_2"><i><font color="Green" title="E96">A105</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_9_2_14_1"><b>suppose </b></a><a NAME="E1:50_9_2_14_1"/><i><font color="Green" title="E97">A106</font></i>: 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_9_2_14_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_9_2_14_1_1"/>
 <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1
 ;<br/><a NAME="E2:50_9_2_14_1_1"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="int_5.html#E49:50_9_2"><i><font color="Green" title="E95">A104</font></i></a>, <a class="txt" href="int_5.html#E1:50_9_2_14_1"><i><font color="Green" title="E97">A106</font></i></a>, <a class="ref" href="int_2.html#T19" title="INT_2:th.19">INT_2:19</a></i>;<br/><b>hence </b><a NAME="E3:50_9_2_14_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="int_5.html#E2:50"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:50_9_2_14_1"/>
 <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">m</font>
 <b>by </b><i><a class="txt" href="int_5.html#E1:50_9_2_14_1"><i><font color="Green" title="E97">A106</font></i></a>, <a class="ref" href="int_5.html#T25" target="_self" title="INT_5:th.25">Th25</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_9_2_14_2"><b>suppose </b></a><a NAME="E1:50_9_2_14_2"/><i><font color="Green" title="E97">A107</font></i>: 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_9_2_14_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_9_2_14_2_1"/>
 <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><a NAME="E2:50_9_2_14_2_1"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="txt" href="int_5.html#E49:50_9_2"><i><font color="Green" title="E95">A104</font></i></a>, <a class="txt" href="int_5.html#E1:50_9_2_14_2"><i><font color="Green" title="E97">A107</font></i></a>, <a class="ref" href="int_2.html#T19" title="INT_2:th.19">INT_2:19</a></i>;<br/><a NAME="E3:50_9_2_14_2_1"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 2
 ;<br/><a NAME="E4:50_9_2_14_2_1"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 2
 <b>by </b><i><a class="ref" href="int_2.html#T14" title="INT_2:th.14">INT_2:14</a></i>;<br/><b>hence </b><a NAME="E5:50_9_2_14_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="int_5.html#E2:50"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:50_9_2_14_2"/>
 <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">m</font>
 <b>by </b><i><a class="txt" href="int_5.html#E1:50_9_2_14_2"><i><font color="Green" title="E97">A107</font></i></a>, <a class="ref" href="int_5.html#T25" target="_self" title="INT_5:th.25">Th25</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div>
