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<b>let </b><font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c2">q</font> be   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>;<br/>



<b>assume </b><a NAME="E1:59"/><i><font color="Green" title="E43">A1</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">q</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 &amp; <font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">q</font> )
 ;<br/>

<a NAME="E2:59"/><b>then </b><i><font color="Green" title="E44">A2</font></i>: 
<font color="Maroon" title="c2">q</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="ref" href="int_2.html#T47" title="INT_2:th.47">INT_2:47</a></i>;<br/>
<a NAME="E3:59"/><b>then </b><i><font color="Green" title="E45">A3</font></i>: 
<font color="Maroon" title="c2">q</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
 
<b>by </b><i><a class="ref" href="int_2.html#D6" title="INT_2:def.6">INT_2:def 6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c3">p</font> = <font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c4">q</font> = <font color="Maroon" title="c2">q</font> as  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <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="E5:59"/><i><font color="Green" title="E46">A4</font></i>: 
( 2,<font color="Maroon" title="c3">p</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a>  &amp; 2,<font color="Maroon" title="c4">q</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#E1:59"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="euler_1.html#T3" title="EULER_1:th.3">EULER_1:3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">p'</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">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">q'</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</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="c7">f1</font> = <font color="Maroon" title="c4">q</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="c3">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>;<br/>
<a NAME="E6:59"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="newton.html#K4" title="NEWTON:func.4">idseq</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</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="c3">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="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>;<br/>
<a NAME="E7:59"/><b>then </b><i><font color="Green" title="E47">A5</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">q</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="newton.html#K4" title="NEWTON:func.4">idseq</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">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="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"><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="c3">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="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E8:59"/><b>then </b><i><font color="Green" title="E48">A6</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">q</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="newton.html#K4" title="NEWTON:func.4">idseq</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">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="E9:59"/><i><font color="Green" title="E49">A7</font></i>: 
( <font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <font color="Maroon" title="c4">q</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="E10:59"/><b>then </b><i><font color="Green" title="E50">A8</font></i>: 
( <font color="Maroon" title="c3">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="c3">p</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Maroon" title="c4">q</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="c4">q</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="E11:59"/>
( <font color="Maroon" title="c3">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 &amp; <font color="Maroon" title="c4">q</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#E1:59"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E12:59"/><b>then </b>
( <font color="Maroon" title="c3">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 &amp; <font color="Maroon" title="c4">q</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="E13:59"/><b>then </b><i><font color="Green" title="E51">A9</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">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 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</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#E10:59"><i><font color="Green" title="E50">A8</font></i></a>, <a class="ref" href="nat_2.html#T15" title="NAT_2:th.15">NAT_2:15</a></i>;<br/>
<a NAME="E14:59"/>
( not <font color="Maroon" title="c3">p</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; not <font color="Maroon" title="c4">q</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> )
 
<b>by </b><i><a class="txt" href="int_5.html#E1:59"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="pepin.html#T17" title="PEPIN:th.17">PEPIN:17</a></i>;<br/>
<a NAME="E15:59"/><b>then </b><i><font color="Green" title="E52">A10</font></i>: 
( <font color="Maroon" title="c3">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> &amp; <font color="Maroon" title="c4">q</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#E10:59"><i><font color="Green" title="E50">A8</font></i></a>, <a class="ref" href="hilbert3.html#T2" title="HILBERT3:th.2">HILBERT3:2</a></i>;<br/>
<a NAME="E16:59"/><b>then </b><i><font color="Green" title="E53">A11</font></i>: 
( 2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 &amp; 2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">q</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="E17:59"/><b>then </b><i><font color="Green" title="E54">A12</font></i>: 
( <font color="Maroon" title="c3">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="c3">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> &amp; <font color="Maroon" title="c4">q</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="c4">q</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="E18:59"/><b>then </b><i><font color="Green" title="E55">A13</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">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="c3">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</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="c4">q</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="E19:59"/>
( <font color="Maroon" title="c3">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>  &amp; <font color="Maroon" title="c4">q</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#E9:59"><i><font color="Green" title="E49">A7</font></i></a>, <a class="txt" href="int_5.html#E10:59"><i><font color="Green" title="E50">A8</font></i></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/>
<a NAME="E20:59"/><b>then </b><i><font color="Green" title="E56">A14</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">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="c3">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</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="c4">q</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 )
 
<b>by </b><i><a class="txt" href="int_5.html#E18:59"><i><font color="Green" title="E55">A13</font></i></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></i>;<br/>
<a NAME="E21:59"/><i><font color="Green" title="E57">A15</font></i>: 
( <font color="Maroon" title="c3">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">p</font> &amp; <font color="Maroon" title="c4">q</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">q</font> )
 
<b>by </b><i><a class="txt" href="int_5.html#E10:59"><i><font color="Green" title="E50">A8</font></i></a>, <a class="ref" href="xreal_1.html#T148" title="XREAL_1:th.148">XREAL_1:148</a></i>;<br/>
<a NAME="E22:59"/><b>then </b><i><font color="Green" title="E58">A16</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">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="c3">p</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</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="c4">q</font> )
 
<b>by </b><i><a class="txt" href="int_5.html#E20:59"><i><font color="Green" title="E56">A14</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="E59">A17</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c3">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="c3">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#E15:59"><i><font color="Green" title="E52">A10</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="c3">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#E9:59"><i><font color="Green" title="E49">A7</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>
;<br/>
<i><font color="Green" title="E60">A18</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</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="c4">q</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#E15:59"><i><font color="Green" title="E52">A10</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="c4">q</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#E9:59"><i><font color="Green" title="E49">A7</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>
;<br/>
<a NAME="E25:59"/><i><font color="Green" title="E61">A19</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"><font color="Maroon" title="c4">q</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="newton.html#K4" title="NEWTON:func.4">idseq</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">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> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="newton.html#K4" title="NEWTON:func.4">idseq</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">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="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="c4">q</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b1">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:59_3"/><i><font color="Green" title="E62">A20</font></i>: 
<font color="Maroon" title="c8">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">q</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="newton.html#K4" title="NEWTON:func.4">idseq</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">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/>

<a NAME="E2:59_3"/><b>then </b><i><font color="Green" title="E63">A21</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="newton.html#K4" title="NEWTON:func.4">idseq</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> <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="newton.html#K4" title="NEWTON:func.4">idseq</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">d</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>;<br/>
<a NAME="E3:59_3"/>
<font color="Maroon" title="c8">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"><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="c3">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#E1:59_3"><i><font color="Green" title="E62">A20</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="E4:59_3"/><b>then </b>
<font color="Maroon" title="c8">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"><a href="newton.html#K4" title="NEWTON:func.4">idseq</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">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="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:59_3"/><b>then </b>
<font color="Maroon" title="c8">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="c3">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>hence </b><a NAME="E6:59_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="newton.html#K4" title="NEWTON:func.4">idseq</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">d</font>
 <b>by </b><i><a class="txt" href="int_5.html#E13:59"><i><font color="Green" title="E51">A9</font></i></a>, <a class="txt" href="int_5.html#E2:59_3"><i><font color="Green" title="E63">A21</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="E26:59"/>
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">q</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="newton.html#K4" title="NEWTON:func.4">idseq</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">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> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="newton.html#K4" title="NEWTON:func.4">idseq</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">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="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="59_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:59_4"/>
<font color="Maroon" title="c8">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">q</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="newton.html#K4" title="NEWTON:func.4">idseq</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">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/>

<a NAME="E2:59_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="newton.html#K4" title="NEWTON:func.4">idseq</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">d</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E25:59"><i><font color="Green" title="E61">A19</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:59_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="newton.html#K4" title="NEWTON:func.4">idseq</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">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="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c8">f1</font> = <font color="Maroon" title="c4">q</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="c3">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> 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="E28:59"/><i><font color="Green" title="E62">A22</font></i>: 
for <font color="Olive" title="b1">d</font>, <font color="Olive" title="b2">e</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="c8">f1</font> &amp; <font color="Olive" title="b2">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="c8">f1</font> &amp; <font color="Maroon" title="c3">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="c8">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</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="c8">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">e</font></span>)</span> holds <br/><font color="Olive" title="b1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">e</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">d</font>, <font color="Maroon" title="c10">e</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:59_5"/><i><font color="Green" title="E63">A23</font></i>: 
( <font color="Maroon" title="c9">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="c8">f1</font> &amp; <font color="Maroon" title="c10">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="c8">f1</font> &amp; <font color="Maroon" title="c3">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="c8">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">d</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="c8">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">e</font></span>)</span> )
 ;<br/>

<a NAME="E2:59_5"/><b>then </b>
( <font color="Maroon" title="c8">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c9">d</font> &amp; <font color="Maroon" title="c8">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">e</font> )
 
<b>by </b><i><a class="txt" href="int_5.html#E25:59"><i><font color="Green" title="E61">A19</font></i></a></i>;<br/>
<a NAME="E3:59_5"/><b>then </b><i><font color="Green" title="E64">A24</font></i>: 
<font color="Maroon" title="c3">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="c9">d</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c10">e</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">q</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E1:59_5"><i><font color="Green" title="E63">A23</font></i></a></i>;<br/>
<a NAME="E4:59_5"/>
<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">p</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 
<b>by </b><i><a class="txt" href="int_5.html#E2:59"><i><font color="Green" title="E44">A2</font></i></a>, <a class="ref" href="euler_2.html#T1" title="EULER_2:th.1">EULER_2:1</a></i>;<br/>
<a NAME="E5:59_5"/><b>then </b><i><font color="Green" title="E65">A25</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c9">d</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c10">e</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E3:59_5"><i><font color="Green" title="E64">A24</font></i></a>, <a class="ref" href="int_2.html#T40" title="INT_2:th.40">INT_2:40</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_5_1"/>
<font color="Maroon" title="c9">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">e</font>
 ;<br/><a NAME="E2:59_5_1"/><b>then </b>
<font color="Maroon" title="c9">d</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c10">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E3:59_5_1"/><b>then </b>
 <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">d</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c10">e</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_5.html#E5:59_5"><i><font color="Green" title="E65">A25</font></i></a>, <a class="ref" href="int_4.html#T6" title="INT_4:th.6">INT_4:6</a></i>;<br/><a NAME="E4:59_5_1"/><b>then </b><i><font color="Green" title="E66">A26</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">d</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c10">e</font></span>)</span>
 <b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><a NAME="E5:59_5_1"/>
( <font color="Maroon" title="c9">d</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c9">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="c3">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="c10">e</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c10">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">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#E8:59"><i><font color="Green" title="E48">A6</font></i></a>, <a class="txt" href="int_5.html#E1:59_5"><i><font color="Green" title="E63">A23</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:59_5_1"/><b>then </b><i><font color="Green" title="E67">A27</font></i>: 
( <font color="Maroon" title="c9">d</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c10">e</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="c3">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 &amp; <font color="Maroon" title="c9">d</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c10">e</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</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"><font color="Maroon" title="c3">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#T15" title="XREAL_1:th.15">XREAL_1:15</a></i>;<br/><a NAME="E7:59_5_1"/><i><font color="Green" title="E68">A28</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">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="c3">p</font>
 <b>by </b><i><a class="txt" href="int_5.html#E22:59"><i><font color="Green" title="E58">A16</font></i></a>, <a class="ref" href="xreal_1.html#T149" title="XREAL_1:th.149">XREAL_1:149</a></i>;<br/><a NAME="E8:59_5_1"/><b>then </b><i><font color="Green" title="E69">A29</font></i>: 
<font color="Maroon" title="c9">d</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c10">e</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">p</font>
 <b>by </b><i><a class="txt" href="int_5.html#E6:59_5_1"><i><font color="Green" title="E67">A27</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E9:59_5_1"/>
 <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"><font color="Maroon" title="c3">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</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">p</font>
 <b>by </b><i><a class="txt" href="int_5.html#E7:59_5_1"><i><font color="Green" title="E68">A28</font></i></a>, <a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></i>;<br/><a NAME="E10:59_5_1"/><b>then </b>
<font color="Maroon" title="c9">d</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c10">e</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">p</font>
 <b>by </b><i><a class="txt" href="int_5.html#E6:59_5_1"><i><font color="Green" title="E67">A27</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="E11:59_5_1"/>
contradiction
 <b>by </b><i><a class="txt" href="int_5.html#E4:59_5_1"><i><font color="Green" title="E66">A26</font></i></a>, <a class="txt" href="int_5.html#E8:59_5_1"><i><font color="Green" title="E69">A29</font></i></a>, <a class="ref" href="seq_2.html#T9" title="SEQ_2:th.9">SEQ_2:9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:59_5"/>
<font color="Maroon" title="c9">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">e</font>
 ;<br/>


</div><b>end;</b></div>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  = <span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">p</font>;<br/>
<b>consider </b><font color="Maroon" title="c9">f2</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E30:59"/><i><font color="Green" title="E63">A30</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">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; ( 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">f2</font> holds <br/><font color="Maroon" title="c9">f2</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">H<sub>1</sub></font>(<font color="Olive" title="b1">d</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_5.html#S1" title="FINSEQ_5:sch.1">FINSEQ_5:sch 1</a>();<br/></i>
<a NAME="E31:59"/>
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="c9">f2</font> holds <br/><font color="Maroon" title="c9">f2</font> <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="59_6"><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="E1:59_6"/>
<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">f2</font>
 ;<br/>

<a NAME="E2:59_6"/><b>then </b>
<font color="Maroon" title="c9">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">d</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">p</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E30:59"><i><font color="Green" title="E63">A30</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:59_6"/>
<font color="Maroon" title="c9">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c10">f2</font> = <font color="Maroon" title="c9">f2</font> 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/>
<b>set </b><font color="Maroon" title="c11">f3</font> = <font color="Maroon" title="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font>;<br/>
<a NAME="E33:59"/><i><font color="Green" title="E64">A31</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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">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="c8">f1</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="E34:59"/><b>then </b><i><font color="Green" title="E65">A32</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</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/>
<a NAME="E35:59"/><i><font color="Green" title="E66">A33</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">f1</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="c10">f2</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E8:59"><i><font color="Green" title="E48">A6</font></i></a>, <a class="txt" href="int_5.html#E30:59"><i><font color="Green" title="E63">A30</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<i><font color="Green" title="E67">A34</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c10">f2</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</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"><font color="Maroon" title="c3">p</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c10">f2</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"><font color="Maroon" title="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D3" title="SEQ_1:def.3">SEQ_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">f2</font></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"><font color="Maroon" title="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">f1</font>
<b>by </b><i><a class="txt" href="int_5.html#E34:59"><i><font color="Green" title="E65">A32</font></i></a>, <a class="txt" href="int_5.html#E35:59"><i><font color="Green" title="E66">A33</font></i></a></i>
;<br/>
<a NAME="E37:59"/><i><font color="Green" title="E68">A35</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="c8">f1</font> holds <br/><font color="Maroon" title="c8">f1</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c3">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</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="59_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:59_8"/>
<font color="Maroon" title="c12">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="c8">f1</font>
 ;<br/>

<a NAME="E2:59_8"/><b>then </b>
( <font color="Maroon" title="c10">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">d</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">p</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">d</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">p</font> )
 
<b>by </b><i><a class="txt" href="int_5.html#E30:59"><i><font color="Green" title="E63">A30</font></i></a>, <a class="txt" href="int_5.html#E35:59"><i><font color="Green" title="E66">A33</font></i></a>, <a class="ref" href="euler_2.html#D1" title="EULER_2:def.1">EULER_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:59_8"/>
<font color="Maroon" title="c8">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">d</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c3">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">d</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_d.html#T2" title="NAT_D:th.2">NAT_D:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E38:59"/>
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="c8">f1</font> holds <br/><font color="Maroon" title="c8">f1</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c10">f2</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:59_9"/><i><font color="Green" title="E69">A36</font></i>: 
<font color="Maroon" title="c12">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="c8">f1</font>
 ;<br/>

<a NAME="E2:59_9"/><b>then </b><i><font color="Green" title="E70">A37</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c10">f2</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c10">f2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">d</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">d</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E36:59"><i><font color="Green" title="E67">A34</font></i></a>, <a class="ref" href="seq_1.html#D3" title="SEQ_1:def.3">SEQ_1:def 3</a></i>;<br/>
<a NAME="E3:59_9"/>
<font color="Maroon" title="c12">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="c3">p</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c10">f2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E35:59"><i><font color="Green" title="E66">A33</font></i></a>, <a class="txt" href="int_5.html#E1:59_9"><i><font color="Green" title="E69">A36</font></i></a>, <a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c10">f2</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">d</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">d</font></span>)</span></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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">d</font></span>)</span>
<b>by </b><i><a class="txt" href="int_5.html#E2:59_9"><i><font color="Green" title="E70">A37</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="c8">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">d</font>
<b>by </b><i><a class="txt" href="int_5.html#E37:59"><i><font color="Green" title="E68">A35</font></i></a>, <a class="txt" href="int_5.html#E1:59_9"><i><font color="Green" title="E69">A36</font></i></a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E39:59"/><b>then </b><i><font color="Green" title="E69">A38</font></i>: 
<font color="Maroon" title="c8">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c10">f2</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E36:59"><i><font color="Green" title="E67">A34</font></i></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></i>;<br/>
<a NAME="E40:59"/><i><font color="Green" title="E70">A39</font></i>: 
( <font color="Maroon" title="c3">p</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c10">f2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a>  &amp; <font color="Maroon" title="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a>  )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E41:59"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c10">f2</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="c10">f2</font>
 
<b>by </b><i><a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>;<br/>
<a NAME="E42:59"/><b>then </b>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c10">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">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;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">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#E7:59"><i><font color="Green" title="E47">A5</font></i></a>, <a class="txt" href="int_5.html#E30:59"><i><font color="Green" title="E63">A30</font></i></a>, <a class="txt" href="int_5.html#E33:59"><i><font color="Green" title="E64">A31</font></i></a>, <a class="ref" href="finseq_2.html#T55" title="FINSEQ_2:th.55">FINSEQ_2:55</a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E43:59"/><b>then </b>
( <font color="Maroon" title="c3">p</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c10">f2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">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>  &amp; <font color="Maroon" title="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">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="txt" href="int_5.html#E40:59"><i><font color="Green" title="E70">A39</font></i></a></i>;<br/>
<a NAME="E44:59"/><b>then </b>
( <font color="Maroon" title="c3">p</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c10">f2</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="c3">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>  &amp; <font color="Maroon" title="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</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="c3">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="E71">A40</font></i>:  <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c8">f1</font> = 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c10">f2</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="int_5.html#E39:59"><i><font color="Green" title="E69">A38</font></i></a>, <a class="ref" href="rvsum_1.html#T119" title="RVSUM_1:th.119">RVSUM_1:119</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c10">f2</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T117" title="RVSUM_1:th.117">RVSUM_1:117</a></i>
;<br/>
<a NAME="E46:59"/><b>then </b><i><font color="Green" title="E72">A41</font></i>: 
<font color="Maroon" title="c4">q</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="newton.html#K4" title="NEWTON:func.4">idseq</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c10">f2</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T117" title="RVSUM_1:th.117">RVSUM_1:117</a></i>;<br/>
<a NAME="E47:59"/>
<font color="Maroon" title="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">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#E8:59"><i><font color="Green" title="E48">A6</font></i></a>, <a class="txt" href="int_5.html#E13:59"><i><font color="Green" title="E51">A9</font></i></a>, <a class="txt" href="int_5.html#E33:59"><i><font color="Green" title="E64">A31</font></i></a>, <a class="ref" href="finseq_1.html#T28" title="FINSEQ_1:th.28">FINSEQ_1:28</a></i>;<br/>
<a NAME="E48:59"/><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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <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="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="c12">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="E50:59"/><i><font color="Green" title="E73">A42</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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">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="c12">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="E51:59"/>
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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_11"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:59_11"/>
 <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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c13">a</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:59_11"/><i><font color="Green" title="E74">A43</font></i>: 
( <font color="Maroon" title="c13">a</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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">a</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/>
<a NAME="E4:59_11"/>
<font color="Maroon" title="c8">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">a</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="int_5.html#E34:59"><i><font color="Green" title="E65">A32</font></i></a>, <a class="txt" href="int_5.html#E37:59"><i><font color="Green" title="E68">A35</font></i></a>, <a class="txt" href="int_5.html#E3:59_11"><i><font color="Green" title="E74">A43</font></i></a></i>;<br/>
<a NAME="E5:59_11"/><b>then </b>
<font color="Maroon" title="c4">q</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c13">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">a</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c3">p</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E25:59"><i><font color="Green" title="E61">A19</font></i></a>, <a class="txt" href="int_5.html#E34:59"><i><font color="Green" title="E65">A32</font></i></a>, <a class="txt" href="int_5.html#E3:59_11"><i><font color="Green" title="E74">A43</font></i></a></i>;<br/>
<a NAME="E6:59_11"/><b>then </b>
<font color="Maroon" title="c3">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">q</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c13">a</font>
 
<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="E7:59_11"/><b>then </b><i><font color="Green" title="E75">A44</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c13">a</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E2:59"><i><font color="Green" title="E44">A2</font></i></a>, <a class="ref" href="pepin.html#T3" title="PEPIN:th.3">PEPIN:3</a></i>;<br/>
<a NAME="E8:59_11"/><i><font color="Green" title="E76">A45</font></i>: 
( <font color="Maroon" title="c13">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c13">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">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#E8:59"><i><font color="Green" title="E48">A6</font></i></a>, <a class="txt" href="int_5.html#E33:59"><i><font color="Green" title="E64">A31</font></i></a>, <a class="txt" href="int_5.html#E3:59_11"><i><font color="Green" title="E74">A43</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E9:59_11"/><b>then </b>
<font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">a</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E7:59_11"><i><font color="Green" title="E75">A44</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="E10:59_11"/>
contradiction
 <b>by </b><i><a class="txt" href="int_5.html#E22:59"><i><font color="Green" title="E58">A16</font></i></a>, <a class="txt" href="int_5.html#E8:59_11"><i><font color="Green" title="E76">A45</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="E52:59"/><b>then </b><i><font color="Green" title="E74">A46</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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">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="E53:59"/><b>then </b><i><font color="Green" title="E75">A47</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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">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="c12">n1</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E50:59"><i><font color="Green" title="E73">A42</font></i></a>, <a class="ref" href="xboole_1.html#T73" title="XBOOLE_1:th.73">XBOOLE_1:73</a></i>;<br/>
<a NAME="E54:59"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span> &amp; <font color="Olive" title="b2">y</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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">x</font>, <font color="Maroon" title="c14">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:59_12"/><i><font color="Green" title="E76">A48</font></i>: 
( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span> &amp; <font color="Maroon" title="c14">y</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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c15">x</font> = <font color="Maroon" title="c13">x</font>, <font color="Maroon" title="c16">y</font> = <font color="Maroon" title="c14">y</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>  ;<br/>
<a NAME="E3:59_12"/>
( <font color="Maroon" title="c8">f1</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span>)</span> &amp; <font color="Maroon" title="c8">f1</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c16">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c16">y</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c16">y</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="int_5.html#E34:59"><i><font color="Green" title="E65">A32</font></i></a>, <a class="txt" href="int_5.html#E37:59"><i><font color="Green" title="E68">A35</font></i></a>, <a class="txt" href="int_5.html#E1:59_12"><i><font color="Green" title="E76">A48</font></i></a></i>;<br/>
<a NAME="E4:59_12"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c15">x</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="c8">f1</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c16">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c16">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">p</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E1:59_12"><i><font color="Green" title="E76">A48</font></i></a></i>;<br/>
<a NAME="E5:59_12"/><b>then </b>
<font color="Maroon" title="c3">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="c8">f1</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c15">x</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="c8">f1</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c16">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="int_1.html#D9" title="INT_1:def.9">INT_1:def 9</a></i>;<br/>
<b>hence </b><a NAME="E6:59_12"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">y</font>
 <b>by </b><i><a class="txt" href="int_5.html#E28:59"><i><font color="Green" title="E62">A22</font></i></a>, <a class="txt" href="int_5.html#E34:59"><i><font color="Green" title="E65">A32</font></i></a>, <a class="txt" href="int_5.html#E1:59_12"><i><font color="Green" title="E76">A48</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E55:59"/><b>then </b><i><font color="Green" title="E76">A49</font></i>: 
<font color="Maroon" title="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">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="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<b>set </b><font color="Maroon" title="c13">f4</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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c14">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"><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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></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="c3">p</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 ) </span>}</span> ;<br/>
<a NAME="E56:59"/><i><font color="Green" title="E77">A50</font></i>: 
 <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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></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"><font color="Maroon" title="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E53:59"><i><font color="Green" title="E75">A47</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="E57:59"/><i><font color="Green" title="E78">A51</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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">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#E53:59"><i><font color="Green" title="E75">A47</font></i></a>, <a class="ref" href="finseq_3.html#T99" title="FINSEQ_3:th.99">FINSEQ_3:99</a></i>;<br/>
<a NAME="E58:59"/><b>then </b><i><font color="Green" title="E79">A52</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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</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#E55:59"><i><font color="Green" title="E76">A49</font></i></a>, <a class="txt" href="int_5.html#E56:59"><i><font color="Green" title="E77">A50</font></i></a>, <a class="ref" href="rfinseq.html#T39" title="RFINSEQ:th.39">RFINSEQ:39</a></i>;<br/>
<a NAME="E59:59"/>
(  <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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 
<b>by </b><i><a class="ref" href="finseq_2.html#T27" title="FINSEQ_2:th.27">FINSEQ_2:27</a></i>;<br/>
<a NAME="E60:59"/><b>then </b><i><font color="Green" title="E80">A53</font></i>: 
 <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E58:59"><i><font color="Green" title="E79">A52</font></i></a>, <a class="ref" href="rfinseq.html#T22" title="RFINSEQ:th.22">RFINSEQ:22</a></i>;<br/>
<a NAME="E61:59"/>
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"><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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></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="c3">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"><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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:59_13"/>
<font color="Maroon" title="c15">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"><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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></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="c3">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="c16">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="E3:59_13"/><i><font color="Green" title="E81">A54</font></i>: 
( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">k</font> &amp; <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"><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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon" title="c16">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">p</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 )
 ;<br/>
<b>thus </b><a NAME="E4:59_13"/>
<font color="Maroon" title="c15">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"><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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="int_5.html#E3:59_13"><i><font color="Green" title="E81">A54</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E62:59"/><b>then </b><i><font color="Green" title="E81">A55</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"><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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></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="c3">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"><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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></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="c15">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"><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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></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="c3">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="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="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E64:59"/><i><font color="Green" title="E82">A56</font></i>: 
<font color="Maroon" title="c15">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="c12">n1</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E53:59"><i><font color="Green" title="E75">A47</font></i></a>, <a class="txt" href="int_5.html#E56:59"><i><font color="Green" title="E77">A50</font></i></a>, <a class="txt" href="int_5.html#E62:59"><i><font color="Green" title="E81">A55</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">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"><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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c15">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>  ;<br/>
<a NAME="E66:59"/>
<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"><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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c15">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"><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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></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/>
<a NAME="E67:59"/><b>then </b><i><font color="Green" title="E83">A57</font></i>: 
<font color="Maroon" title="c16">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="c12">n1</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E53:59"><i><font color="Green" title="E75">A47</font></i></a>, <a class="txt" href="int_5.html#E56:59"><i><font color="Green" title="E77">A50</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>set </b><font color="Maroon" title="c17">m</font> =  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">X</font>;<br/>
<a NAME="E68:59"/><i><font color="Green" title="E84">A58</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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">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"><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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E55:59"><i><font color="Green" title="E76">A49</font></i></a>, <a class="txt" href="int_5.html#E56:59"><i><font color="Green" title="E77">A50</font></i></a>, <a class="ref" href="finseq_4.html#T77" title="FINSEQ_4:th.77">FINSEQ_4:77</a></i>;<br/>
<a NAME="E69:59"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">X</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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E62:59"><i><font color="Green" title="E81">A55</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="c18">n</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">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> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">X</font></span>)</span> 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#E8:59"><i><font color="Green" title="E48">A6</font></i></a>, <a class="txt" href="int_5.html#E33:59"><i><font color="Green" title="E64">A31</font></i></a>, <a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a></i>;<br/>
<a NAME="E71:59"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E55:59"><i><font color="Green" title="E76">A49</font></i></a>, <a class="ref" href="finseq_4.html#T77" title="FINSEQ_4:th.77">FINSEQ_4:77</a></i>;<br/>
<a NAME="E72:59"/><b>then </b><i><font color="Green" title="E85">A59</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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">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="c3">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#E8:59"><i><font color="Green" title="E48">A6</font></i></a>, <a class="txt" href="int_5.html#E33:59"><i><font color="Green" title="E64">A31</font></i></a>, <a class="txt" href="int_5.html#E53:59"><i><font color="Green" title="E75">A47</font></i></a>, <a class="ref" href="finseq_3.html#T44" title="FINSEQ_3:th.44">FINSEQ_3:44</a></i>;<br/>
<a NAME="E73:59"/><i><font color="Green" title="E86">A60</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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">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="c16">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="c15">X</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_14"><b>proof </b></a><div class="add">

<a NAME="E1:59_14"/>
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="c16">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="c15">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="59_14_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">k</font>, <font color="Maroon" title="c20">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="E1:59_14_1"/><i><font color="Green" title="E87">A61</font></i>: 
( <font color="Maroon" title="c19">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Y</font> &amp; <font color="Maroon" title="c20">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">X</font> )
 ;<br/>

<a NAME="E2:59_14_1"/><b>then </b>
( <font color="Maroon" title="c19">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"><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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> &amp; not <font color="Maroon" title="c19">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">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="E3:59_14_1"/><b>then </b><i><font color="Green" title="E88">A62</font></i>: 
<font color="Maroon" title="c19">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">p</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 
;<br/>
<a NAME="E4:59_14_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="c20">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"><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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></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="c3">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#E1:59_14_1"><i><font color="Green" title="E87">A61</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:59_14_1"/>
<font color="Maroon" title="c19">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">l</font>
 <b>by </b><i><a class="txt" href="int_5.html#E3:59_14_1"><i><font color="Green" title="E88">A62</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="E2:59_14"/><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="c16">Y</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c15">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="c16">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="c15">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E64:59"><i><font color="Green" title="E82">A56</font></i></a>, <a class="txt" href="int_5.html#E67:59"><i><font color="Green" title="E83">A57</font></i></a>, <a class="ref" href="finseq_3.html#T48" title="FINSEQ_3:th.48">FINSEQ_3:48</a></i>;<br/>
<a NAME="E3:59_14"/><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"><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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c15">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="c16">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="c15">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="E4:59_14"/>
 <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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">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="c16">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="c15">X</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_5.html#E56:59"><i><font color="Green" title="E77">A50</font></i></a>, <a class="txt" href="int_5.html#E62:59"><i><font color="Green" title="E81">A55</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="E74:59"/>
 <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c16">Y</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c15">X</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E73:59"><i><font color="Green" title="E86">A60</font></i></a>, <a class="ref" href="rvsum_1.html#T105" title="RVSUM_1:th.105">RVSUM_1:105</a></i>;<br/>
<a NAME="E75:59"/><b>then </b><i><font color="Green" title="E87">A63</font></i>: 
<font color="Maroon" title="c4">q</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="newton.html#K4" title="NEWTON:func.4">idseq</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c10">f2</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c16">Y</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c15">X</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E45:59"><i><font color="Green" title="E71">A40</font></i></a>, <a class="txt" href="int_5.html#E60:59"><i><font color="Green" title="E80">A53</font></i></a>, <a class="ref" href="rvsum_1.html#T117" title="RVSUM_1:th.117">RVSUM_1:117</a></i>;<br/>
<a NAME="E76:59"/><i><font color="Green" title="E88">A64</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="c16">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_15"><b>proof </b></a><div class="add">

 <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="c16">Y</font></span>)</span> = 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c16">Y</font>
<b>by </b><i><a class="txt" href="int_5.html#E67:59"><i><font color="Green" title="E83">A57</font></i></a>, <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="c3">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> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">X</font></span>)</span>
<b>by </b><i><a class="txt" href="int_5.html#E8:59"><i><font color="Green" title="E48">A6</font></i></a>, <a class="txt" href="int_5.html#E33:59"><i><font color="Green" title="E64">A31</font></i></a>, <a class="txt" href="int_5.html#E62:59"><i><font color="Green" title="E81">A55</font></i></a>, <a class="txt" href="int_5.html#E68:59"><i><font color="Green" title="E84">A58</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="E2:59_15"/>
 <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="c16">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">n</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E77:59"/><b>then </b><i><font color="Green" title="E89">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">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="c15">X</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E73:59"><i><font color="Green" title="E86">A60</font></i></a>, <a class="ref" href="finseq_5.html#T40" title="FINSEQ_5:th.40">FINSEQ_5:40</a></i>;<br/>
<a NAME="E78:59"/>
 <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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c18">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">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="E79:59"/><b>then </b><i><font color="Green" title="E90">A66</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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c18">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">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#E57:59"><i><font color="Green" title="E78">A51</font></i></a>, <a class="ref" href="finseq_3.html#T98" title="FINSEQ_3:th.98">FINSEQ_3:98</a></i>;<br/>
<a NAME="E80:59"/>
 <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c16">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E73:59"><i><font color="Green" title="E86">A60</font></i></a>, <a class="ref" href="finseq_6.html#T12" title="FINSEQ_6:th.12">FINSEQ_6:12</a></i>;<br/>
<a NAME="E81:59"/><b>then </b><i><font color="Green" title="E91">A67</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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c18">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="c16">Y</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E76:59"><i><font color="Green" title="E88">A64</font></i></a>, <a class="ref" href="finseq_3.html#T122" title="FINSEQ_3:th.122">FINSEQ_3:122</a></i>;<br/>
<a NAME="E82:59"/>
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"><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="c3">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> holds <br/><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="c3">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="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="59_16"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:59_16"/>
<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> <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="c3">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>
 ;<br/>

<a NAME="E2:59_16"/><b>then </b><i><font color="Green" title="E92">A68</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#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"><a href="newton.html#K4" title="NEWTON:func.4">idseq</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">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="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:59_16"/><b>then </b>
<font color="Maroon" title="c19">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="c3">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="E4:59_16"/><b>then </b>
<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="c3">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="funct_1.html#K1" title="FUNCT_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="c19">d</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E13:59"><i><font color="Green" title="E51">A9</font></i></a>, <a class="ref" href="finseq_2.html#T57" title="FINSEQ_2:th.57">FINSEQ_2:57</a></i>;<br/>
<b>hence </b><a NAME="E5:59_16"/>
<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="c3">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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">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="txt" href="int_5.html#E2:59_16"><i><font color="Green" title="E92">A68</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E83:59"/><b>then </b>
 <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="c3">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> 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> 
 
<b>by </b><i><a class="ref" href="finseq_2.html#T14" title="FINSEQ_2:th.14">FINSEQ_2:14</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c19">M</font> =  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</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="c3">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> 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#E55"><i><font color="Green" title="E41">Lm4</font></i></a></i>;<br/>
<a NAME="E85:59"/><i><font color="Green" title="E92">A69</font></i>: 
 <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">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> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c10">f2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_17"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_17_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_17_1"/>
( <font color="Maroon" title="c15">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  not <font color="Maroon" title="c15">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="59_17_1_1"><b>suppose </b></a><a NAME="E1:59_17_1_1"/><i><font color="Green" title="E93">A70</font></i>: 
<font color="Maroon" title="c15">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><a NAME="E2:59_17_1_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"><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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> holds <br/>( <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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></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"><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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></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="c3">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="59_17_1_1_1"><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="E1:59_17_1_1_1"/><i><font color="Green" title="E94">A71</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"><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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:59_17_1_1_1"/><i><font color="Green" title="E95">A72</font></i>: 
not <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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">X</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E1:59_17_1_1"><i><font color="Green" title="E93">A70</font></i></a></i>;<br/>
<a NAME="E3:59_17_1_1_1"/>
<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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">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> <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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E1:59_17_1_1_1"><i><font color="Green" title="E94">A71</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E4:59_17_1_1_1"/><b>then </b><i><font color="Green" title="E96">A73</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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <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> <font color="Maroon" title="c3">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#E2:59_17_1_1_1"><i><font color="Green" title="E95">A72</font></i></a></i>;<br/>
<a NAME="E5:59_17_1_1_1"/><i><font color="Green" title="E97">A74</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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E56:59"><i><font color="Green" title="E77">A50</font></i></a>, <a class="txt" href="int_5.html#E1:59_17_1_1_1"><i><font color="Green" title="E94">A71</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E6:59_17_1_1_1"/><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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E7:59_17_1_1_1"/><b>then </b>
not <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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">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#E52:59"><i><font color="Green" title="E74">A46</font></i></a>, <a class="txt" href="int_5.html#E5:59_17_1_1_1"><i><font color="Green" title="E97">A74</font></i></a>, <a class="ref" href="xboole_0.html#T5" title="XBOOLE_0:th.5">XBOOLE_0:5</a></i>;<br/>
<a NAME="E8:59_17_1_1_1"/><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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">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="E9:59_17_1_1_1"/>
( <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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <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; <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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <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="c3">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#E23:59"><i><font color="Green" title="E59">A17</font></i></a>, <a class="txt" href="int_5.html#E4:59_17_1_1_1"><i><font color="Green" title="E96">A73</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="E3:59_17_1_1"/><b>then </b>
 <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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <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> <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"><span class="p3">(<span class="default"><font color="Maroon" title="c3">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#E56:59"><i><font color="Green" title="E77">A50</font></i></a>, <a class="txt" href="int_5.html#E57:59"><i><font color="Green" title="E78">A51</font></i></a>, <a class="txt" href="int_5.html#E72:59"><i><font color="Green" title="E85">A59</font></i></a>, <a class="ref" href="int_5.html#T40" target="_self" title="INT_5:th.40">Th40</a></i>;<br/><a NAME="E4:59_17_1_1"/><b>then </b>
<font color="Maroon" title="c4">q</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="newton.html#K4" title="NEWTON:func.4">idseq</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c10">f2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="newton.html#K4" title="NEWTON:func.4">idseq</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">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#E46:59"><i><font color="Green" title="E72">A41</font></i></a>, <a class="txt" href="int_5.html#E60:59"><i><font color="Green" title="E80">A53</font></i></a>, <a class="ref" href="finseq_3.html#T54" title="FINSEQ_3:th.54">FINSEQ_3:54</a></i>;<br/><a NAME="E5:59_17_1_1"/><b>then </b><i><font color="Green" title="E94">A75</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="newton.html#K4" title="NEWTON:func.4">idseq</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">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#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="newton.html#K4" title="NEWTON:func.4">idseq</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c10">f2</font></span>)</span>
 ;<br/><a NAME="E6:59_17_1_1"/>
2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">M</font>
 <b>by </b><i><a class="txt" href="int_5.html#E16:59"><i><font color="Green" title="E53">A11</font></i></a>, <a class="ref" href="nat_d.html#T9" title="NAT_D:th.9">NAT_D:9</a></i>;<br/><a NAME="E7:59_17_1_1"/><b>then </b>
2 <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c10">f2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="int_5.html#E5:59"><i><font color="Green" title="E46">A4</font></i></a>, <a class="txt" href="int_5.html#E10:59"><i><font color="Green" title="E50">A8</font></i></a>, <a class="txt" href="int_5.html#E5:59_17_1_1"><i><font color="Green" title="E94">A75</font></i></a>, <a class="ref" href="pepin.html#T3" title="PEPIN:th.3">PEPIN:3</a></i>;<br/><a NAME="E8:59_17_1_1"/><b>then </b>
 <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c10">f2</font>, <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> 2
 <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="E9:59_17_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c10">f2</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2
 <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="E10:59_17_1_1"/><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> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">X</font></span>)</span> <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> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c10">f2</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_5.html#E1:59_17_1_1"><i><font color="Green" title="E93">A70</font></i></a>, <a class="ref" href="int_5.html#T45" target="_self" title="INT_5:th.45">Th45</a>, <a class="ref" href="card_1.html#T78" title="CARD_1:th.78">CARD_1:78</a></i>;<br/><b>hence </b><a NAME="E11:59_17_1_1"/>
 <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">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> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c10">f2</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_5.html#E1:59"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="int_5.html#E3:59"><i><font color="Green" title="E45">A3</font></i></a>, <a class="txt" href="int_5.html#E56:59"><i><font color="Green" title="E77">A50</font></i></a>, <a class="ref" href="int_5.html#T41" target="_self" title="INT_5:th.41">Th41</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_17_1_2"><b>suppose </b></a><a NAME="E1:59_17_1_2"/>
not <font color="Maroon" title="c15">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><a NAME="E2:59_17_1_2"/><b>then </b><i><font color="Green" title="E93">A76</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">X</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="card_4.html#T17" title="CARD_4:th.17">CARD_4:17</a></i>;<br/><a NAME="E3:59_17_1_2"/><i><font color="Green" title="E94">A77</font></i>: 
<font color="Maroon" title="c18">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="int_5.html#E72:59"><i><font color="Green" title="E85">A59</font></i></a>, <a class="ref" href="xreal_1.html#T45" title="XREAL_1:th.45">XREAL_1:45</a></i>;<br/><a NAME="E4:59_17_1_2"/><b>then </b><i><font color="Green" title="E95">A78</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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">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/><i><font color="Green" title="E96">A79</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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c18">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">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="c18">n</font>
<b>by </b><i><a class="txt" href="int_5.html#E3:59_17_1_2"><i><font color="Green" title="E94">A77</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>
<br/>.= 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">X</font>
<b>by </b><i><a class="txt" href="int_5.html#E72:59"><i><font color="Green" title="E85">A59</font></i></a></i>
;<br/><b>set </b><font color="Maroon" title="c20">f5</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">X</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c3">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">n</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c21">f6</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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c18">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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">X</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c3">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span>;<br/><i><font color="Green" title="E97">A80</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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">X</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c3">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">X</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c3">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">X</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c3">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="int_5.html#E5:59_17_1_2"><i><font color="Green" title="E96">A79</font></i></a>, <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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">n</font></span>)</span>

;<br/><a NAME="E7:59_17_1_2"/><b>then </b><i><font color="Green" title="E98">A81</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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">X</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c3">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">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/><a NAME="E8:59_17_1_2"/><i><font color="Green" title="E99">A82</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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">X</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c3">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">X</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c3">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">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="c3">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">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="59_17_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c22">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:59_17_1_2_3"/><i><font color="Green" title="E100">A83</font></i>: 
<font color="Maroon" title="c22">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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">X</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c3">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:59_17_1_2_3"/><b>then </b>
<font color="Maroon" title="c22">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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E5:59_17_1_2"><i><font color="Green" title="E96">A79</font></i></a>, <a class="txt" href="int_5.html#E6:59_17_1_2"><i><font color="Green" title="E97">A80</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="E3:59_17_1_2_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">X</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E2:59_17_1_2"><i><font color="Green" title="E93">A76</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="E4:59_17_1_2_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">X</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c3">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">d</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_5.html#E1:59_17_1_2_3"><i><font color="Green" title="E100">A83</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="E9:59_17_1_2"/><i><font color="Green" title="E100">A84</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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">X</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c3">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">X</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c3">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">X</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c3">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">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="c3">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="59_17_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c22">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:59_17_1_2_4"/><i><font color="Green" title="E101">A85</font></i>: 
<font color="Maroon" title="c22">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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">X</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c3">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:59_17_1_2_4"/><b>then </b><i><font color="Green" title="E102">A86</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">X</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c3">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">d</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E8:59_17_1_2"><i><font color="Green" title="E99">A82</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c23">w</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">X</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c3">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">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="E4:59_17_1_2_4"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c15">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">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="c15">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E77:59"><i><font color="Green" title="E89">A65</font></i></a>, <a class="txt" href="int_5.html#E6:59_17_1_2"><i><font color="Green" title="E97">A80</font></i></a>, <a class="txt" href="int_5.html#E1:59_17_1_2_4"><i><font color="Green" title="E101">A85</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E5:59_17_1_2_4"/><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="c15">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">X</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E64:59"><i><font color="Green" title="E82">A56</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="c24">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="E7:59_17_1_2_4"/><i><font color="Green" title="E103">A87</font></i>: 
( <font color="Maroon" title="c24">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="c15">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">d</font> &amp; <font color="Maroon" title="c24">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span> &amp; <font color="Maroon" title="c24">ll</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">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#E56:59"><i><font color="Green" title="E77">A50</font></i></a></i>;<br/>
<a NAME="E8:59_17_1_2_4"/><i><font color="Green" title="E104">A88</font></i>: 
<font color="Maroon" title="c23">w</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">p</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">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#E77:59"><i><font color="Green" title="E89">A65</font></i></a>, <a class="txt" href="int_5.html#E2:59_17_1_2_4"><i><font color="Green" title="E102">A86</font></i></a>, <a class="txt" href="int_5.html#E7:59_17_1_2_4"><i><font color="Green" title="E103">A87</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="c25">e</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E10:59_17_1_2_4"/><i><font color="Green" title="E105">A89</font></i>: 
( <font color="Maroon" title="c25">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">d</font> )
 <b>by </b><i><a class="txt" href="int_5.html#E77:59"><i><font color="Green" title="E89">A65</font></i></a>, <a class="txt" href="int_5.html#E7:59_17_1_2_4"><i><font color="Green" title="E103">A87</font></i></a>, <a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E11:59_17_1_2_4"/>
<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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">e</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">p</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E34:59"><i><font color="Green" title="E65">A32</font></i></a>, <a class="txt" href="int_5.html#E10:59_17_1_2_4"><i><font color="Green" title="E105">A89</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="E12:59_17_1_2_4"/><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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">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="E13:59_17_1_2_4"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">X</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c3">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">X</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c3">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">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="c3">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#E23:59"><i><font color="Green" title="E59">A17</font></i></a>, <a class="txt" href="int_5.html#E2:59_17_1_2_4"><i><font color="Green" title="E102">A86</font></i></a>, <a class="txt" href="int_5.html#E8:59_17_1_2_4"><i><font color="Green" title="E104">A88</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="E10:59_17_1_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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">X</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c3">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">X</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c3">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">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="59_17_1_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c22">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:59_17_1_2_5"/>
<font color="Maroon" title="c22">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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">X</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c3">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:59_17_1_2_5"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">X</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c3">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">X</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c3">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">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#E8:59_17_1_2"><i><font color="Green" title="E99">A82</font></i></a>, <a class="txt" href="int_5.html#E9:59_17_1_2"><i><font color="Green" title="E100">A84</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:59_17_1_2_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">X</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c3">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">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="c22">f5</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">X</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c3">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">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="E12:59_17_1_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="c22">f5</font> holds <br/><font color="Maroon" title="c22">f5</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="c22">f5</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="59_17_1_2_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">d</font>, <font color="Maroon" title="c24">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="E1:59_17_1_2_6"/><i><font color="Green" title="E101">A90</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">d</font> &amp; <font color="Maroon" title="c23">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">e</font> &amp; <font color="Maroon" title="c24">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="c22">f5</font> )
 ;<br/>

<a NAME="E2:59_17_1_2_6"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">d</font> &amp; <font color="Maroon" title="c23">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="c22">f5</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">e</font> &amp; <font color="Maroon" title="c24">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="c22">f5</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="E3:59_17_1_2_6"/><b>then </b><i><font color="Green" title="E102">A91</font></i>: 
( <font color="Maroon" title="c23">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="c22">f5</font> &amp; <font color="Maroon" title="c24">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="c22">f5</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="E4:59_17_1_2_6"/><b>then </b>
( <font color="Maroon" title="c22">f5</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c23">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c23">d</font></span>)</span> &amp; <font color="Maroon" title="c22">f5</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c24">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c24">e</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="int_5.html#E8:59_17_1_2"><i><font color="Green" title="E99">A82</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:59_17_1_2_6"/>
<font color="Maroon" title="c22">f5</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c23">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c22">f5</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c24">e</font>
 <b>by </b><i><a class="txt" href="int_5.html#E79:59"><i><font color="Green" title="E90">A66</font></i></a>, <a class="txt" href="int_5.html#E6:59_17_1_2"><i><font color="Green" title="E97">A80</font></i></a>, <a class="txt" href="int_5.html#E1:59_17_1_2_6"><i><font color="Green" title="E101">A90</font></i></a>, <a class="txt" href="int_5.html#E3:59_17_1_2_6"><i><font color="Green" title="E102">A91</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="E13:59_17_1_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">f5</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="c22">f5</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="E14:59_17_1_2"/><b>then </b><i><font color="Green" title="E101">A92</font></i>: 
<font color="Maroon" title="c22">f5</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="E15:59_17_1_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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c18">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="c22">f5</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="c23">f6</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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c18">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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">X</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c3">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">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="E102">A93</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">f6</font> = 
<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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">f5</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">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#E4:59_17_1_2"><i><font color="Green" title="E95">A78</font></i></a>, <a class="txt" href="int_5.html#E5:59_17_1_2"><i><font color="Green" title="E96">A79</font></i></a>, <a class="txt" href="int_5.html#E7:59_17_1_2"><i><font color="Green" title="E98">A81</font></i></a></i>
;<br/><a NAME="E18:59_17_1_2"/><i><font color="Green" title="E103">A94</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="c3">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="c3">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/><a NAME="E19:59_17_1_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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c18">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="c22">f5</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_17_1_2_8"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:59_17_1_2_8"/>
not  <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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c18">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="c22">f5</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c24">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:59_17_1_2_8"/><i><font color="Green" title="E104">A95</font></i>: 
( <font color="Maroon" title="c24">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c18">n</font></span>)</span> &amp; <font color="Maroon" title="c24">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="c22">f5</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="E4:59_17_1_2_8"/>
 <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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c18">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">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="c25">d1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E6:59_17_1_2_8"/><i><font color="Green" title="E105">A96</font></i>: 
( <font color="Maroon" title="c25">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">x</font> )
 <b>by </b><i><a class="txt" href="int_5.html#E56:59"><i><font color="Green" title="E77">A50</font></i></a>, <a class="txt" href="int_5.html#E3:59_17_1_2_8"><i><font color="Green" title="E104">A95</font></i></a>, <a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E7:59_17_1_2_8"/><i><font color="Green" title="E106">A97</font></i>: 
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">d1</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">p</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E34:59"><i><font color="Green" title="E65">A32</font></i></a>, <a class="txt" href="int_5.html#E6:59_17_1_2_8"><i><font color="Green" title="E105">A96</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="c26">e</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E9:59_17_1_2_8"/><i><font color="Green" title="E107">A98</font></i>: 
( <font color="Maroon" title="c26">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="c22">f5</font> &amp; <font color="Maroon" title="c22">f5</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">x</font> )
 <b>by </b><i><a class="txt" href="int_5.html#E3:59_17_1_2_8"><i><font color="Green" title="E104">A95</font></i></a>, <a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E10:59_17_1_2_8"/>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">e</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E8:59_17_1_2"><i><font color="Green" title="E99">A82</font></i></a>, <a class="txt" href="int_5.html#E9:59_17_1_2_8"><i><font color="Green" title="E107">A98</font></i></a></i>;<br/>
<a NAME="E11:59_17_1_2_8"/><b>then </b>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">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="c26">e</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E3:59_17_1_2"><i><font color="Green" title="E94">A77</font></i></a>, <a class="txt" href="int_5.html#E6:59_17_1_2"><i><font color="Green" title="E97">A80</font></i></a>, <a class="txt" href="int_5.html#E9:59_17_1_2_8"><i><font color="Green" title="E107">A98</font></i></a>, <a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></i>;<br/>
<a NAME="E12:59_17_1_2_8"/><b>then </b><i><font color="Green" title="E108">A99</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">d1</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">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="c26">e</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E7:59_17_1_2_8"><i><font color="Green" title="E106">A97</font></i></a></i>;<br/>
<a NAME="E13:59_17_1_2_8"/>
<font color="Maroon" title="c26">e</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c18">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E6:59_17_1_2"><i><font color="Green" title="E97">A80</font></i></a>, <a class="txt" href="int_5.html#E9:59_17_1_2_8"><i><font color="Green" title="E107">A98</font></i></a>, <a class="ref" href="finseq_5.html#T29" title="FINSEQ_5:th.29">FINSEQ_5:29</a></i>;<br/>
<a NAME="E14:59_17_1_2_8"/><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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">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="c26">e</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c18">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E56:59"><i><font color="Green" title="E77">A50</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="c27">e1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E16:59_17_1_2_8"/><i><font color="Green" title="E109">A100</font></i>: 
( <font color="Maroon" title="c27">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">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="c26">e</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c18">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="E17:59_17_1_2_8"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">d1</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">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="c8">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">e1</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E34:59"><i><font color="Green" title="E65">A32</font></i></a>, <a class="txt" href="int_5.html#E12:59_17_1_2_8"><i><font color="Green" title="E108">A99</font></i></a>, <a class="txt" href="int_5.html#E16:59_17_1_2_8"><i><font color="Green" title="E109">A100</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="E18:59_17_1_2_8"/><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="c8">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">d1</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">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="c8">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">e1</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">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="E19:59_17_1_2_8"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">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="c8">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">e1</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">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="E20:59_17_1_2_8"/><b>then </b>
<font color="Maroon" title="c3">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="c8">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">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="c8">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">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="E21:59_17_1_2_8"/><b>then </b>
<font color="Maroon" title="c3">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="c25">d1</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">e1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E25:59"><i><font color="Green" title="E61">A19</font></i></a>, <a class="txt" href="int_5.html#E34:59"><i><font color="Green" title="E65">A32</font></i></a>, <a class="txt" href="int_5.html#E6:59_17_1_2_8"><i><font color="Green" title="E105">A96</font></i></a></i>;<br/>
<a NAME="E22:59_17_1_2_8"/><b>then </b>
<font color="Maroon" title="c3">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="c25">d1</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">e1</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c4">q</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E25:59"><i><font color="Green" title="E61">A19</font></i></a>, <a class="txt" href="int_5.html#E34:59"><i><font color="Green" title="E65">A32</font></i></a>, <a class="txt" href="int_5.html#E16:59_17_1_2_8"><i><font color="Green" title="E109">A100</font></i></a></i>;<br/>
<a NAME="E23:59_17_1_2_8"/><b>then </b>
<font color="Maroon" title="c3">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="c25">d1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c27">e1</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c4">q</font>
 
;<br/>
<a NAME="E24:59_17_1_2_8"/><b>then </b><i><font color="Green" title="E110">A101</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c25">d1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c27">e1</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E2:59"><i><font color="Green" title="E44">A2</font></i></a>, <a class="ref" href="pepin.html#T3" title="PEPIN:th.3">PEPIN:3</a></i>;<br/>
<a NAME="E25:59_17_1_2_8"/>
( <font color="Maroon" title="c25">d1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c25">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="c3">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="c27">e1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c27">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="c3">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#E8:59"><i><font color="Green" title="E48">A6</font></i></a>, <a class="txt" href="int_5.html#E33:59"><i><font color="Green" title="E64">A31</font></i></a>, <a class="txt" href="int_5.html#E6:59_17_1_2_8"><i><font color="Green" title="E105">A96</font></i></a>, <a class="txt" href="int_5.html#E16:59_17_1_2_8"><i><font color="Green" title="E109">A100</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E26:59_17_1_2_8"/><b>then </b>
( <font color="Maroon" title="c25">d1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c27">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="c25">d1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c27">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="c3">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="c3">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="E27:59_17_1_2_8"/><b>then </b>
( <font color="Maroon" title="c25">d1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c27">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="c25">d1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c27">e1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">p</font> )
 
<b>by </b><i><a class="txt" href="int_5.html#E17:59"><i><font color="Green" title="E54">A12</font></i></a>, <a class="txt" href="int_5.html#E21:59"><i><font color="Green" title="E57">A15</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="E28:59_17_1_2_8"/>
contradiction
 <b>by </b><i><a class="txt" href="int_5.html#E24:59_17_1_2_8"><i><font color="Green" title="E110">A101</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="E20:59_17_1_2"/><b>then </b><i><font color="Green" title="E104">A102</font></i>: 
<font color="Maroon" title="c23">f6</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#E79:59"><i><font color="Green" title="E90">A66</font></i></a>, <a class="txt" href="int_5.html#E14:59_17_1_2"><i><font color="Green" title="E101">A92</font></i></a>, <a class="ref" href="finseq_3.html#T98" title="FINSEQ_3:th.98">FINSEQ_3:98</a></i>;<br/><a NAME="E21:59_17_1_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="c23">f6</font> holds <br/>( <font color="Maroon" title="c23">f6</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="c23">f6</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="c3">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="59_17_1_2_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c24">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:59_17_1_2_9"/><i><font color="Green" title="E105">A103</font></i>: 
<font color="Maroon" title="c24">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="c23">f6</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_17_1_2_9_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_17_1_2_9_1"/>
( <font color="Maroon" title="c24">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c18">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="c22">f5</font> &amp; <font color="Maroon" title="c24">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c18">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#E1:59_17_1_2_9"><i><font color="Green" title="E105">A103</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="59_17_1_2_9_1_1"><b>suppose </b></a><a NAME="E1:59_17_1_2_9_1_1"/><i><font color="Green" title="E106">A104</font></i>: 
<font color="Maroon" title="c24">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c18">n</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:59_17_1_2_9_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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">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="c16">Y</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_5.html#E81:59"><i><font color="Green" title="E91">A67</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><a NAME="E3:59_17_1_2_9_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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Y</font>
 <b>by </b><i><a class="txt" href="int_5.html#E67:59"><i><font color="Green" title="E83">A57</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="E4:59_17_1_2_9_1_1"/><b>then </b><i><font color="Green" title="E107">A105</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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">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> <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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">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="E5:59_17_1_2_9_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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">p</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 ;<br/><a NAME="E6:59_17_1_2_9_1_1"/><b>then </b><i><font color="Green" title="E108">A106</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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">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="c3">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#E23:59"><i><font color="Green" title="E59">A17</font></i></a>, <a class="ref" href="int_1.html#T81" title="INT_1:th.81">INT_1:81</a></i>;<br/><a NAME="E7:59_17_1_2_9_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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">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#E52:59"><i><font color="Green" title="E74">A46</font></i></a>, <a class="txt" href="int_5.html#E56:59"><i><font color="Green" title="E77">A50</font></i></a>, <a class="txt" href="int_5.html#E4:59_17_1_2_9_1_1"><i><font color="Green" title="E107">A105</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><a NAME="E8:59_17_1_2_9_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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">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="E9:59_17_1_2_9_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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">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="E10:59_17_1_2_9_1_1"/>
( <font color="Maroon" title="c23">f6</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">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="c23">f6</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">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="c3">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#E1:59_17_1_2_9_1_1"><i><font color="Green" title="E106">A104</font></i></a>, <a class="txt" href="int_5.html#E6:59_17_1_2_9_1_1"><i><font color="Green" title="E108">A106</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="59_17_1_2_9_1_2"><b>suppose </b></a><a NAME="E1:59_17_1_2_9_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="c22">f5</font> &amp; <font color="Maroon" title="c24">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c18">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="c25">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="E3:59_17_1_2_9_1_2"/><i><font color="Green" title="E106">A107</font></i>: 
( <font color="Maroon" title="c25">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="c22">f5</font> &amp; <font color="Maroon" title="c24">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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c25">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="E4:59_17_1_2_9_1_2"/>
<font color="Maroon" title="c23">f6</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">f5</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c25">l</font>
 <b>by </b><i><a class="txt" href="int_5.html#E3:59_17_1_2_9_1_2"><i><font color="Green" title="E106">A107</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="E5:59_17_1_2_9_1_2"/>
( <font color="Maroon" title="c23">f6</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">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="c23">f6</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">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="c3">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#E9:59_17_1_2"><i><font color="Green" title="E100">A84</font></i></a>, <a class="txt" href="int_5.html#E3:59_17_1_2_9_1_2"><i><font color="Green" title="E106">A107</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E22:59_17_1_2"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c23">f6</font> <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="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="c3">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#E17:59_17_1_2"><i><font color="Green" title="E102">A93</font></i></a>, <a class="txt" href="int_5.html#E18:59_17_1_2"><i><font color="Green" title="E103">A94</font></i></a>, <a class="txt" href="int_5.html#E20:59_17_1_2"><i><font color="Green" title="E104">A102</font></i></a>, <a class="ref" href="int_5.html#T40" target="_self" title="INT_5:th.40">Th40</a></i>;<br/><a NAME="E23:59_17_1_2"/><b>then </b><i><font color="Green" title="E105">A108</font></i>: 
<font color="Maroon" title="c23">f6</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="c3">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="rfinseq.html#R1" title="RFINSEQ:pred.1">are_fiberwise_equipotent</a> 
 <b>by </b><i><a class="txt" href="int_5.html#E20:59_17_1_2"><i><font color="Green" title="E104">A102</font></i></a>, <a class="ref" href="rfinseq.html#T39" title="RFINSEQ:th.39">RFINSEQ:39</a></i>;<br/><a NAME="E24:59_17_1_2"/><i><font color="Green" title="E106">A109</font></i>: 
( <font color="Maroon" title="c23">f6</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">n</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 <b>by </b><i><a class="ref" href="finseq_2.html#T27" title="FINSEQ_2:th.27">FINSEQ_2:27</a></i>;<br/><b>then </b><font color="Maroon" title="c19">M</font> = 
 <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c23">f6</font>
<b>by </b><i><a class="txt" href="int_5.html#E23:59_17_1_2"><i><font color="Green" title="E105">A108</font></i></a>, <a class="ref" href="rfinseq.html#T22" title="RFINSEQ:th.22">RFINSEQ:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c22">f5</font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T105" title="RVSUM_1:th.105">RVSUM_1:105</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">X</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="int_5.html#E5:59_17_1_2"><i><font color="Green" title="E96">A79</font></i></a>, <a class="txt" href="int_5.html#E24:59_17_1_2"><i><font color="Green" title="E106">A109</font></i></a>, <a class="ref" href="int_5.html#T47" target="_self" title="INT_5:th.47">Th47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c18">n</font></span>)</span></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"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">X</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</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="c8">f1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c18">n</font></span>)</span></span>)</span>

;<br/><a NAME="E26:59_17_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c10">f2</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c15">X</font></span>)</span></span>)</span></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"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">X</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_5.html#E75:59"><i><font color="Green" title="E87">A63</font></i></a>, <a class="txt" href="int_5.html#E77:59"><i><font color="Green" title="E89">A65</font></i></a>, <a class="txt" href="int_5.html#E81:59"><i><font color="Green" title="E91">A67</font></i></a></i>;<br/><b>then </b><i><font color="Green" title="E107">A110</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">M</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c10">f2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">X</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c15">X</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2
<b>by </b><i><a class="txt" href="int_5.html#E9:59"><i><font color="Green" title="E49">A7</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c10">f2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">X</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2
<b>by </b><i><a class="ref" href="euler_1.html#T13" title="EULER_1:th.13">EULER_1:13</a></i>
;<br/><a NAME="E28:59_17_1_2"/>
2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">M</font>
 <b>by </b><i><a class="txt" href="int_5.html#E16:59"><i><font color="Green" title="E53">A11</font></i></a>, <a class="ref" href="nat_d.html#T9" title="NAT_D:th.9">NAT_D:9</a></i>;<br/><a NAME="E29:59_17_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c19">M</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <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="pepin.html#T6" title="PEPIN:th.6">PEPIN:6</a></i>;<br/><a NAME="E30:59_17_1_2"/><b>then </b><i><font color="Green" title="E108">A111</font></i>: 
2 <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c3">p</font> <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#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c10">f2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">X</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="int_5.html#E27:59_17_1_2"><i><font color="Green" title="E107">A110</font></i></a>, <a class="txt" href="int_5.html#E4"><i><font color="Green" title="E4">Lm2</font></i></a></i>;<br/><a NAME="E31:59_17_1_2"/>
2,<font color="Maroon" title="c3">p</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 <b>by </b><i><a class="txt" href="int_5.html#E5:59"><i><font color="Green" title="E46">A4</font></i></a>, <a class="ref" href="euler_2.html#T1" title="EULER_2:th.1">EULER_2:1</a></i>;<br/><a NAME="E32:59_17_1_2"/><b>then </b>
2 <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c10">f2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">X</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_5.html#E30:59_17_1_2"><i><font color="Green" title="E108">A111</font></i></a>, <a class="ref" href="int_2.html#T40" title="INT_2:th.40">INT_2:40</a></i>;<br/><a NAME="E33:59_17_1_2"/><b>then </b>
 <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c10">f2</font>, <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">X</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> 2
 <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="E34:59_17_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c10">f2</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">X</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2
 <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="E35:59_17_1_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> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c10">f2</font></span>)</span> <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> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="int_5.html#T45" target="_self" title="INT_5:th.45">Th45</a></i>;<br/><b>hence </b><a NAME="E36:59_17_1_2"/>
 <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">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> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c10">f2</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_5.html#E1:59"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="int_5.html#E3:59"><i><font color="Green" title="E45">A3</font></i></a>, <a class="txt" href="int_5.html#E56:59"><i><font color="Green" title="E77">A50</font></i></a>, <a class="ref" href="int_5.html#T41" target="_self" title="INT_5:th.41">Th41</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c20">g1</font> = <font color="Maroon" title="c3">p</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="c4">q</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>;<br/>
<a NAME="E86:59"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="newton.html#K4" title="NEWTON:func.4">idseq</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">q</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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</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="c4">q</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="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>;<br/>
<a NAME="E87:59"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="newton.html#K4" title="NEWTON:func.4">idseq</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">q</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="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"><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="c4">q</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="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E88:59"/><b>then </b><i><font color="Green" title="E93">A112</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="c3">p</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="newton.html#K4" title="NEWTON:func.4">idseq</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">q</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</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="E89:59"/><i><font color="Green" title="E94">A113</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"><font color="Maroon" title="c3">p</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="newton.html#K4" title="NEWTON:func.4">idseq</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">q</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> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="newton.html#K4" title="NEWTON:func.4">idseq</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">q</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="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="c3">p</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b1">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_18"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c21">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:59_18"/><i><font color="Green" title="E95">A114</font></i>: 
<font color="Maroon" title="c21">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="c3">p</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="newton.html#K4" title="NEWTON:func.4">idseq</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">q</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/>

<a NAME="E2:59_18"/><b>then </b><i><font color="Green" title="E96">A115</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="newton.html#K4" title="NEWTON:func.4">idseq</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">q</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <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="newton.html#K4" title="NEWTON:func.4">idseq</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">q</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">d</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>;<br/>
<a NAME="E3:59_18"/>
<font color="Maroon" title="c21">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"><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="c4">q</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#E1:59_18"><i><font color="Green" title="E95">A114</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="E4:59_18"/><b>then </b>
<font color="Maroon" title="c21">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"><a href="newton.html#K4" title="NEWTON:func.4">idseq</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">q</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="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:59_18"/><b>then </b>
<font color="Maroon" title="c21">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="c4">q</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>hence </b><a NAME="E6:59_18"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="newton.html#K4" title="NEWTON:func.4">idseq</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">q</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c21">d</font>
 <b>by </b><i><a class="txt" href="int_5.html#E13:59"><i><font color="Green" title="E51">A9</font></i></a>, <a class="txt" href="int_5.html#E2:59_18"><i><font color="Green" title="E96">A115</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="E90:59"/>
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="c3">p</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="newton.html#K4" title="NEWTON:func.4">idseq</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">q</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> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="newton.html#K4" title="NEWTON:func.4">idseq</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">q</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="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="59_19"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c21">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:59_19"/>
<font color="Maroon" title="c21">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="c3">p</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="newton.html#K4" title="NEWTON:func.4">idseq</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">q</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/>

<a NAME="E2:59_19"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="newton.html#K4" title="NEWTON:func.4">idseq</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">q</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c21">d</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E89:59"><i><font color="Green" title="E94">A113</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:59_19"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="newton.html#K4" title="NEWTON:func.4">idseq</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">q</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">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="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c21">g1</font> = <font color="Maroon" title="c3">p</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="c4">q</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> 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="E92:59"/><i><font color="Green" title="E95">A116</font></i>: 
for <font color="Olive" title="b1">d</font>, <font color="Olive" title="b2">e</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="c21">g1</font> &amp; <font color="Olive" title="b2">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="c21">g1</font> &amp; <font color="Maroon" title="c4">q</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</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="c21">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">e</font></span>)</span> holds <br/><font color="Olive" title="b1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">e</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_20"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c22">d</font>, <font color="Maroon" title="c23">e</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:59_20"/><i><font color="Green" title="E96">A117</font></i>: 
( <font color="Maroon" title="c22">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="c21">g1</font> &amp; <font color="Maroon" title="c23">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="c21">g1</font> &amp; <font color="Maroon" title="c4">q</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">d</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="c21">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">e</font></span>)</span> )
 ;<br/>

<a NAME="E2:59_20"/><b>then </b>
( <font color="Maroon" title="c21">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c22">d</font> &amp; <font color="Maroon" title="c21">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c23">e</font> )
 
<b>by </b><i><a class="txt" href="int_5.html#E89:59"><i><font color="Green" title="E94">A113</font></i></a></i>;<br/>
<a NAME="E3:59_20"/><b>then </b><i><font color="Green" title="E97">A118</font></i>: 
<font color="Maroon" title="c4">q</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">d</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c23">e</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">p</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E1:59_20"><i><font color="Green" title="E96">A117</font></i></a></i>;<br/>
<a NAME="E4:59_20"/>
<font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c3">p</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 
<b>by </b><i><a class="txt" href="int_5.html#E2:59"><i><font color="Green" title="E44">A2</font></i></a>, <a class="ref" href="euler_2.html#T1" title="EULER_2:th.1">EULER_2:1</a></i>;<br/>
<a NAME="E5:59_20"/><b>then </b><i><font color="Green" title="E98">A119</font></i>: 
<font color="Maroon" title="c4">q</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c22">d</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c23">e</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E3:59_20"><i><font color="Green" title="E97">A118</font></i></a>, <a class="ref" href="int_2.html#T40" title="INT_2:th.40">INT_2:40</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_20_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_20_1"/>
<font color="Maroon" title="c22">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">e</font>
 ;<br/><a NAME="E2:59_20_1"/><b>then </b>
<font color="Maroon" title="c22">d</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c23">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E3:59_20_1"/><b>then </b>
 <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c4">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">d</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c23">e</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_5.html#E5:59_20"><i><font color="Green" title="E98">A119</font></i></a>, <a class="ref" href="int_4.html#T6" title="INT_4:th.6">INT_4:6</a></i>;<br/><a NAME="E4:59_20_1"/><b>then </b><i><font color="Green" title="E99">A120</font></i>: 
<font color="Maroon" title="c4">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">d</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c23">e</font></span>)</span>
 <b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><a NAME="E5:59_20_1"/>
( <font color="Maroon" title="c22">d</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c22">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="c4">q</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">e</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c23">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</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#E88:59"><i><font color="Green" title="E93">A112</font></i></a>, <a class="txt" href="int_5.html#E1:59_20"><i><font color="Green" title="E96">A117</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:59_20_1"/><b>then </b><i><font color="Green" title="E100">A121</font></i>: 
( <font color="Maroon" title="c22">d</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c23">e</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="c4">q</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 &amp; <font color="Maroon" title="c22">d</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c23">e</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</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"><font color="Maroon" title="c4">q</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#T15" title="XREAL_1:th.15">XREAL_1:15</a></i>;<br/><a NAME="E7:59_20_1"/><i><font color="Green" title="E101">A122</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</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="c4">q</font>
 <b>by </b><i><a class="txt" href="int_5.html#E22:59"><i><font color="Green" title="E58">A16</font></i></a>, <a class="ref" href="xreal_1.html#T149" title="XREAL_1:th.149">XREAL_1:149</a></i>;<br/><a NAME="E8:59_20_1"/><b>then </b><i><font color="Green" title="E102">A123</font></i>: 
<font color="Maroon" title="c22">d</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c23">e</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">q</font>
 <b>by </b><i><a class="txt" href="int_5.html#E6:59_20_1"><i><font color="Green" title="E100">A121</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E9:59_20_1"/>
 <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"><font color="Maroon" title="c4">q</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</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c4">q</font>
 <b>by </b><i><a class="txt" href="int_5.html#E7:59_20_1"><i><font color="Green" title="E101">A122</font></i></a>, <a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></i>;<br/><a NAME="E10:59_20_1"/><b>then </b>
<font color="Maroon" title="c22">d</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c23">e</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c4">q</font>
 <b>by </b><i><a class="txt" href="int_5.html#E6:59_20_1"><i><font color="Green" title="E100">A121</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="E11:59_20_1"/>
contradiction
 <b>by </b><i><a class="txt" href="int_5.html#E4:59_20_1"><i><font color="Green" title="E99">A120</font></i></a>, <a class="txt" href="int_5.html#E8:59_20_1"><i><font color="Green" title="E102">A123</font></i></a>, <a class="ref" href="seq_2.html#T9" title="SEQ_2:th.9">SEQ_2:9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:59_20"/>
<font color="Maroon" title="c22">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">e</font>
 ;<br/>


</div><b>end;</b></div>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  = <span class="p1">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">q</font>;<br/>
<b>consider </b><font color="Maroon" title="c22">g2</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E94:59"/><i><font color="Green" title="E96">A124</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</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; ( 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="c22">g2</font> holds <br/><font color="Maroon" title="c22">g2</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">H<sub>2</sub></font>(<font color="Olive" title="b1">d</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_5.html#S1" title="FINSEQ_5:sch.1">FINSEQ_5:sch 1</a>();<br/></i>
<a NAME="E95:59"/>
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="c22">g2</font> holds <br/><font color="Maroon" title="c22">g2</font> <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="59_21"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:59_21"/>
<font color="Maroon" title="c23">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="c22">g2</font>
 ;<br/>

<a NAME="E2:59_21"/><b>then </b>
<font color="Maroon" title="c22">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">d</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">q</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E94:59"><i><font color="Green" title="E96">A124</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:59_21"/>
<font color="Maroon" title="c22">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c23">g2</font> = <font color="Maroon" title="c22">g2</font> 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/>
<b>set </b><font color="Maroon" title="c24">g3</font> = <font color="Maroon" title="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font>;<br/>
<a NAME="E97:59"/><i><font color="Green" title="E97">A125</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</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="c21">g1</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="E98:59"/><b>then </b><i><font color="Green" title="E98">A126</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</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/>
<a NAME="E99:59"/><i><font color="Green" title="E99">A127</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g1</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="c23">g2</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E88:59"><i><font color="Green" title="E93">A112</font></i></a>, <a class="txt" href="int_5.html#E94:59"><i><font color="Green" title="E96">A124</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<i><font color="Green" title="E100">A128</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c23">g2</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</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"><font color="Maroon" title="c4">q</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c23">g2</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"><font color="Maroon" title="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D3" title="SEQ_1:def.3">SEQ_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">g2</font></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"><font color="Maroon" title="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g1</font>
<b>by </b><i><a class="txt" href="int_5.html#E98:59"><i><font color="Green" title="E98">A126</font></i></a>, <a class="txt" href="int_5.html#E99:59"><i><font color="Green" title="E99">A127</font></i></a></i>
;<br/>
<a NAME="E101:59"/><i><font color="Green" title="E101">A129</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="c21">g1</font> holds <br/><font color="Maroon" title="c21">g1</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c4">q</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</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="59_23"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c25">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:59_23"/>
<font color="Maroon" title="c25">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="c21">g1</font>
 ;<br/>

<a NAME="E2:59_23"/><b>then </b>
( <font color="Maroon" title="c23">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">d</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">q</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">d</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">q</font> )
 
<b>by </b><i><a class="txt" href="int_5.html#E94:59"><i><font color="Green" title="E96">A124</font></i></a>, <a class="txt" href="int_5.html#E99:59"><i><font color="Green" title="E99">A127</font></i></a>, <a class="ref" href="euler_2.html#D1" title="EULER_2:def.1">EULER_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:59_23"/>
<font color="Maroon" title="c21">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">d</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c4">q</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">d</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_d.html#T2" title="NAT_D:th.2">NAT_D:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E102:59"/>
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="c21">g1</font> holds <br/><font color="Maroon" title="c21">g1</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c23">g2</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_24"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c25">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:59_24"/><i><font color="Green" title="E102">A130</font></i>: 
<font color="Maroon" title="c25">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="c21">g1</font>
 ;<br/>

<a NAME="E2:59_24"/><b>then </b><i><font color="Green" title="E103">A131</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c23">g2</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c23">g2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">d</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">d</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E100:59"><i><font color="Green" title="E100">A128</font></i></a>, <a class="ref" href="seq_1.html#D3" title="SEQ_1:def.3">SEQ_1:def 3</a></i>;<br/>
<a NAME="E3:59_24"/>
<font color="Maroon" title="c25">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">q</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c23">g2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E99:59"><i><font color="Green" title="E99">A127</font></i></a>, <a class="txt" href="int_5.html#E1:59_24"><i><font color="Green" title="E102">A130</font></i></a>, <a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c23">g2</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">d</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">d</font></span>)</span></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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">d</font></span>)</span>
<b>by </b><i><a class="txt" href="int_5.html#E2:59_24"><i><font color="Green" title="E103">A131</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="c21">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">d</font>
<b>by </b><i><a class="txt" href="int_5.html#E101:59"><i><font color="Green" title="E101">A129</font></i></a>, <a class="txt" href="int_5.html#E1:59_24"><i><font color="Green" title="E102">A130</font></i></a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E103:59"/><b>then </b><i><font color="Green" title="E102">A132</font></i>: 
<font color="Maroon" title="c21">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c23">g2</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E100:59"><i><font color="Green" title="E100">A128</font></i></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></i>;<br/>
<a NAME="E104:59"/><i><font color="Green" title="E103">A133</font></i>: 
( <font color="Maroon" title="c4">q</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c23">g2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a>  &amp; <font color="Maroon" title="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a>  )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E105:59"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c23">g2</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="c23">g2</font>
 
<b>by </b><i><a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>;<br/>
<a NAME="E106:59"/><b>then </b>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c23">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</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;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</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#E88:59"><i><font color="Green" title="E93">A112</font></i></a>, <a class="txt" href="int_5.html#E94:59"><i><font color="Green" title="E96">A124</font></i></a>, <a class="ref" href="euler_2.html#D1" title="EULER_2:def.1">EULER_2:def 1</a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E107:59"/><b>then </b><i><font color="Green" title="E104">A134</font></i>: 
( <font color="Maroon" title="c4">q</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c23">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</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>  &amp; <font color="Maroon" title="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</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="txt" href="int_5.html#E104:59"><i><font color="Green" title="E103">A133</font></i></a></i>;<br/>
<a NAME="E108:59"/>
( <font color="Maroon" title="c4">q</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c23">g2</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="c4">q</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>  &amp; <font color="Maroon" title="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</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="c4">q</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="txt" href="int_5.html#E107:59"><i><font color="Green" title="E104">A134</font></i></a>, <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="E105">A135</font></i>:  <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c21">g1</font> = 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c23">g2</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="int_5.html#E103:59"><i><font color="Green" title="E102">A132</font></i></a>, <a class="ref" href="rvsum_1.html#T119" title="RVSUM_1:th.119">RVSUM_1:119</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c23">g2</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T117" title="RVSUM_1:th.117">RVSUM_1:117</a></i>
;<br/>
<a NAME="E110:59"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span> <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#E13:59"><i><font color="Green" title="E51">A9</font></i></a>, <a class="txt" href="int_5.html#E88:59"><i><font color="Green" title="E93">A112</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="E111:59"/><b>then </b>
<font color="Maroon" title="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</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="ref" href="finseq_1.html#T28" title="FINSEQ_1:th.28">FINSEQ_1:28</a></i>;<br/>
<a NAME="E112:59"/><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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <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="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="c25">n2</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="E114:59"/><i><font color="Green" title="E106">A136</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</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="c25">n2</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="E115:59"/>
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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_26"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:59_26"/>
 <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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c26">a</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:59_26"/><i><font color="Green" title="E107">A137</font></i>: 
( <font color="Maroon" title="c26">a</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">a</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/>
<a NAME="E4:59_26"/>
<font color="Maroon" title="c26">a</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="c21">g1</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E97:59"><i><font color="Green" title="E97">A125</font></i></a>, <a class="txt" href="int_5.html#E3:59_26"><i><font color="Green" title="E107">A137</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E5:59_26"/><b>then </b><i><font color="Green" title="E108">A138</font></i>: 
<font color="Maroon" title="c21">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">a</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="int_5.html#E101:59"><i><font color="Green" title="E101">A129</font></i></a>, <a class="txt" href="int_5.html#E3:59_26"><i><font color="Green" title="E107">A137</font></i></a></i>;<br/>
<a NAME="E6:59_26"/>
<font color="Maroon" title="c26">a</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="c21">g1</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E97:59"><i><font color="Green" title="E97">A125</font></i></a>, <a class="txt" href="int_5.html#E3:59_26"><i><font color="Green" title="E107">A137</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E7:59_26"/><b>then </b>
<font color="Maroon" title="c3">p</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c26">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">a</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c4">q</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E89:59"><i><font color="Green" title="E94">A113</font></i></a>, <a class="txt" href="int_5.html#E5:59_26"><i><font color="Green" title="E108">A138</font></i></a></i>;<br/>
<a NAME="E8:59_26"/><b>then </b>
<font color="Maroon" title="c4">q</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">p</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c26">a</font>
 
<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="E9:59_26"/><b>then </b><i><font color="Green" title="E109">A139</font></i>: 
<font color="Maroon" title="c4">q</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c26">a</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E2:59"><i><font color="Green" title="E44">A2</font></i></a>, <a class="ref" href="pepin.html#T3" title="PEPIN:th.3">PEPIN:3</a></i>;<br/>
<a NAME="E10:59_26"/><i><font color="Green" title="E110">A140</font></i>: 
( <font color="Maroon" title="c26">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c26">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</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#E88:59"><i><font color="Green" title="E93">A112</font></i></a>, <a class="txt" href="int_5.html#E97:59"><i><font color="Green" title="E97">A125</font></i></a>, <a class="txt" href="int_5.html#E3:59_26"><i><font color="Green" title="E107">A137</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E11:59_26"/><b>then </b>
<font color="Maroon" title="c4">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">a</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E9:59_26"><i><font color="Green" title="E109">A139</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="E12:59_26"/>
contradiction
 <b>by </b><i><a class="txt" href="int_5.html#E22:59"><i><font color="Green" title="E58">A16</font></i></a>, <a class="txt" href="int_5.html#E10:59_26"><i><font color="Green" title="E110">A140</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="E116:59"/><b>then </b><i><font color="Green" title="E107">A141</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</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="E117:59"/><b>then </b><i><font color="Green" title="E108">A142</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</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="c25">n2</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E114:59"><i><font color="Green" title="E106">A136</font></i></a>, <a class="ref" href="xboole_1.html#T73" title="XBOOLE_1:th.73">XBOOLE_1:73</a></i>;<br/>
<a NAME="E118:59"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span> &amp; <font color="Olive" title="b2">y</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_27"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c26">x</font>, <font color="Maroon" title="c27">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:59_27"/><i><font color="Green" title="E109">A143</font></i>: 
( <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span> &amp; <font color="Maroon" title="c27">y</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">y</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c28">x</font> = <font color="Maroon" title="c26">x</font>, <font color="Maroon" title="c29">y</font> = <font color="Maroon" title="c27">y</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>  ;<br/>
<a NAME="E3:59_27"/>
( <font color="Maroon" title="c21">g1</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c28">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">g2</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c28">x</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c28">x</font></span>)</span> &amp; <font color="Maroon" title="c21">g1</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c29">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">g2</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c29">y</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c29">y</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="int_5.html#E98:59"><i><font color="Green" title="E98">A126</font></i></a>, <a class="txt" href="int_5.html#E101:59"><i><font color="Green" title="E101">A129</font></i></a>, <a class="txt" href="int_5.html#E1:59_27"><i><font color="Green" title="E109">A143</font></i></a></i>;<br/>
<a NAME="E4:59_27"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c28">x</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="c21">g1</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c29">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">g2</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c28">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">g2</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c29">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">q</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E1:59_27"><i><font color="Green" title="E109">A143</font></i></a></i>;<br/>
<a NAME="E5:59_27"/><b>then </b>
<font color="Maroon" title="c4">q</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c28">x</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="c21">g1</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c29">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="int_1.html#D9" title="INT_1:def.9">INT_1:def 9</a></i>;<br/>
<b>hence </b><a NAME="E6:59_27"/>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">y</font>
 <b>by </b><i><a class="txt" href="int_5.html#E92:59"><i><font color="Green" title="E95">A116</font></i></a>, <a class="txt" href="int_5.html#E98:59"><i><font color="Green" title="E98">A126</font></i></a>, <a class="txt" href="int_5.html#E1:59_27"><i><font color="Green" title="E109">A143</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E119:59"/><b>then </b><i><font color="Green" title="E109">A144</font></i>: 
<font color="Maroon" title="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</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="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<b>set </b><font color="Maroon" title="c26">g4</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c27">XX</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"><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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></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="c4">q</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 ) </span>}</span> ;<br/>
<a NAME="E120:59"/><i><font color="Green" title="E110">A145</font></i>: 
 <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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></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"><font color="Maroon" title="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E117:59"><i><font color="Green" title="E108">A142</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="E121:59"/><i><font color="Green" title="E111">A146</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</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#E117:59"><i><font color="Green" title="E108">A142</font></i></a>, <a class="ref" href="finseq_3.html#T99" title="FINSEQ_3:th.99">FINSEQ_3:99</a></i>;<br/>
<a NAME="E122:59"/><b>then </b><i><font color="Green" title="E112">A147</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>,<font color="Maroon" title="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</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#E119:59"><i><font color="Green" title="E109">A144</font></i></a>, <a class="txt" href="int_5.html#E120:59"><i><font color="Green" title="E110">A145</font></i></a>, <a class="ref" href="rfinseq.html#T39" title="RFINSEQ:th.39">RFINSEQ:39</a></i>;<br/>
<a NAME="E123:59"/>
(  <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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 
<b>by </b><i><a class="ref" href="finseq_2.html#T27" title="FINSEQ_2:th.27">FINSEQ_2:27</a></i>;<br/>
<a NAME="E124:59"/><b>then </b><i><font color="Green" title="E113">A148</font></i>: 
 <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E122:59"><i><font color="Green" title="E112">A147</font></i></a>, <a class="ref" href="rfinseq.html#T22" title="RFINSEQ:th.22">RFINSEQ:22</a></i>;<br/>
<a NAME="E125:59"/>
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"><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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></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="c4">q</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"><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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_28"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c28">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:59_28"/>
<font color="Maroon" title="c28">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"><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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></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="c4">q</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="c29">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="E3:59_28"/><i><font color="Green" title="E114">A149</font></i>: 
( <font color="Maroon" title="c28">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">k</font> &amp; <font color="Maroon" title="c29">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"><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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">q</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 )
 ;<br/>
<b>thus </b><a NAME="E4:59_28"/>
<font color="Maroon" title="c28">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"><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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="int_5.html#E3:59_28"><i><font color="Green" title="E114">A149</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E126:59"/><b>then </b><i><font color="Green" title="E114">A150</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"><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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></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="c4">q</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"><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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></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="c28">XX</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"><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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></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="c4">q</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="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="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c29">YY</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"><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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c28">XX</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>  ;<br/>
<a NAME="E129:59"/>
<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"><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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c28">XX</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"><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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></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/>
<a NAME="E130:59"/><b>then </b><i><font color="Green" title="E115">A151</font></i>: 
<font color="Maroon" title="c29">YY</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="c25">n2</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E117:59"><i><font color="Green" title="E108">A142</font></i></a>, <a class="txt" href="int_5.html#E120:59"><i><font color="Green" title="E110">A145</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>set </b><font color="Maroon" title="c30">mm</font> =  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c28">XX</font>;<br/>
<a NAME="E131:59"/><i><font color="Green" title="E116">A152</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</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"><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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E119:59"><i><font color="Green" title="E109">A144</font></i></a>, <a class="txt" href="int_5.html#E120:59"><i><font color="Green" title="E110">A145</font></i></a>, <a class="ref" href="finseq_4.html#T77" title="FINSEQ_4:th.77">FINSEQ_4:77</a></i>;<br/>
<a NAME="E132:59"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c28">XX</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"><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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E126:59"><i><font color="Green" title="E114">A150</font></i></a>, <a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a></i>;<br/>
<a NAME="E133:59"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c28">XX</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</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#E88:59"><i><font color="Green" title="E93">A112</font></i></a>, <a class="txt" href="int_5.html#E131:59"><i><font color="Green" title="E116">A152</font></i></a>, <a class="ref" href="euler_2.html#D1" title="EULER_2:def.1">EULER_2:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c31">nn</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</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> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c28">XX</font></span>)</span> 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="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a></i>;<br/>
<a NAME="E135:59"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E119:59"><i><font color="Green" title="E109">A144</font></i></a>, <a class="ref" href="finseq_4.html#T77" title="FINSEQ_4:th.77">FINSEQ_4:77</a></i>;<br/>
<a NAME="E136:59"/><b>then </b><i><font color="Green" title="E117">A153</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</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="c4">q</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#E88:59"><i><font color="Green" title="E93">A112</font></i></a>, <a class="txt" href="int_5.html#E97:59"><i><font color="Green" title="E97">A125</font></i></a>, <a class="txt" href="int_5.html#E117:59"><i><font color="Green" title="E108">A142</font></i></a>, <a class="ref" href="finseq_3.html#T44" title="FINSEQ_3:th.44">FINSEQ_3:44</a></i>;<br/>
<a NAME="E137:59"/><i><font color="Green" title="E118">A154</font></i>: 
<font color="Maroon" title="c28">XX</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="c25">n2</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E117:59"><i><font color="Green" title="E108">A142</font></i></a>, <a class="txt" href="int_5.html#E120:59"><i><font color="Green" title="E110">A145</font></i></a>, <a class="txt" href="int_5.html#E126:59"><i><font color="Green" title="E114">A150</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E138:59"/><i><font color="Green" title="E119">A155</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</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="c29">YY</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="c28">XX</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_29"><b>proof </b></a><div class="add">

<a NAME="E1:59_29"/>
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="c29">YY</font> &amp; <font color="Olive" title="b2">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">XX</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="59_29_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c32">k</font>, <font color="Maroon" title="c33">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="E1:59_29_1"/><i><font color="Green" title="E120">A157</font></i>: 
( <font color="Maroon" title="c32">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c29">YY</font> &amp; <font color="Maroon" title="c33">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">XX</font> )
 ;<br/>

<a NAME="E2:59_29_1"/><b>then </b>
( <font color="Maroon" title="c32">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"><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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> &amp; not <font color="Maroon" title="c32">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">XX</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="E3:59_29_1"/><b>then </b><i><font color="Green" title="E121">A158</font></i>: 
<font color="Maroon" title="c32">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">q</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 
;<br/>
<a NAME="E4:59_29_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="c33">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"><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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></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="c4">q</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#E1:59_29_1"><i><font color="Green" title="E120">A157</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:59_29_1"/>
<font color="Maroon" title="c32">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c33">l</font>
 <b>by </b><i><a class="txt" href="int_5.html#E3:59_29_1"><i><font color="Green" title="E121">A158</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="E2:59_29"/><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="c29">YY</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c28">XX</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="c29">YY</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="c28">XX</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E130:59"><i><font color="Green" title="E115">A151</font></i></a>, <a class="txt" href="int_5.html#E137:59"><i><font color="Green" title="E118">A154</font></i></a>, <a class="ref" href="finseq_3.html#T48" title="FINSEQ_3:th.48">FINSEQ_3:48</a></i>;<br/>
<a NAME="E3:59_29"/><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"><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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c28">XX</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="c29">YY</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="c28">XX</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="E4:59_29"/>
 <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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</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="c29">YY</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="c28">XX</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_5.html#E120:59"><i><font color="Green" title="E110">A145</font></i></a>, <a class="txt" href="int_5.html#E126:59"><i><font color="Green" title="E114">A150</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="E139:59"/>
 <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c29">YY</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c28">XX</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E138:59"><i><font color="Green" title="E119">A155</font></i></a>, <a class="ref" href="rvsum_1.html#T105" title="RVSUM_1:th.105">RVSUM_1:105</a></i>;<br/>
<a NAME="E140:59"/><b>then </b><i><font color="Green" title="E120">A159</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="newton.html#K4" title="NEWTON:func.4">idseq</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">q</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c23">g2</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c29">YY</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c28">XX</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E109:59"><i><font color="Green" title="E105">A135</font></i></a>, <a class="txt" href="int_5.html#E124:59"><i><font color="Green" title="E113">A148</font></i></a>, <a class="ref" href="rvsum_1.html#T117" title="RVSUM_1:th.117">RVSUM_1:117</a></i>;<br/>
<a NAME="E141:59"/><i><font color="Green" title="E121">A160</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="c29">YY</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">nn</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_30"><b>proof </b></a><div class="add">

 <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="c29">YY</font></span>)</span> = 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c29">YY</font>
<b>by </b><i><a class="txt" href="int_5.html#E130:59"><i><font color="Green" title="E115">A151</font></i></a>, <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="c4">q</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> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c28">XX</font></span>)</span>
<b>by </b><i><a class="txt" href="int_5.html#E88:59"><i><font color="Green" title="E93">A112</font></i></a>, <a class="txt" href="int_5.html#E97:59"><i><font color="Green" title="E97">A125</font></i></a>, <a class="txt" href="int_5.html#E126:59"><i><font color="Green" title="E114">A150</font></i></a>, <a class="txt" href="int_5.html#E131:59"><i><font color="Green" title="E116">A152</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="E2:59_30"/>
 <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="c29">YY</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">nn</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E142:59"/><b>then </b><i><font color="Green" title="E122">A161</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</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="c28">XX</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E138:59"><i><font color="Green" title="E119">A155</font></i></a>, <a class="ref" href="finseq_5.html#T40" title="FINSEQ_5:th.40">FINSEQ_5:40</a></i>;<br/>
<a NAME="E143:59"/>
 <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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c31">nn</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</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="E144:59"/><b>then </b><i><font color="Green" title="E123">A162</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c31">nn</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</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#E121:59"><i><font color="Green" title="E111">A146</font></i></a>, <a class="ref" href="finseq_3.html#T98" title="FINSEQ_3:th.98">FINSEQ_3:98</a></i>;<br/>
<a NAME="E145:59"/>
 <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c29">YY</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E138:59"><i><font color="Green" title="E119">A155</font></i></a>, <a class="ref" href="finseq_6.html#T12" title="FINSEQ_6:th.12">FINSEQ_6:12</a></i>;<br/>
<a NAME="E146:59"/><b>then </b><i><font color="Green" title="E124">A163</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c31">nn</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="c29">YY</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E141:59"><i><font color="Green" title="E121">A160</font></i></a>, <a class="ref" href="finseq_3.html#T122" title="FINSEQ_3:th.122">FINSEQ_3:122</a></i>;<br/>
<a NAME="E147:59"/>
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"><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="c4">q</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> holds <br/><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="c4">q</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="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="59_31"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c32">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:59_31"/>
<font color="Maroon" title="c32">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"><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="c4">q</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>
 ;<br/>

<a NAME="E2:59_31"/><b>then </b>
<font color="Maroon" title="c32">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"><a href="newton.html#K4" title="NEWTON:func.4">idseq</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">q</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="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:59_31"/><b>then </b>
<font color="Maroon" title="c32">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="c4">q</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="E4:59_31"/><b>then </b>
<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="c4">q</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">d</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E13:59"><i><font color="Green" title="E51">A9</font></i></a>, <a class="ref" href="finseq_2.html#T57" title="FINSEQ_2:th.57">FINSEQ_2:57</a></i>;<br/>
<b>hence </b><a NAME="E5:59_31"/>
<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="c4">q</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">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="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E148:59"/><b>then </b>
 <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="c4">q</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> 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> 
 
<b>by </b><i><a class="ref" href="finseq_2.html#T14" title="FINSEQ_2:th.14">FINSEQ_2:14</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c32">N</font> =  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</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="c4">q</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> 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#E55"><i><font color="Green" title="E41">Lm4</font></i></a></i>;<br/>
<a NAME="E150:59"/><i><font color="Green" title="E125">A164</font></i>: 
 <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</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> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c23">g2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_32"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_32_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_32_1"/>
( <font color="Maroon" title="c28">XX</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  not <font color="Maroon" title="c28">XX</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="59_32_1_1"><b>suppose </b></a><a NAME="E1:59_32_1_1"/><i><font color="Green" title="E126">A165</font></i>: 
<font color="Maroon" title="c28">XX</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><a NAME="E2:59_32_1_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"><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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> holds <br/>( <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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></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"><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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></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="c4">q</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="59_32_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c33">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:59_32_1_1_1"/>
<font color="Maroon" title="c33">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"><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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:59_32_1_1_1"/><b>then </b><i><font color="Green" title="E127">A166</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">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> <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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E3:59_32_1_1_1"/>
not <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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">XX</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E1:59_32_1_1"><i><font color="Green" title="E126">A165</font></i></a></i>;<br/>
<a NAME="E4:59_32_1_1_1"/><b>then </b><i><font color="Green" title="E128">A167</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">q</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#E2:59_32_1_1_1"><i><font color="Green" title="E127">A166</font></i></a></i>;<br/>
<a NAME="E5:59_32_1_1_1"/>
<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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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#E120:59"><i><font color="Green" title="E110">A145</font></i></a>, <a class="txt" href="int_5.html#E2:59_32_1_1_1"><i><font color="Green" title="E127">A166</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E6:59_32_1_1_1"/><b>then </b>
not <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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">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#E116:59"><i><font color="Green" title="E107">A141</font></i></a>, <a class="txt" href="int_5.html#E120:59"><i><font color="Green" title="E110">A145</font></i></a>, <a class="txt" href="int_5.html#E2:59_32_1_1_1"><i><font color="Green" title="E127">A166</font></i></a>, <a class="ref" href="xboole_0.html#T5" title="XBOOLE_0:th.5">XBOOLE_0:5</a></i>;<br/>
<a NAME="E7:59_32_1_1_1"/><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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">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="E8:59_32_1_1_1"/>
( <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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">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"><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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">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="c4">q</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#E24:59"><i><font color="Green" title="E60">A18</font></i></a>, <a class="txt" href="int_5.html#E4:59_32_1_1_1"><i><font color="Green" title="E128">A167</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="E3:59_32_1_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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></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="c4">q</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="txt" href="int_5.html#E120:59"><i><font color="Green" title="E110">A145</font></i></a>, <a class="txt" href="int_5.html#E121:59"><i><font color="Green" title="E111">A146</font></i></a>, <a class="txt" href="int_5.html#E136:59"><i><font color="Green" title="E117">A153</font></i></a>, <a class="ref" href="int_5.html#T40" target="_self" title="INT_5:th.40">Th40</a></i>;<br/><a NAME="E4:59_32_1_1"/><b>then </b>
 <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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c4">q</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_3.html#T54" title="FINSEQ_3:th.54">FINSEQ_3:54</a></i>;<br/><a NAME="E5:59_32_1_1"/><b>then </b>
<font color="Maroon" title="c3">p</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c32">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c23">g2</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c32">N</font>
 <b>by </b><i><a class="txt" href="int_5.html#E109:59"><i><font color="Green" title="E105">A135</font></i></a>, <a class="txt" href="int_5.html#E124:59"><i><font color="Green" title="E113">A148</font></i></a>, <a class="ref" href="rvsum_1.html#T117" title="RVSUM_1:th.117">RVSUM_1:117</a></i>;<br/><a NAME="E6:59_32_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c32">N</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c32">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c23">g2</font></span>)</span>
 ;<br/><a NAME="E7:59_32_1_1"/><b>then </b><i><font color="Green" title="E127">A168</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c32">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c23">g2</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_5.html#E10:59"><i><font color="Green" title="E50">A8</font></i></a></i>;<br/><a NAME="E8:59_32_1_1"/>
2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">q</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c23">g2</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_5.html#E16:59"><i><font color="Green" title="E53">A11</font></i></a>, <a class="txt" href="int_5.html#E7:59_32_1_1"><i><font color="Green" title="E127">A168</font></i></a>, <a class="ref" href="nat_d.html#T9" title="NAT_D:th.9">NAT_D:9</a></i>;<br/><a NAME="E9:59_32_1_1"/><b>then </b>
2 <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c23">g2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="int_5.html#E5:59"><i><font color="Green" title="E46">A4</font></i></a>, <a class="ref" href="pepin.html#T3" title="PEPIN:th.3">PEPIN:3</a></i>;<br/><a NAME="E10:59_32_1_1"/><b>then </b>
 <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c23">g2</font>, <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> 2
 <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="E11:59_32_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c23">g2</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2
 <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="E12:59_32_1_1"/><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> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c28">XX</font></span>)</span> <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> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c23">g2</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_5.html#E1:59_32_1_1"><i><font color="Green" title="E126">A165</font></i></a>, <a class="ref" href="int_5.html#T45" target="_self" title="INT_5:th.45">Th45</a>, <a class="ref" href="card_1.html#T78" title="CARD_1:th.78">CARD_1:78</a></i>;<br/><b>hence </b><a NAME="E13:59_32_1_1"/>
 <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</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> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c23">g2</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_5.html#E1:59"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="int_5.html#E3:59"><i><font color="Green" title="E45">A3</font></i></a>, <a class="txt" href="int_5.html#E120:59"><i><font color="Green" title="E110">A145</font></i></a>, <a class="ref" href="int_5.html#T41" target="_self" title="INT_5:th.41">Th41</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_32_1_2"><b>suppose </b></a><a NAME="E1:59_32_1_2"/>
not <font color="Maroon" title="c28">XX</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><a NAME="E2:59_32_1_2"/><b>then </b><i><font color="Green" title="E126">A169</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c28">XX</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="card_4.html#T17" title="CARD_4:th.17">CARD_4:17</a></i>;<br/><a NAME="E3:59_32_1_2"/><i><font color="Green" title="E127">A170</font></i>: 
<font color="Maroon" title="c31">nn</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="int_5.html#E136:59"><i><font color="Green" title="E117">A153</font></i></a>, <a class="ref" href="xreal_1.html#T45" title="XREAL_1:th.45">XREAL_1:45</a></i>;<br/><a NAME="E4:59_32_1_2"/><b>then </b><i><font color="Green" title="E128">A171</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c31">nn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">nn</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/><i><font color="Green" title="E129">A172</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c31">nn</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</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="c31">nn</font>
<b>by </b><i><a class="txt" href="int_5.html#E3:59_32_1_2"><i><font color="Green" title="E127">A170</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>
<br/>.= 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c28">XX</font>
<b>by </b><i><a class="txt" href="int_5.html#E136:59"><i><font color="Green" title="E117">A153</font></i></a></i>
;<br/><b>set </b><font color="Maroon" title="c33">g5</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c28">XX</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c4">q</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c34">g6</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c31">nn</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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c28">XX</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c4">q</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</font></span>)</span></span>)</span>;<br/><i><font color="Green" title="E130">A173</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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c28">XX</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c4">q</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c28">XX</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c4">q</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c28">XX</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c4">q</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="int_5.html#E5:59_32_1_2"><i><font color="Green" title="E129">A172</font></i></a>, <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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</font></span>)</span>

;<br/><a NAME="E7:59_32_1_2"/><b>then </b><i><font color="Green" title="E131">A174</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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c28">XX</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c4">q</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</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/><a NAME="E8:59_32_1_2"/><i><font color="Green" title="E132">A175</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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c28">XX</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c4">q</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c28">XX</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c4">q</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</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="c4">q</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</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="59_32_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c35">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:59_32_1_2_3"/><i><font color="Green" title="E133">A176</font></i>: 
<font color="Maroon" title="c35">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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c28">XX</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c4">q</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:59_32_1_2_3"/><b>then </b>
<font color="Maroon" title="c35">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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c28">XX</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E5:59_32_1_2"><i><font color="Green" title="E129">A172</font></i></a>, <a class="txt" href="int_5.html#E6:59_32_1_2"><i><font color="Green" title="E130">A173</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="E3:59_32_1_2_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c28">XX</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E2:59_32_1_2"><i><font color="Green" title="E126">A169</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="E4:59_32_1_2_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c28">XX</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c4">q</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">d</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_5.html#E1:59_32_1_2_3"><i><font color="Green" title="E133">A176</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="E9:59_32_1_2"/><i><font color="Green" title="E133">A177</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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c28">XX</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c4">q</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c28">XX</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c4">q</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c28">XX</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c4">q</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</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="c4">q</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="59_32_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c35">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:59_32_1_2_4"/><i><font color="Green" title="E134">A178</font></i>: 
<font color="Maroon" title="c35">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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c28">XX</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c4">q</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:59_32_1_2_4"/><b>then </b><i><font color="Green" title="E135">A179</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c28">XX</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c4">q</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">d</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E8:59_32_1_2"><i><font color="Green" title="E132">A175</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c36">w</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c28">XX</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c4">q</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">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="E4:59_32_1_2_4"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c28">XX</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">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="c28">XX</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E142:59"><i><font color="Green" title="E122">A161</font></i></a>, <a class="txt" href="int_5.html#E6:59_32_1_2"><i><font color="Green" title="E130">A173</font></i></a>, <a class="txt" href="int_5.html#E1:59_32_1_2_4"><i><font color="Green" title="E134">A178</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E5:59_32_1_2_4"/><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="c28">XX</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">XX</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E137:59"><i><font color="Green" title="E118">A154</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="c37">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="E7:59_32_1_2_4"/><i><font color="Green" title="E136">A180</font></i>: 
( <font color="Maroon" title="c37">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="c28">XX</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">d</font> &amp; <font color="Maroon" title="c37">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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span> &amp; <font color="Maroon" title="c37">ll</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">q</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#E120:59"><i><font color="Green" title="E110">A145</font></i></a></i>;<br/>
<a NAME="E8:59_32_1_2_4"/><i><font color="Green" title="E137">A181</font></i>: 
<font color="Maroon" title="c36">w</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">q</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</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#E142:59"><i><font color="Green" title="E122">A161</font></i></a>, <a class="txt" href="int_5.html#E2:59_32_1_2_4"><i><font color="Green" title="E135">A179</font></i></a>, <a class="txt" href="int_5.html#E7:59_32_1_2_4"><i><font color="Green" title="E136">A180</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="c38">e</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E10:59_32_1_2_4"/><i><font color="Green" title="E138">A182</font></i>: 
( <font color="Maroon" title="c38">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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c38">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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">d</font> )
 <b>by </b><i><a class="txt" href="int_5.html#E142:59"><i><font color="Green" title="E122">A161</font></i></a>, <a class="txt" href="int_5.html#E7:59_32_1_2_4"><i><font color="Green" title="E136">A180</font></i></a>, <a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E11:59_32_1_2_4"/>
<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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c38">e</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">q</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E98:59"><i><font color="Green" title="E98">A126</font></i></a>, <a class="txt" href="int_5.html#E10:59_32_1_2_4"><i><font color="Green" title="E138">A182</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="E12:59_32_1_2_4"/><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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">q</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="E13:59_32_1_2_4"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c28">XX</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c4">q</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c28">XX</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c4">q</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">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="c4">q</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#E24:59"><i><font color="Green" title="E60">A18</font></i></a>, <a class="txt" href="int_5.html#E2:59_32_1_2_4"><i><font color="Green" title="E135">A179</font></i></a>, <a class="txt" href="int_5.html#E8:59_32_1_2_4"><i><font color="Green" title="E137">A181</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="E10:59_32_1_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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c28">XX</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c4">q</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c28">XX</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c4">q</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</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="59_32_1_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c35">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:59_32_1_2_5"/>
<font color="Maroon" title="c35">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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c28">XX</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c4">q</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:59_32_1_2_5"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c28">XX</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c4">q</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c28">XX</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c4">q</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">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#E8:59_32_1_2"><i><font color="Green" title="E132">A175</font></i></a>, <a class="txt" href="int_5.html#E9:59_32_1_2"><i><font color="Green" title="E133">A177</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:59_32_1_2_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c28">XX</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c4">q</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">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="c35">g5</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c28">XX</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c4">q</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</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="E12:59_32_1_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="c35">g5</font> holds <br/><font color="Maroon" title="c35">g5</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="c35">g5</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="59_32_1_2_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c36">d</font>, <font color="Maroon" title="c37">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="E1:59_32_1_2_6"/><i><font color="Green" title="E134">A183</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">d</font> &amp; <font color="Maroon" title="c36">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c37">e</font> &amp; <font color="Maroon" title="c37">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="c35">g5</font> )
 ;<br/>

<a NAME="E2:59_32_1_2_6"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">d</font> &amp; <font color="Maroon" title="c36">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="c35">g5</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c37">e</font> &amp; <font color="Maroon" title="c37">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="c35">g5</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="E3:59_32_1_2_6"/><b>then </b><i><font color="Green" title="E135">A184</font></i>: 
( <font color="Maroon" title="c36">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="c35">g5</font> &amp; <font color="Maroon" title="c37">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="c35">g5</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="E4:59_32_1_2_6"/><b>then </b>
( <font color="Maroon" title="c35">g5</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c36">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c36">d</font></span>)</span> &amp; <font color="Maroon" title="c35">g5</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c37">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c37">e</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="int_5.html#E8:59_32_1_2"><i><font color="Green" title="E132">A175</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:59_32_1_2_6"/>
<font color="Maroon" title="c35">g5</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c36">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c35">g5</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c37">e</font>
 <b>by </b><i><a class="txt" href="int_5.html#E144:59"><i><font color="Green" title="E123">A162</font></i></a>, <a class="txt" href="int_5.html#E6:59_32_1_2"><i><font color="Green" title="E130">A173</font></i></a>, <a class="txt" href="int_5.html#E1:59_32_1_2_6"><i><font color="Green" title="E134">A183</font></i></a>, <a class="txt" href="int_5.html#E3:59_32_1_2_6"><i><font color="Green" title="E135">A184</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="E13:59_32_1_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c35">g5</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="c35">g5</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="E14:59_32_1_2"/><b>then </b><i><font color="Green" title="E134">A185</font></i>: 
<font color="Maroon" title="c35">g5</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="E15:59_32_1_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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c31">nn</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="c35">g5</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="c36">g6</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c31">nn</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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c28">XX</font></span>)</span> <a href="gobrd10.html#K1" title="GOBRD10:func.1">|-&gt;</a> <font color="Maroon" title="c4">q</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</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="E135">A186</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c36">g6</font> = 
<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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c31">nn</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c35">g5</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</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#E4:59_32_1_2"><i><font color="Green" title="E128">A171</font></i></a>, <a class="txt" href="int_5.html#E5:59_32_1_2"><i><font color="Green" title="E129">A172</font></i></a>, <a class="txt" href="int_5.html#E7:59_32_1_2"><i><font color="Green" title="E131">A174</font></i></a></i>
;<br/><a NAME="E18:59_32_1_2"/><i><font color="Green" title="E136">A187</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="c4">q</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="c4">q</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/><a NAME="E19:59_32_1_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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c31">nn</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="c35">g5</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_32_1_2_8"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:59_32_1_2_8"/>
not  <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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c31">nn</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="c35">g5</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c37">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:59_32_1_2_8"/><i><font color="Green" title="E137">A188</font></i>: 
( <font color="Maroon" title="c37">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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c31">nn</font></span>)</span> &amp; <font color="Maroon" title="c37">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="c35">g5</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="E4:59_32_1_2_8"/>
 <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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c31">nn</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</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="c38">d1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E6:59_32_1_2_8"/><i><font color="Green" title="E138">A189</font></i>: 
( <font color="Maroon" title="c38">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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c38">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">x</font> )
 <b>by </b><i><a class="txt" href="int_5.html#E120:59"><i><font color="Green" title="E110">A145</font></i></a>, <a class="txt" href="int_5.html#E3:59_32_1_2_8"><i><font color="Green" title="E137">A188</font></i></a>, <a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E7:59_32_1_2_8"/><i><font color="Green" title="E139">A190</font></i>: 
<font color="Maroon" title="c38">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> <font color="Maroon" title="c21">g1</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E97:59"><i><font color="Green" title="E97">A125</font></i></a>, <a class="txt" href="int_5.html#E6:59_32_1_2_8"><i><font color="Green" title="E138">A189</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E8:59_32_1_2_8"/><b>then </b><i><font color="Green" title="E140">A191</font></i>: 
<font color="Maroon" title="c37">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c38">d1</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">q</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E6:59_32_1_2_8"><i><font color="Green" title="E138">A189</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="c39">e</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E10:59_32_1_2_8"/><i><font color="Green" title="E141">A192</font></i>: 
( <font color="Maroon" title="c39">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="c35">g5</font> &amp; <font color="Maroon" title="c35">g5</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">x</font> )
 <b>by </b><i><a class="txt" href="int_5.html#E3:59_32_1_2_8"><i><font color="Green" title="E137">A188</font></i></a>, <a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E11:59_32_1_2_8"/>
<font color="Maroon" title="c37">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">e</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E8:59_32_1_2"><i><font color="Green" title="E132">A175</font></i></a>, <a class="txt" href="int_5.html#E10:59_32_1_2_8"><i><font color="Green" title="E141">A192</font></i></a></i>;<br/>
<a NAME="E12:59_32_1_2_8"/><b>then </b><i><font color="Green" title="E142">A193</font></i>: 
<font color="Maroon" title="c37">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</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="c39">e</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c31">nn</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E3:59_32_1_2"><i><font color="Green" title="E127">A170</font></i></a>, <a class="txt" href="int_5.html#E6:59_32_1_2"><i><font color="Green" title="E130">A173</font></i></a>, <a class="txt" href="int_5.html#E10:59_32_1_2_8"><i><font color="Green" title="E141">A192</font></i></a>, <a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></i>;<br/>
<a NAME="E13:59_32_1_2_8"/>
<font color="Maroon" title="c39">e</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c31">nn</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E6:59_32_1_2"><i><font color="Green" title="E130">A173</font></i></a>, <a class="txt" href="int_5.html#E10:59_32_1_2_8"><i><font color="Green" title="E141">A192</font></i></a>, <a class="ref" href="finseq_5.html#T29" title="FINSEQ_5:th.29">FINSEQ_5:29</a></i>;<br/>
<a NAME="E14:59_32_1_2_8"/><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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</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="c39">e</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c31">nn</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E120:59"><i><font color="Green" title="E110">A145</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="c40">e1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E16:59_32_1_2_8"/><i><font color="Green" title="E143">A194</font></i>: 
( <font color="Maroon" title="c40">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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c40">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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</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="c39">e</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c31">nn</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="E17:59_32_1_2_8"/><i><font color="Green" title="E144">A195</font></i>: 
<font color="Maroon" title="c40">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="c21">g1</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E97:59"><i><font color="Green" title="E97">A125</font></i></a>, <a class="txt" href="int_5.html#E16:59_32_1_2_8"><i><font color="Green" title="E143">A194</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E18:59_32_1_2_8"/><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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</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="c39">e</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c31">nn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c40">e1</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">q</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E16:59_32_1_2_8"><i><font color="Green" title="E143">A194</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="E19:59_32_1_2_8"/><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="c21">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c38">d1</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">q</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="c21">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c40">e1</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">q</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="txt" href="int_5.html#E8:59_32_1_2_8"><i><font color="Green" title="E140">A191</font></i></a>, <a class="txt" href="int_5.html#E12:59_32_1_2_8"><i><font color="Green" title="E142">A193</font></i></a>, <a class="ref" href="nat_d.html#T25" title="NAT_D:th.25">NAT_D:25</a></i>;<br/>
<a NAME="E20:59_32_1_2_8"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c38">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="c21">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c40">e1</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">q</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="E21:59_32_1_2_8"/><b>then </b>
<font color="Maroon" title="c4">q</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">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c38">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="c21">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c40">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="E22:59_32_1_2_8"/><b>then </b>
<font color="Maroon" title="c4">q</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">d1</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c40">e1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E89:59"><i><font color="Green" title="E94">A113</font></i></a>, <a class="txt" href="int_5.html#E7:59_32_1_2_8"><i><font color="Green" title="E139">A190</font></i></a></i>;<br/>
<a NAME="E23:59_32_1_2_8"/><b>then </b>
<font color="Maroon" title="c4">q</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">d1</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c40">e1</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c3">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E89:59"><i><font color="Green" title="E94">A113</font></i></a>, <a class="txt" href="int_5.html#E17:59_32_1_2_8"><i><font color="Green" title="E144">A195</font></i></a></i>;<br/>
<a NAME="E24:59_32_1_2_8"/><b>then </b>
<font color="Maroon" title="c4">q</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">d1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c40">e1</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c3">p</font>
 
;<br/>
<a NAME="E25:59_32_1_2_8"/><b>then </b><i><font color="Green" title="E145">A196</font></i>: 
<font color="Maroon" title="c4">q</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c38">d1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c40">e1</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E2:59"><i><font color="Green" title="E44">A2</font></i></a>, <a class="ref" href="pepin.html#T3" title="PEPIN:th.3">PEPIN:3</a></i>;<br/>
<a NAME="E26:59_32_1_2_8"/>
( <font color="Maroon" title="c38">d1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c38">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="c4">q</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="c40">e1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c40">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="c4">q</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#E88:59"><i><font color="Green" title="E93">A112</font></i></a>, <a class="txt" href="int_5.html#E97:59"><i><font color="Green" title="E97">A125</font></i></a>, <a class="txt" href="int_5.html#E6:59_32_1_2_8"><i><font color="Green" title="E138">A189</font></i></a>, <a class="txt" href="int_5.html#E16:59_32_1_2_8"><i><font color="Green" title="E143">A194</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E27:59_32_1_2_8"/><b>then </b>
( <font color="Maroon" title="c38">d1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c40">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="c38">d1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c40">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="c4">q</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="c4">q</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="E28:59_32_1_2_8"/><b>then </b>
( <font color="Maroon" title="c38">d1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c40">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="c38">d1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c40">e1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">q</font> )
 
<b>by </b><i><a class="txt" href="int_5.html#E17:59"><i><font color="Green" title="E54">A12</font></i></a>, <a class="txt" href="int_5.html#E21:59"><i><font color="Green" title="E57">A15</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="E29:59_32_1_2_8"/>
contradiction
 <b>by </b><i><a class="txt" href="int_5.html#E25:59_32_1_2_8"><i><font color="Green" title="E145">A196</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="E20:59_32_1_2"/><b>then </b><i><font color="Green" title="E137">A197</font></i>: 
<font color="Maroon" title="c36">g6</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#E144:59"><i><font color="Green" title="E123">A162</font></i></a>, <a class="txt" href="int_5.html#E14:59_32_1_2"><i><font color="Green" title="E134">A185</font></i></a>, <a class="ref" href="finseq_3.html#T98" title="FINSEQ_3:th.98">FINSEQ_3:98</a></i>;<br/><a NAME="E21:59_32_1_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="c36">g6</font> holds <br/>( <font color="Maroon" title="c36">g6</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="c36">g6</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="c4">q</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="59_32_1_2_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c37">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:59_32_1_2_9"/><i><font color="Green" title="E138">A198</font></i>: 
<font color="Maroon" title="c37">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="c36">g6</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_32_1_2_9_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_32_1_2_9_1"/>
( <font color="Maroon" title="c37">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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c31">nn</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="c35">g5</font> &amp; <font color="Maroon" title="c37">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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c31">nn</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#E1:59_32_1_2_9"><i><font color="Green" title="E138">A198</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="59_32_1_2_9_1_1"><b>suppose </b></a><a NAME="E1:59_32_1_2_9_1_1"/><i><font color="Green" title="E139">A199</font></i>: 
<font color="Maroon" title="c37">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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c31">nn</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:59_32_1_2_9_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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c31">nn</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">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="c29">YY</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_5.html#E146:59"><i><font color="Green" title="E124">A163</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><a NAME="E3:59_32_1_2_9_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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c31">nn</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c29">YY</font>
 <b>by </b><i><a class="txt" href="int_5.html#E130:59"><i><font color="Green" title="E115">A151</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="E4:59_32_1_2_9_1_1"/><b>then </b><i><font color="Green" title="E140">A200</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c31">nn</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">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> <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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c31">nn</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">XX</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="E5:59_32_1_2_9_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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c31">nn</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">q</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 ;<br/><a NAME="E6:59_32_1_2_9_1_1"/><b>then </b><i><font color="Green" title="E141">A201</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c31">nn</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">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="c4">q</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#E24:59"><i><font color="Green" title="E60">A18</font></i></a>, <a class="ref" href="int_1.html#T81" title="INT_1:th.81">INT_1:81</a></i>;<br/><a NAME="E7:59_32_1_2_9_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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c31">nn</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">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#E116:59"><i><font color="Green" title="E107">A141</font></i></a>, <a class="txt" href="int_5.html#E120:59"><i><font color="Green" title="E110">A145</font></i></a>, <a class="txt" href="int_5.html#E4:59_32_1_2_9_1_1"><i><font color="Green" title="E140">A200</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><a NAME="E8:59_32_1_2_9_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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c31">nn</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">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="E9:59_32_1_2_9_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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c31">nn</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">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="E10:59_32_1_2_9_1_1"/>
( <font color="Maroon" title="c36">g6</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">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="c36">g6</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">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="c4">q</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#E1:59_32_1_2_9_1_1"><i><font color="Green" title="E139">A199</font></i></a>, <a class="txt" href="int_5.html#E6:59_32_1_2_9_1_1"><i><font color="Green" title="E141">A201</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="59_32_1_2_9_1_2"><b>suppose </b></a><a NAME="E1:59_32_1_2_9_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="c35">g5</font> &amp; <font color="Maroon" title="c37">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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c31">nn</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="c38">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="E3:59_32_1_2_9_1_2"/><i><font color="Green" title="E139">A202</font></i>: 
( <font color="Maroon" title="c38">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="c35">g5</font> &amp; <font color="Maroon" title="c37">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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c31">nn</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c38">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="E4:59_32_1_2_9_1_2"/>
<font color="Maroon" title="c36">g6</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c35">g5</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c38">l</font>
 <b>by </b><i><a class="txt" href="int_5.html#E3:59_32_1_2_9_1_2"><i><font color="Green" title="E139">A202</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="E5:59_32_1_2_9_1_2"/>
( <font color="Maroon" title="c36">g6</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">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="c36">g6</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">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="c4">q</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#E9:59_32_1_2"><i><font color="Green" title="E133">A177</font></i></a>, <a class="txt" href="int_5.html#E3:59_32_1_2_9_1_2"><i><font color="Green" title="E139">A202</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E22:59_32_1_2"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c36">g6</font> <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="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="c4">q</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#E17:59_32_1_2"><i><font color="Green" title="E135">A186</font></i></a>, <a class="txt" href="int_5.html#E18:59_32_1_2"><i><font color="Green" title="E136">A187</font></i></a>, <a class="txt" href="int_5.html#E20:59_32_1_2"><i><font color="Green" title="E137">A197</font></i></a>, <a class="ref" href="int_5.html#T40" target="_self" title="INT_5:th.40">Th40</a></i>;<br/><a NAME="E23:59_32_1_2"/><b>then </b><i><font color="Green" title="E138">A203</font></i>: 
<font color="Maroon" title="c36">g6</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="c4">q</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="rfinseq.html#R1" title="RFINSEQ:pred.1">are_fiberwise_equipotent</a> 
 <b>by </b><i><a class="txt" href="int_5.html#E20:59_32_1_2"><i><font color="Green" title="E137">A197</font></i></a>, <a class="ref" href="rfinseq.html#T39" title="RFINSEQ:th.39">RFINSEQ:39</a></i>;<br/><a NAME="E24:59_32_1_2"/><i><font color="Green" title="E139">A204</font></i>: 
( <font color="Maroon" title="c36">g6</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 <b>by </b><i><a class="ref" href="finseq_2.html#T27" title="FINSEQ_2:th.27">FINSEQ_2:27</a></i>;<br/><b>then </b><font color="Maroon" title="c32">N</font> = 
 <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c36">g6</font>
<b>by </b><i><a class="txt" href="int_5.html#E23:59_32_1_2"><i><font color="Green" title="E138">A203</font></i></a>, <a class="ref" href="rfinseq.html#T22" title="RFINSEQ:th.22">RFINSEQ:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c31">nn</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c35">g5</font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T105" title="RVSUM_1:th.105">RVSUM_1:105</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c31">nn</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c28">XX</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="int_5.html#E5:59_32_1_2"><i><font color="Green" title="E129">A172</font></i></a>, <a class="txt" href="int_5.html#E24:59_32_1_2"><i><font color="Green" title="E139">A204</font></i></a>, <a class="ref" href="int_5.html#T47" target="_self" title="INT_5:th.47">Th47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c31">nn</font></span>)</span></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"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c28">XX</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</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="c21">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c31">nn</font></span>)</span></span>)</span>

;<br/><a NAME="E26:59_32_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c32">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c23">g2</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c28">XX</font></span>)</span></span>)</span></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"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c28">XX</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_5.html#E140:59"><i><font color="Green" title="E120">A159</font></i></a>, <a class="txt" href="int_5.html#E142:59"><i><font color="Green" title="E122">A161</font></i></a>, <a class="txt" href="int_5.html#E146:59"><i><font color="Green" title="E124">A163</font></i></a></i>;<br/><b>then </b><i><font color="Green" title="E140">A205</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c32">N</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c23">g2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c28">XX</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c28">XX</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2
<b>by </b><i><a class="txt" href="int_5.html#E9:59"><i><font color="Green" title="E49">A7</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c23">g2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c28">XX</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2
<b>by </b><i><a class="ref" href="euler_1.html#T13" title="EULER_1:th.13">EULER_1:13</a></i>
;<br/><a NAME="E28:59_32_1_2"/>
2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c32">N</font>
 <b>by </b><i><a class="txt" href="int_5.html#E16:59"><i><font color="Green" title="E53">A11</font></i></a>, <a class="ref" href="nat_d.html#T9" title="NAT_D:th.9">NAT_D:9</a></i>;<br/><a NAME="E29:59_32_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c23">g2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c28">XX</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <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="txt" href="int_5.html#E27:59_32_1_2"><i><font color="Green" title="E140">A205</font></i></a>, <a class="ref" href="pepin.html#T6" title="PEPIN:th.6">PEPIN:6</a></i>;<br/><a NAME="E30:59_32_1_2"/><b>then </b><i><font color="Green" title="E141">A206</font></i>: 
2 <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c4">q</font> <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#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c23">g2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c28">XX</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="int_5.html#E4"><i><font color="Green" title="E4">Lm2</font></i></a></i>;<br/><a NAME="E31:59_32_1_2"/>
2,<font color="Maroon" title="c4">q</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 <b>by </b><i><a class="txt" href="int_5.html#E5:59"><i><font color="Green" title="E46">A4</font></i></a>, <a class="ref" href="euler_2.html#T1" title="EULER_2:th.1">EULER_2:1</a></i>;<br/><a NAME="E32:59_32_1_2"/><b>then </b>
2 <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c23">g2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c28">XX</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_5.html#E30:59_32_1_2"><i><font color="Green" title="E141">A206</font></i></a>, <a class="ref" href="int_2.html#T40" title="INT_2:th.40">INT_2:40</a></i>;<br/><a NAME="E33:59_32_1_2"/><b>then </b>
 <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c23">g2</font>, <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c28">XX</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> 2
 <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="E34:59_32_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c23">g2</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c28">XX</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2
 <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="E35:59_32_1_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> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c23">g2</font></span>)</span> <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> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c28">XX</font></span>)</span>
 <b>by </b><i><a class="ref" href="int_5.html#T45" target="_self" title="INT_5:th.45">Th45</a></i>;<br/><b>hence </b><a NAME="E36:59_32_1_2"/>
 <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</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> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c23">g2</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_5.html#E1:59"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="int_5.html#E3:59"><i><font color="Green" title="E45">A3</font></i></a>, <a class="txt" href="int_5.html#E120:59"><i><font color="Green" title="E110">A145</font></i></a>, <a class="ref" href="int_5.html#T41" target="_self" title="INT_5:th.41">Th41</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E151:59"/>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c10">f2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c23">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">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#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_33"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c33">A</font> =  <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="c3">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>, <font color="Maroon" title="c34">B</font> =  <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="c4">q</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  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <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#E13:59"><i><font color="Green" title="E51">A9</font></i></a>, <a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c33">A</font>,   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c34">B</font>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">(<span class="default">$1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default">$2 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">q</font></span>)</span>;<br/>
<a NAME="E2:59_33"/><i><font color="Green" title="E126">A207</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c33">A</font><br/> for <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c34">B</font> holds  <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c35">z</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c33">A</font>,<font color="Maroon" title="c34">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E4:59_33"/><i><font color="Green" title="E127">A208</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c33">A</font><br/> for <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c34">B</font> holds  <font color="Maroon" title="c35">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>)
 <b>from </b><i><a class="ref" href="funct_7.html#S1" title="FUNCT_7:sch.1">FUNCT_7:sch 1</a>(<a class="txt" href="int_5.html#E2:59_33"><i><font color="Green" title="E126">A207</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c33">A</font> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c34">B</font> : <font color="Maroon" title="c35">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</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>  </span>}</span>  );<br/>
<a NAME="E5:59_33"/><i><font color="Green" title="E128">A209</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#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">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> holds <br/> ex <font color="Olive" title="b2">x1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c35">z</font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">d</font>,<font color="Olive" title="b2">x1</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_33_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c36">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:59_33_1"/>
<font color="Maroon" title="c36">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="c3">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>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c37">d</font> = <font color="Maroon" title="c36">d</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c33">A</font> ;<br/>
<b>take </b><font color="Maroon" title="c38">x1</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c37">d</font>,<font color="Olive" title="b1">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c34">B</font> : <font color="Maroon" title="c35">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c37">d</font>,<font color="Olive" title="b1">y</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>  </span>}</span> ;<br/>

<a NAME="E3:59_33_1"/>
<font color="Maroon" title="c38">x1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c35">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_33_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c39">l</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:59_33_1_1"/>
<font color="Maroon" title="c39">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c38">x1</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c40">yy</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c34">B</font><b> such that </b><br/><a NAME="E3:59_33_1_1"/><i><font color="Green" title="E129">A210</font></i>: 
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c37">d</font>,<font color="Maroon" title="c40">yy</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">l</font> &amp; <font color="Maroon" title="c35">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c37">d</font>,<font color="Maroon" title="c40">yy</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/>
<a NAME="E4:59_33_1_1"/>
<font color="Maroon" title="c39">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c33">A</font>,<font color="Maroon" title="c34">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="int_5.html#E3:59_33_1_1"><i><font color="Green" title="E129">A210</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:59_33_1_1"/>
<font color="Maroon" title="c39">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c35">z</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:59_33_1"/>
( <font color="Maroon" title="c38">x1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c35">z</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c36">d</font>,<font color="Maroon" title="c38">x1</font>] )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c36">Pr</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c35">z</font></span>)</span><b> such that </b><br/><a NAME="E7:59_33"/><i><font color="Green" title="E129">A211</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c36">Pr</font> <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="c3">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> &amp; ( 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#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">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> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">d</font>,<font color="Maroon" title="c36">Pr</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Olive" title="b1">d</font>] ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S5" title="FINSEQ_1:sch.5">FINSEQ_1:sch 5</a>(<a class="txt" href="int_5.html#E5:59_33"><i><font color="Green" title="E128">A209</font></i></a>);<br/></i>
<a NAME="E8:59_33"/><i><font color="Green" title="E130">A212</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c36">Pr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">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#E7:59_33"><i><font color="Green" title="E129">A211</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="E9:59_33"/>
for <font color="Olive" title="b1">d</font>, <font color="Olive" title="b2">e</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="c36">Pr</font> &amp; <font color="Olive" title="b2">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="c36">Pr</font> &amp; <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">e</font> holds <br/><font color="Maroon" title="c36">Pr</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Olive" title="b1">d</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c36">Pr</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Olive" title="b2">e</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_33_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c37">d</font>, <font color="Maroon" title="c38">e</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:59_33_2"/><i><font color="Green" title="E131">A213</font></i>: 
( <font color="Maroon" title="c37">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="c36">Pr</font> &amp; <font color="Maroon" title="c38">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="c36">Pr</font> &amp; <font color="Maroon" title="c37">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c38">e</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c39">x1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c33">A</font><b> such that </b><br/><a NAME="E3:59_33_2"/><i><font color="Green" title="E132">A214</font></i>: 
( <font color="Maroon" title="c39">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">d</font> &amp; <font color="Maroon" title="c36">Pr</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c37">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c39">x1</font>,<font color="Olive" title="b1">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c34">B</font> : <font color="Maroon" title="c35">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c39">x1</font>,<font color="Olive" title="b1">y</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>  </span>}</span>  )
 <b>by </b><i><a class="txt" href="int_5.html#E7:59_33"><i><font color="Green" title="E129">A211</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c40">x2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c33">A</font><b> such that </b><br/><a NAME="E5:59_33_2"/><i><font color="Green" title="E133">A215</font></i>: 
( <font color="Maroon" title="c40">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">e</font> &amp; <font color="Maroon" title="c36">Pr</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c38">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c40">x2</font>,<font color="Olive" title="b1">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c34">B</font> : <font color="Maroon" title="c35">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c40">x2</font>,<font color="Olive" title="b1">y</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>  </span>}</span>  )
 <b>by </b><i><a class="txt" href="int_5.html#E7:59_33"><i><font color="Green" title="E129">A211</font></i></a>, <a class="txt" href="int_5.html#E1:59_33_2"><i><font color="Green" title="E131">A213</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_33_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_33_2_1"/>
not <font color="Maroon" title="c36">Pr</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c37">d</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c36">Pr</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c38">e</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c41">l</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:59_33_2_1"/><i><font color="Green" title="E134">A216</font></i>: 
( <font color="Maroon" title="c41">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c36">Pr</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c37">d</font> &amp; <font color="Maroon" title="c41">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c36">Pr</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c38">e</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><b>consider </b><font color="Maroon" title="c42">y1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c34">B</font><b> such that </b><br/><a NAME="E5:59_33_2_1"/><i><font color="Green" title="E135">A217</font></i>: 
( <font color="Maroon" title="c41">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c39">x1</font>,<font color="Maroon" title="c42">y1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c35">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c39">x1</font>,<font color="Maroon" title="c42">y1</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#E3:59_33_2"><i><font color="Green" title="E132">A214</font></i></a>, <a class="txt" href="int_5.html#E3:59_33_2_1"><i><font color="Green" title="E134">A216</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c43">y2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c34">B</font><b> such that </b><br/><a NAME="E7:59_33_2_1"/><i><font color="Green" title="E136">A218</font></i>: 
( <font color="Maroon" title="c41">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c40">x2</font>,<font color="Maroon" title="c43">y2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c35">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c40">x2</font>,<font color="Maroon" title="c43">y2</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#E5:59_33_2"><i><font color="Green" title="E133">A215</font></i></a>, <a class="txt" href="int_5.html#E3:59_33_2_1"><i><font color="Green" title="E134">A216</font></i></a></i>;<br/><b>thus </b><a NAME="E8:59_33_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="int_5.html#E1:59_33_2"><i><font color="Green" title="E131">A213</font></i></a>, <a class="txt" href="int_5.html#E3:59_33_2"><i><font color="Green" title="E132">A214</font></i></a>, <a class="txt" href="int_5.html#E5:59_33_2"><i><font color="Green" title="E133">A215</font></i></a>, <a class="txt" href="int_5.html#E5:59_33_2_1"><i><font color="Green" title="E135">A217</font></i></a>, <a class="txt" href="int_5.html#E7:59_33_2_1"><i><font color="Green" title="E136">A218</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:59_33_2"/>
<font color="Maroon" title="c36">Pr</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c37">d</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c36">Pr</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c38">e</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E10:59_33"/><b>then </b><i><font color="Green" title="E131">A219</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c36">Pr</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <span class="p1">(<span class="default"><a href="int_5.html#K3" title="INT_5:func.3">Card</a> <font color="Maroon" title="c36">Pr</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E8:59_33"><i><font color="Green" title="E130">A212</font></i></a>, <a class="ref" href="int_5.html#T48" target="_self" title="INT_5:th.48">Th48</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c34">B</font> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b2">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c33">A</font> : <font color="Maroon" title="c35">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span>  );<br/>
<a NAME="E11:59_33"/><i><font color="Green" title="E132">A220</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#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</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> holds <br/> ex <font color="Olive" title="b2">x1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c35">z</font></span>)</span> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">d</font>,<font color="Olive" title="b2">x1</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_33_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c37">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:59_33_3"/>
<font color="Maroon" title="c37">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="c4">q</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>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c38">d</font> = <font color="Maroon" title="c37">d</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c34">B</font> ;<br/>
<b>take </b><font color="Maroon" title="c39">x1</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Maroon" title="c38">d</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c33">A</font> : <font color="Maroon" title="c35">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c38">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span> ;<br/>

<a NAME="E3:59_33_3"/>
<font color="Maroon" title="c39">x1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c35">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_33_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c40">l</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:59_33_3_1"/>
<font color="Maroon" title="c40">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c39">x1</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c41">xx</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c33">A</font><b> such that </b><br/><a NAME="E3:59_33_3_1"/><i><font color="Green" title="E133">A221</font></i>: 
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c41">xx</font>,<font color="Maroon" title="c38">d</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c40">l</font> &amp; <font color="Maroon" title="c35">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c41">xx</font>,<font color="Maroon" title="c38">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/>
<a NAME="E4:59_33_3_1"/>
<font color="Maroon" title="c40">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c33">A</font>,<font color="Maroon" title="c34">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="int_5.html#E3:59_33_3_1"><i><font color="Green" title="E133">A221</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:59_33_3_1"/>
<font color="Maroon" title="c40">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c35">z</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:59_33_3"/>
( <font color="Maroon" title="c39">x1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c35">z</font></span>)</span> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c37">d</font>,<font color="Maroon" title="c39">x1</font>] )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c37">Pk</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c35">z</font></span>)</span><b> such that </b><br/><a NAME="E13:59_33"/><i><font color="Green" title="E133">A222</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c37">Pk</font> <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="c4">q</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> &amp; ( 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#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</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> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">d</font>,<font color="Maroon" title="c37">Pk</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Olive" title="b1">d</font>] ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S5" title="FINSEQ_1:sch.5">FINSEQ_1:sch 5</a>(<a class="txt" href="int_5.html#E11:59_33"><i><font color="Green" title="E132">A220</font></i></a>);<br/></i>
<a NAME="E14:59_33"/><i><font color="Green" title="E134">A223</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c37">Pk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</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#E13:59_33"><i><font color="Green" title="E133">A222</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="E15:59_33"/>
for <font color="Olive" title="b1">d</font>, <font color="Olive" title="b2">e</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="c37">Pk</font> &amp; <font color="Olive" title="b2">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="c37">Pk</font> &amp; <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">e</font> holds <br/><font color="Maroon" title="c37">Pk</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Olive" title="b1">d</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c37">Pk</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Olive" title="b2">e</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_33_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c38">d</font>, <font color="Maroon" title="c39">e</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:59_33_4"/><i><font color="Green" title="E135">A224</font></i>: 
( <font color="Maroon" title="c38">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="c37">Pk</font> &amp; <font color="Maroon" title="c39">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="c37">Pk</font> &amp; <font color="Maroon" title="c38">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c39">e</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c40">y1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c34">B</font><b> such that </b><br/><a NAME="E3:59_33_4"/><i><font color="Green" title="E136">A225</font></i>: 
( <font color="Maroon" title="c40">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">d</font> &amp; <font color="Maroon" title="c37">Pk</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c38">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Maroon" title="c40">y1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c33">A</font> : <font color="Maroon" title="c35">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c40">y1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span>  )
 <b>by </b><i><a class="txt" href="int_5.html#E13:59_33"><i><font color="Green" title="E133">A222</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c41">y2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c34">B</font><b> such that </b><br/><a NAME="E5:59_33_4"/><i><font color="Green" title="E137">A226</font></i>: 
( <font color="Maroon" title="c41">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">e</font> &amp; <font color="Maroon" title="c37">Pk</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c39">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Maroon" title="c41">y2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c33">A</font> : <font color="Maroon" title="c35">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c41">y2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span>  )
 <b>by </b><i><a class="txt" href="int_5.html#E13:59_33"><i><font color="Green" title="E133">A222</font></i></a>, <a class="txt" href="int_5.html#E1:59_33_4"><i><font color="Green" title="E135">A224</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_33_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_33_4_1"/>
not <font color="Maroon" title="c37">Pk</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c38">d</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c37">Pk</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c39">e</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c42">l</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:59_33_4_1"/><i><font color="Green" title="E138">A227</font></i>: 
( <font color="Maroon" title="c42">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c37">Pk</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c38">d</font> &amp; <font color="Maroon" title="c42">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c37">Pk</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c39">e</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><b>consider </b><font color="Maroon" title="c43">x1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c33">A</font><b> such that </b><br/><a NAME="E5:59_33_4_1"/><i><font color="Green" title="E139">A228</font></i>: 
( <font color="Maroon" title="c42">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c43">x1</font>,<font color="Maroon" title="c40">y1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c35">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c43">x1</font>,<font color="Maroon" title="c40">y1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="int_5.html#E3:59_33_4"><i><font color="Green" title="E136">A225</font></i></a>, <a class="txt" href="int_5.html#E3:59_33_4_1"><i><font color="Green" title="E138">A227</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c44">x2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c33">A</font><b> such that </b><br/><a NAME="E7:59_33_4_1"/><i><font color="Green" title="E140">A229</font></i>: 
( <font color="Maroon" title="c42">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c44">x2</font>,<font color="Maroon" title="c41">y2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c35">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c44">x2</font>,<font color="Maroon" title="c41">y2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="int_5.html#E5:59_33_4"><i><font color="Green" title="E137">A226</font></i></a>, <a class="txt" href="int_5.html#E3:59_33_4_1"><i><font color="Green" title="E138">A227</font></i></a></i>;<br/><b>thus </b><a NAME="E8:59_33_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="int_5.html#E1:59_33_4"><i><font color="Green" title="E135">A224</font></i></a>, <a class="txt" href="int_5.html#E3:59_33_4"><i><font color="Green" title="E136">A225</font></i></a>, <a class="txt" href="int_5.html#E5:59_33_4"><i><font color="Green" title="E137">A226</font></i></a>, <a class="txt" href="int_5.html#E5:59_33_4_1"><i><font color="Green" title="E139">A228</font></i></a>, <a class="txt" href="int_5.html#E7:59_33_4_1"><i><font color="Green" title="E140">A229</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:59_33_4"/>
<font color="Maroon" title="c37">Pk</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c38">d</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c37">Pk</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c39">e</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E16:59_33"/><b>then </b><i><font color="Green" title="E135">A230</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c37">Pk</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <span class="p1">(<span class="default"><a href="int_5.html#K3" title="INT_5:func.3">Card</a> <font color="Maroon" title="c37">Pk</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E14:59_33"><i><font color="Green" title="E134">A223</font></i></a>, <a class="ref" href="int_5.html#T48" target="_self" title="INT_5:th.48">Th48</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c38">U1</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c36">Pr</font></span>)</span>, <font color="Maroon" title="c39">U2</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c37">Pk</font></span>)</span> 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 <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c35">z</font></span>)</span> <b>by </b><i><a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a>, <a class="ref" href="prob_3.html#T53" title="PROB_3:th.53">PROB_3:53</a></i>;<br/>
<a NAME="E18:59_33"/>
<font color="Maroon" title="c38">U1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c39">U2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_33_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:59_33_5"/>
<font color="Maroon" title="c38">U1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c39">U2</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c40">l</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:59_33_5"/><i><font color="Green" title="E136">A231</font></i>: 
( <font color="Maroon" title="c40">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c38">U1</font> &amp; <font color="Maroon" title="c40">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c39">U2</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="E4:59_33_5"/>
<font color="Maroon" title="c40">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_3.html#K8" title="PROB_3:func.8">Union</a> <font color="Maroon" title="c36">Pr</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E3:59_33_5"><i><font color="Green" title="E136">A231</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c41">k1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E6:59_33_5"/><i><font color="Green" title="E137">A232</font></i>: 
( <font color="Maroon" title="c41">k1</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="c36">Pr</font> &amp; <font color="Maroon" title="c40">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c36">Pr</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c41">k1</font> )
 <b>by </b><i><a class="ref" href="prob_3.html#T54" title="PROB_3:th.54">PROB_3:54</a></i>;<br/>
<a NAME="E7:59_33_5"/>
<font color="Maroon" title="c40">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_3.html#K8" title="PROB_3:func.8">Union</a> <font color="Maroon" title="c37">Pk</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E3:59_33_5"><i><font color="Green" title="E136">A231</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c42">k2</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E9:59_33_5"/><i><font color="Green" title="E138">A233</font></i>: 
( <font color="Maroon" title="c42">k2</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="c37">Pk</font> &amp; <font color="Maroon" title="c40">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c37">Pk</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c42">k2</font> )
 <b>by </b><i><a class="ref" href="prob_3.html#T54" title="PROB_3:th.54">PROB_3:54</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c43">k1</font> = <font color="Maroon" title="c41">k1</font>, <font color="Maroon" title="c44">k2</font> = <font color="Maroon" title="c42">k2</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/>
<b>consider </b><font color="Maroon" title="c45">m1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c33">A</font><b> such that </b><br/><a NAME="E12:59_33_5"/><i><font color="Green" title="E139">A234</font></i>: 
( <font color="Maroon" title="c45">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c43">k1</font> &amp; <font color="Maroon" title="c36">Pr</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c43">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c45">m1</font>,<font color="Olive" title="b1">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c34">B</font> : <font color="Maroon" title="c35">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c45">m1</font>,<font color="Olive" title="b1">y</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>  </span>}</span>  )
 <b>by </b><i><a class="txt" href="int_5.html#E7:59_33"><i><font color="Green" title="E129">A211</font></i></a>, <a class="txt" href="int_5.html#E6:59_33_5"><i><font color="Green" title="E137">A232</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c46">m2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c34">B</font><b> such that </b><br/><a NAME="E14:59_33_5"/><i><font color="Green" title="E140">A235</font></i>: 
( <font color="Maroon" title="c40">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c45">m1</font>,<font color="Maroon" title="c46">m2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c35">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c45">m1</font>,<font color="Maroon" title="c46">m2</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#E6:59_33_5"><i><font color="Green" title="E137">A232</font></i></a>, <a class="txt" href="int_5.html#E12:59_33_5"><i><font color="Green" title="E139">A234</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c47">n1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c34">B</font><b> such that </b><br/><a NAME="E16:59_33_5"/><i><font color="Green" title="E141">A236</font></i>: 
( <font color="Maroon" title="c47">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c44">k2</font> &amp; <font color="Maroon" title="c37">Pk</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c44">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Maroon" title="c47">n1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c33">A</font> : <font color="Maroon" title="c35">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c47">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span>  )
 <b>by </b><i><a class="txt" href="int_5.html#E13:59_33"><i><font color="Green" title="E133">A222</font></i></a>, <a class="txt" href="int_5.html#E9:59_33_5"><i><font color="Green" title="E138">A233</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c48">n2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c33">A</font><b> such that </b><br/><a NAME="E18:59_33_5"/><i><font color="Green" title="E142">A237</font></i>: 
( <font color="Maroon" title="c40">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c48">n2</font>,<font color="Maroon" title="c47">n1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c35">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c48">n2</font>,<font color="Maroon" title="c47">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="int_5.html#E9:59_33_5"><i><font color="Green" title="E138">A233</font></i></a>, <a class="txt" href="int_5.html#E16:59_33_5"><i><font color="Green" title="E141">A236</font></i></a></i>;<br/>
<a NAME="E19:59_33_5"/>
( <font color="Maroon" title="c45">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c48">n2</font> &amp; <font color="Maroon" title="c46">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c47">n1</font> )
 
<b>by </b><i><a class="txt" href="int_5.html#E14:59_33_5"><i><font color="Green" title="E140">A235</font></i></a>, <a class="txt" href="int_5.html#E18:59_33_5"><i><font color="Green" title="E142">A237</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E20:59_33_5"/>
contradiction
 <b>by </b><i><a class="txt" href="int_5.html#E14:59_33_5"><i><font color="Green" title="E140">A235</font></i></a>, <a class="txt" href="int_5.html#E18:59_33_5"><i><font color="Green" title="E142">A237</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:59_33"/><b>then </b><i><font color="Green" title="E136">A238</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">U1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c39">U2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <span class="p2">(<span class="default"><a href="int_5.html#K3" title="INT_5:func.3">Card</a> <font color="Maroon" title="c36">Pr</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <span class="p2">(<span class="default"><a href="int_5.html#K3" title="INT_5:func.3">Card</a> <font color="Maroon" title="c37">Pk</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E10:59_33"><i><font color="Green" title="E131">A219</font></i></a>, <a class="txt" href="int_5.html#E16:59_33"><i><font color="Green" title="E135">A230</font></i></a>, <a class="ref" href="card_2.html#T53" title="CARD_2:th.53">CARD_2:53</a></i>;<br/>
<a NAME="E20:59_33"/>
<font color="Maroon" title="c38">U1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c39">U2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c35">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_33_6"><b>proof </b></a><div class="add">

<a NAME="E1:59_33_6"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c35">z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c38">U1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c39">U2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_33_6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c40">l</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:59_33_6_1"/>
<font color="Maroon" title="c40">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c35">z</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c41">x</font>, <font color="Maroon" title="c42">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:59_33_6_1"/><i><font color="Green" title="E137">A239</font></i>: 
( <font color="Maroon" title="c41">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c33">A</font> &amp; <font color="Maroon" title="c42">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c34">B</font> &amp; <font color="Maroon" title="c40">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c41">x</font>,<font color="Maroon" title="c42">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c43">x</font> = <font color="Maroon" title="c41">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c33">A</font> <b>by </b><i><a class="txt" href="int_5.html#E3:59_33_6_1"><i><font color="Green" title="E137">A239</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c44">y</font> = <font color="Maroon" title="c42">y</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c34">B</font> <b>by </b><i><a class="txt" href="int_5.html#E3:59_33_6_1"><i><font color="Green" title="E137">A239</font></i></a></i>;<br/>
<a NAME="E6:59_33_6_1"/><i><font color="Green" title="E138">A240</font></i>: 
<font color="Maroon" title="c35">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c43">x</font>,<font color="Maroon" title="c44">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_33_6_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:59_33_6_1_1"/>
<font color="Maroon" title="c35">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c43">x</font>,<font color="Maroon" title="c44">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<a NAME="E2:59_33_6_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c43">x</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">p</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="c44">y</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">q</font></span>)</span> <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="txt" href="int_5.html#E4:59_33"><i><font color="Green" title="E127">A208</font></i></a></i>;<br/>
<a NAME="E3:59_33_6_1_1"/><b>then </b>
<font color="Maroon" title="c43">x</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c44">y</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">p</font>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T96" title="XCMPLX_1:th.96">XCMPLX_1:96</a></i>;<br/>
<a NAME="E4:59_33_6_1_1"/><b>then </b>
<font color="Maroon" title="c3">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c43">x</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</font>
 
<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="E5:59_33_6_1_1"/><b>then </b><i><font color="Green" title="E139">A241</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c43">x</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E3:59"><i><font color="Green" title="E45">A3</font></i></a>, <a class="ref" href="wsierp_1.html#T37" title="WSIERP_1:th.37">WSIERP_1:37</a></i>;<br/>
<a NAME="E6:59_33_6_1_1"/><i><font color="Green" title="E140">A242</font></i>: 
( <font color="Maroon" title="c43">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c43">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">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_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E7:59_33_6_1_1"/><b>then </b>
<font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c43">x</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E5:59_33_6_1_1"><i><font color="Green" title="E139">A241</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="E8:59_33_6_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="int_5.html#E22:59"><i><font color="Green" title="E58">A16</font></i></a>, <a class="txt" href="int_5.html#E6:59_33_6_1_1"><i><font color="Green" title="E140">A242</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>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_33_6_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_33_6_1_2"/>
( <font color="Maroon" title="c35">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c43">x</font>,<font color="Maroon" title="c44">y</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>  or <font color="Maroon" title="c35">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c43">x</font>,<font color="Maroon" title="c44">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span><b>by </b><i><a class="txt" href="int_5.html#E6:59_33_6_1"><i><font color="Green" title="E138">A240</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_33_6_1_2_1"><b>suppose </b></a><a NAME="E1:59_33_6_1_2_1"/><i><font color="Green" title="E139">A243</font></i>: 
<font color="Maroon" title="c35">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c43">x</font>,<font color="Maroon" title="c44">y</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/><div class="add"><a NAME="E2:59_33_6_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c43">x</font>,<font color="Maroon" title="c36">Pr</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c43">x</font>]
 <b>by </b><i><a class="txt" href="int_5.html#E7:59_33"><i><font color="Green" title="E129">A211</font></i></a></i>;<br/><a NAME="E3:59_33_6_1_2_1"/><b>then </b>
<font color="Maroon" title="c40">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c36">Pr</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c43">x</font>
 <b>by </b><i><a class="txt" href="int_5.html#E3:59_33_6_1"><i><font color="Green" title="E137">A239</font></i></a>, <a class="txt" href="int_5.html#E1:59_33_6_1_2_1"><i><font color="Green" title="E139">A243</font></i></a></i>;<br/><a NAME="E4:59_33_6_1_2_1"/><b>then </b>
<font color="Maroon" title="c40">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_3.html#K8" title="PROB_3:func.8">Union</a> <font color="Maroon" title="c36">Pr</font>
 <b>by </b><i><a class="txt" href="int_5.html#E7:59_33"><i><font color="Green" title="E129">A211</font></i></a>, <a class="ref" href="prob_3.html#T54" title="PROB_3:th.54">PROB_3:54</a></i>;<br/><b>hence </b><a NAME="E5:59_33_6_1_2_1"/>
<font color="Maroon" title="c40">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c38">U1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c39">U2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_33_6_1_2_2"><b>suppose </b></a><a NAME="E1:59_33_6_1_2_2"/><i><font color="Green" title="E139">A244</font></i>: 
<font color="Maroon" title="c35">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c43">x</font>,<font color="Maroon" title="c44">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:59_33_6_1_2_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c44">y</font>,<font color="Maroon" title="c37">Pk</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c44">y</font>]
 <b>by </b><i><a class="txt" href="int_5.html#E13:59_33"><i><font color="Green" title="E133">A222</font></i></a></i>;<br/><a NAME="E3:59_33_6_1_2_2"/><b>then </b>
<font color="Maroon" title="c40">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c37">Pk</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c44">y</font>
 <b>by </b><i><a class="txt" href="int_5.html#E3:59_33_6_1"><i><font color="Green" title="E137">A239</font></i></a>, <a class="txt" href="int_5.html#E1:59_33_6_1_2_2"><i><font color="Green" title="E139">A244</font></i></a></i>;<br/><a NAME="E4:59_33_6_1_2_2"/><b>then </b>
<font color="Maroon" title="c40">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_3.html#K8" title="PROB_3:func.8">Union</a> <font color="Maroon" title="c37">Pk</font>
 <b>by </b><i><a class="txt" href="int_5.html#E13:59_33"><i><font color="Green" title="E133">A222</font></i></a>, <a class="ref" href="prob_3.html#T54" title="PROB_3:th.54">PROB_3:54</a></i>;<br/><b>hence </b><a NAME="E5:59_33_6_1_2_2"/>
<font color="Maroon" title="c40">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c38">U1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c39">U2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E2:59_33_6"/>
<font color="Maroon" title="c38">U1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c39">U2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c35">z</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>then </b><i><font color="Green" title="E137">A245</font></i>: <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <span class="p2">(<span class="default"><a href="int_5.html#K3" title="INT_5:func.3">Card</a> <font color="Maroon" title="c36">Pr</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <span class="p2">(<span class="default"><a href="int_5.html#K3" title="INT_5:func.3">Card</a> <font color="Maroon" title="c37">Pk</font></span>)</span></span>)</span> = 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c33">A</font>,<font color="Maroon" title="c34">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="txt" href="int_5.html#E19:59_33"><i><font color="Green" title="E136">A238</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c33">A</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c34">B</font></span>)</span>
<b>by </b><i><a class="ref" href="card_2.html#T65" title="CARD_2:th.65">CARD_2:65</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">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#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c34">B</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T78" title="FINSEQ_1:th.78">FINSEQ_1:78</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">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#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</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_1.html#T78" title="FINSEQ_1:th.78">FINSEQ_1:78</a></i>
;<br/>
<a NAME="E22:59_33"/><i><font color="Green" title="E138">A246</font></i>: 
 <a href="int_5.html#K3" title="INT_5:func.3">Card</a> <font color="Maroon" title="c36">Pr</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c10">f2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_33_8"><b>proof </b></a><div class="add">

<i><font color="Green" title="E139">A247</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="int_5.html#K3" title="INT_5:func.3">Card</a> <font color="Maroon" title="c36">Pr</font></span>)</span> = 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c36">Pr</font>
<b>by </b><i><a class="ref" href="card_3.html#D2" title="CARD_3:def.2">CARD_3:def 2</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">f2</font>
<b>by </b><i><a class="txt" href="int_5.html#E30:59"><i><font color="Green" title="E63">A30</font></i></a>, <a class="txt" href="int_5.html#E7:59_33"><i><font color="Green" title="E129">A211</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="E2:59_33_8"/>
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"><a href="int_5.html#K3" title="INT_5:func.3">Card</a> <font color="Maroon" title="c36">Pr</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="int_5.html#K3" title="INT_5:func.3">Card</a> <font color="Maroon" title="c36">Pr</font></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="c10">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_33_8_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c40">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:59_33_8_2"/><i><font color="Green" title="E140">A248</font></i>: 
<font color="Maroon" title="c40">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"><a href="int_5.html#K3" title="INT_5:func.3">Card</a> <font color="Maroon" title="c36">Pr</font></span>)</span>
 ;<br/>

<a NAME="E2:59_33_8_2"/><b>then </b><i><font color="Green" title="E141">A249</font></i>: 
<font color="Maroon" title="c40">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="c36">Pr</font>
 
<b>by </b><i><a class="ref" href="card_3.html#D2" title="CARD_3:def.2">CARD_3:def 2</a></i>;<br/>
<a NAME="E3:59_33_8_2"/>
<font color="Maroon" title="c40">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="c3">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="txt" href="int_5.html#E30:59"><i><font color="Green" title="E63">A30</font></i></a>, <a class="txt" href="int_5.html#E1:59_33_8"><i><font color="Green" title="E139">A247</font></i></a>, <a class="txt" href="int_5.html#E1:59_33_8_2"><i><font color="Green" title="E140">A248</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c41">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c33">A</font><b> such that </b><br/><a NAME="E5:59_33_8_2"/><i><font color="Green" title="E142">A250</font></i>: 
( <font color="Maroon" title="c41">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c40">d</font> &amp; <font color="Maroon" title="c36">Pr</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c40">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c41">m</font>,<font color="Olive" title="b1">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c34">B</font> : <font color="Maroon" title="c35">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c41">m</font>,<font color="Olive" title="b1">y</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>  </span>}</span>  )
 <b>by </b><i><a class="txt" href="int_5.html#E7:59_33"><i><font color="Green" title="E129">A211</font></i></a></i>;<br/>
<a NAME="E6:59_33_8_2"/>
<font color="Maroon" title="c36">Pr</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c40">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c41">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c41">m</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_33_8_2_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c42">L</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c41">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c41">m</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>
<a NAME="E1:59_33_8_2_1"/><i><font color="Green" title="E143">A251</font></i>: 
<font color="Maroon" title="c36">Pr</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c40">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c41">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c41">m</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_33_8_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c43">l</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:59_33_8_2_1_1"/>
<font color="Maroon" title="c43">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c36">Pr</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c40">d</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c44">y1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c34">B</font><b> such that </b><br/><a NAME="E3:59_33_8_2_1_1"/><i><font color="Green" title="E144">A252</font></i>: 
( <font color="Maroon" title="c43">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c41">m</font>,<font color="Maroon" title="c44">y1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c35">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c41">m</font>,<font color="Maroon" title="c44">y1</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#E5:59_33_8_2"><i><font color="Green" title="E142">A250</font></i></a></i>;<br/>
<a NAME="E4:59_33_8_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c41">m</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">p</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="c44">y1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">q</font></span>)</span> <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#E4:59_33"><i><font color="Green" title="E127">A208</font></i></a>, <a class="txt" href="int_5.html#E3:59_33_8_2_1_1"><i><font color="Green" title="E144">A252</font></i></a></i>;<br/>
<a NAME="E5:59_33_8_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c41">m</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c44">y1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c44">y1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c44">y1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E6:59_33_8_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c41">m</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">q</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c44">y1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">q</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/>
<a NAME="E7:59_33_8_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c41">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c44">y1</font>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>;<br/>
<a NAME="E8:59_33_8_2_1_1"/><b>then </b><i><font color="Green" title="E145">A253</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c41">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c44">y1</font>
 
<b>by </b><i><a class="ref" href="int_1.html#T81" title="INT_1:th.81">INT_1:81</a></i>;<br/>
<a NAME="E9:59_33_8_2_1_1"/>
<font color="Maroon" title="c41">m</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="c3">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>
 
;<br/>
<a NAME="E10:59_33_8_2_1_1"/><b>then </b><i><font color="Green" title="E146">A254</font></i>: 
<font color="Maroon" title="c41">m</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="c8">f1</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E8:59"><i><font color="Green" title="E48">A6</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="E11:59_33_8_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c41">m</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c44">y1</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E25:59"><i><font color="Green" title="E61">A19</font></i></a>, <a class="txt" href="int_5.html#E8:59_33_8_2_1_1"><i><font color="Green" title="E145">A253</font></i></a></i>;<br/>
<a NAME="E12:59_33_8_2_1_1"/><b>then </b><i><font color="Green" title="E147">A255</font></i>: 
<font color="Maroon" title="c44">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">f2</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c41">m</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E30:59"><i><font color="Green" title="E63">A30</font></i></a>, <a class="txt" href="int_5.html#E35:59"><i><font color="Green" title="E66">A33</font></i></a>, <a class="txt" href="int_5.html#E10:59_33_8_2_1_1"><i><font color="Green" title="E146">A254</font></i></a></i>;<br/>
<a NAME="E13:59_33_8_2_1_1"/>
<font color="Maroon" title="c44">y1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E14:59_33_8_2_1_1"/><b>then </b><i><font color="Green" title="E148">A256</font></i>: 
<font color="Maroon" title="c44">y1</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"><font color="Maroon" title="c10">f2</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c41">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E12:59_33_8_2_1_1"><i><font color="Green" title="E147">A255</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E15:59_33_8_2_1_1"/>
<font color="Maroon" title="c41">m</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"><font color="Maroon" title="c41">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<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="E16:59_33_8_2_1_1"/>
<font color="Maroon" title="c43">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c41">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c41">m</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="int_5.html#E3:59_33_8_2_1_1"><i><font color="Green" title="E144">A252</font></i></a>, <a class="txt" href="int_5.html#E14:59_33_8_2_1_1"><i><font color="Green" title="E148">A256</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:59_33_8_2_1"/>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c41">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c41">m</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c36">Pr</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c40">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_33_8_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c43">l</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:59_33_8_2_1_2"/>
<font color="Maroon" title="c43">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c41">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c41">m</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c44">x</font>, <font color="Maroon" title="c45">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:59_33_8_2_1_2"/><i><font color="Green" title="E144">A257</font></i>: 
( <font color="Maroon" title="c44">x</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"><font color="Maroon" title="c41">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c45">y</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"><font color="Maroon" title="c10">f2</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c41">m</font></span>)</span> &amp; <font color="Maroon" title="c43">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c44">x</font>,<font color="Maroon" title="c45">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E4:59_33_8_2_1_2"/>
<font color="Maroon" title="c41">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">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_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E5:59_33_8_2_1_2"/><b>then </b>
<font color="Maroon" title="c41">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</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="c3">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#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/>
<a NAME="E6:59_33_8_2_1_2"/><b>then </b><i><font color="Green" title="E145">A258</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c41">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">p</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"><span class="p3">(<span class="default"><font color="Maroon" title="c3">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#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">p</font>
 
<b>by </b><i><a class="ref" href="nat_2.html#T26" title="NAT_2:th.26">NAT_2:26</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_33_8_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_33_8_2_1_2_1"/>
<font color="Maroon" title="c4">q</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E2:59_33_8_2_1_2_1"/><b>then </b><i><font color="Green" title="E146">A259</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">q</font>
 <b>by </b><i><a class="ref" href="pepin.html#T6" title="PEPIN:th.6">PEPIN:6</a></i>;<br/><a NAME="E3:59_33_8_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">q</font>
 <b>by </b><i><a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></i>;<br/><a NAME="E4:59_33_8_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">q</font>
 <b>by </b><i><a class="txt" href="int_5.html#E1:59"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E5:59_33_8_2_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="int_5.html#E9:59"><i><font color="Green" title="E49">A7</font></i></a>, <a class="txt" href="int_5.html#E2:59_33_8_2_1_2_1"><i><font color="Green" title="E146">A259</font></i></a>, <a class="ref" href="nat_4.html#T12" title="NAT_4:th.12">NAT_4:12</a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:59_33_8_2_1_2"/><b>then </b><i><font color="Green" title="E146">A260</font></i>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">p</font></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="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="wsierp_1.html#T49" title="WSIERP_1:th.49">WSIERP_1:49</a></i>;<br/>
<a NAME="E9:59_33_8_2_1_2"/>
2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E16:59"><i><font color="Green" title="E53">A11</font></i></a>, <a class="ref" href="nat_d.html#T9" title="NAT_D:th.9">NAT_D:9</a></i>;<br/>
<a NAME="E10:59_33_8_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <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="pepin.html#T6" title="PEPIN:th.6">PEPIN:6</a></i>;<br/>
<a NAME="E11:59_33_8_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 
<b>by </b><i><a class="ref" href="real_3.html#T4" title="REAL_3:th.4">REAL_3:4</a></i>;<br/>
<b>then </b><i><font color="Green" title="E147">A261</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">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#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">p</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">p</font></span>)</span>
<b>by </b><i><a class="txt" href="int_5.html#E10:59"><i><font color="Green" title="E50">A8</font></i></a>, <a class="txt" href="int_5.html#E17:59"><i><font color="Green" title="E54">A12</font></i></a>, <a class="ref" href="nat_2.html#T29" title="NAT_2:th.29">NAT_2:29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2
<b>by </b><i><a class="ref" href="pre_ff.html#T5" title="PRE_FF:th.5">PRE_FF:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2
<b>by </b><i><a class="txt" href="int_5.html#E8:59_33_8_2_1_2"><i><font color="Green" title="E146">A260</font></i></a>, <a class="ref" href="int_3.html#T8" title="INT_3:th.8">INT_3:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">q</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="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"><font color="Maroon" title="c4">q</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2
<b>by </b><i><a class="txt" href="int_5.html#E10:59"><i><font color="Green" title="E50">A8</font></i></a>, <a class="txt" href="int_5.html#E17:59"><i><font color="Green" title="E54">A12</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</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#K2" title="XCMPLX_0:func.2">+</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> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2</span>)</span>
<b>by </b><i><a class="ref" href="int_3.html#T8" title="INT_3:th.8">INT_3:8</a></i>
;<br/>
<a NAME="E13:59_33_8_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">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#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</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="59_33_8_2_1_2_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_33_8_2_1_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_33_8_2_1_2_3_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_33_8_2_1_2_3_1_1"><b>suppose </b></a><a NAME="E1:59_33_8_2_1_2_3_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2 = 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">p</font></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="wsierp_1.html#T50" title="WSIERP_1:th.50">WSIERP_1:50</a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="nat_2.html#T29" title="NAT_2:th.29">NAT_2:29</a></i>
;<br/><a NAME="E3:59_33_8_2_1_2_3_1_1"/><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="c3">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#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="int_5.html#E12:59_33_8_2_1_2"><i><font color="Green" title="E147">A261</font></i></a></i>;<br/><b>hence </b><a NAME="E4:59_33_8_2_1_2_3_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">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#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</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="xreal_1.html#T45" title="XREAL_1:th.45">XREAL_1:45</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_33_8_2_1_2_3_1_2"><b>suppose </b></a><a NAME="E1:59_33_8_2_1_2_3_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:59_33_8_2_1_2_3_1_2"/><b>then </b>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</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="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="wsierp_1.html#T49" title="WSIERP_1:th.49">WSIERP_1:49</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2 = 
<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"><font color="Maroon" title="c4">q</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1

<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"><font color="Maroon" title="c4">q</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
<b>by </b><i><a class="ref" href="nat_2.html#T29" title="NAT_2:th.29">NAT_2:29</a></i>
;<br/><a NAME="E4:59_33_8_2_1_2_3_1_2"/><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="c3">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#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="int_5.html#E12:59_33_8_2_1_2"><i><font color="Green" title="E147">A261</font></i></a></i>;<br/><b>hence </b><a NAME="E5:59_33_8_2_1_2_3_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">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#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</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="xreal_1.html#T45" title="XREAL_1:th.45">XREAL_1:45</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E14:59_33_8_2_1_2"/><b>then </b><i><font color="Green" title="E148">A262</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c41">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</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#E6:59_33_8_2_1_2"><i><font color="Green" title="E145">A258</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E15:59_33_8_2_1_2"/>
<font color="Maroon" title="c41">m</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="c3">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>
 
;<br/>
<a NAME="E16:59_33_8_2_1_2"/><b>then </b><i><font color="Green" title="E149">A263</font></i>: 
<font color="Maroon" title="c41">m</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="c8">f1</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E8:59"><i><font color="Green" title="E48">A6</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>then </b><i><font color="Green" title="E150">A264</font></i>: <font color="Maroon" title="c10">f2</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c41">m</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c41">m</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">p</font>
<b>by </b><i><a class="txt" href="int_5.html#E30:59"><i><font color="Green" title="E63">A30</font></i></a>, <a class="txt" href="int_5.html#E35:59"><i><font color="Green" title="E66">A33</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c41">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">p</font>
<b>by </b><i><a class="txt" href="int_5.html#E25:59"><i><font color="Green" title="E61">A19</font></i></a>, <a class="txt" href="int_5.html#E16:59_33_8_2_1_2"><i><font color="Green" title="E149">A263</font></i></a></i>
;<br/>
<b>reconsider </b><font color="Maroon" title="c46">y</font> = <font color="Maroon" title="c45">y</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#E3:59_33_8_2_1_2"><i><font color="Green" title="E144">A257</font></i></a></i>;<br/>
<a NAME="E19:59_33_8_2_1_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c46">y</font> &amp; <font color="Maroon" title="c46">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">f2</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c41">m</font> )
 
<b>by </b><i><a class="txt" href="int_5.html#E3:59_33_8_2_1_2"><i><font color="Green" title="E144">A257</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E20:59_33_8_2_1_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c46">y</font> &amp; <font color="Maroon" title="c46">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</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#E14:59_33_8_2_1_2"><i><font color="Green" title="E148">A262</font></i></a>, <a class="txt" href="int_5.html#E17:59_33_8_2_1_2"><i><font color="Green" title="E150">A264</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c47">y</font> = <font color="Maroon" title="c46">y</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c34">B</font> <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E22:59_33_8_2_1_2"/><i><font color="Green" title="E151">A265</font></i>: 
<font color="Maroon" title="c47">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c41">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">p</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
 
<b>by </b><i><a class="txt" href="int_5.html#E3:59_33_8_2_1_2"><i><font color="Green" title="E144">A257</font></i></a>, <a class="txt" href="int_5.html#E17:59_33_8_2_1_2"><i><font color="Green" title="E150">A264</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_33_8_2_1_2_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_33_8_2_1_2_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c41">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">p</font> is <a href="int_1.html#V1" title="INT_1:attr.1">integer</a>
 ;<br/><a NAME="E2:59_33_8_2_1_2_5"/><b>then </b>
<font color="Maroon" title="c3">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c41">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</font>
 <b>by </b><i><a class="ref" href="wsierp_1.html#T22" title="WSIERP_1:th.22">WSIERP_1:22</a></i>;<br/><a NAME="E3:59_33_8_2_1_2_5"/><b>then </b><i><font color="Green" title="E152">A266</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c41">m</font>
 <b>by </b><i><a class="txt" href="int_5.html#E3:59"><i><font color="Green" title="E45">A3</font></i></a>, <a class="ref" href="wsierp_1.html#T37" title="WSIERP_1:th.37">WSIERP_1:37</a></i>;<br/><a NAME="E4:59_33_8_2_1_2_5"/><i><font color="Green" title="E153">A267</font></i>: 
( <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c41">m</font> &amp; <font color="Maroon" title="c41">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">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_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:59_33_8_2_1_2_5"/><b>then </b>
<font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c41">m</font>
 <b>by </b><i><a class="txt" href="int_5.html#E3:59_33_8_2_1_2_5"><i><font color="Green" title="E152">A266</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="E6:59_33_8_2_1_2_5"/>
contradiction
 <b>by </b><i><a class="txt" href="int_5.html#E22:59"><i><font color="Green" title="E58">A16</font></i></a>, <a class="txt" href="int_5.html#E4:59_33_8_2_1_2_5"><i><font color="Green" title="E153">A267</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="E24:59_33_8_2_1_2"/><b>then </b>
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c41">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">p</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">p</font>
 
<b>by </b><i><a class="ref" href="int_1.html#T48" title="INT_1:th.48">INT_1:48</a></i>;<br/>
<a NAME="E25:59_33_8_2_1_2"/><b>then </b>
<font color="Maroon" title="c47">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">p</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E22:59_33_8_2_1_2"><i><font color="Green" title="E151">A265</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E26:59_33_8_2_1_2"/><b>then </b>
<font color="Maroon" title="c47">y</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c41">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">p</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/>
<a NAME="E27:59_33_8_2_1_2"/><b>then </b>
<font color="Maroon" title="c47">y</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c41">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</font>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>;<br/>
<a NAME="E28:59_33_8_2_1_2"/><b>then </b>
<font color="Maroon" title="c47">y</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">q</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c41">m</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">p</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T108" title="XREAL_1:th.108">XREAL_1:108</a></i>;<br/>
<a NAME="E29:59_33_8_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c41">m</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">p</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="c47">y</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">q</font></span>)</span> <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="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/>
<a NAME="E30:59_33_8_2_1_2"/><b>then </b>
<font color="Maroon" title="c35">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c41">m</font>,<font color="Maroon" title="c47">y</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#E4:59_33"><i><font color="Green" title="E127">A208</font></i></a></i>;<br/>
<a NAME="E31:59_33_8_2_1_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c41">m</font>,<font color="Maroon" title="c47">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c36">Pr</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c40">d</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E5:59_33_8_2"><i><font color="Green" title="E142">A250</font></i></a></i>;<br/>
<b>hence </b><a NAME="E32:59_33_8_2_1_2"/>
<font color="Maroon" title="c43">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c36">Pr</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c40">d</font>
 <b>by </b><i><a class="txt" href="int_5.html#E3:59_33_8_2_1_2"><i><font color="Green" title="E144">A257</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:59_33_8_2_1"/>
<font color="Maroon" title="c36">Pr</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c40">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c41">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c41">m</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="int_5.html#E1:59_33_8_2_1"><i><font color="Green" title="E143">A251</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>then </b> <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">Pr</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c40">d</font></span>)</span> = 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c41">m</font></span>)</span></span>)</span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c41">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="ref" href="card_2.html#T11" title="CARD_2:th.11">CARD_2:11</a></i>
<br/>.= 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <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"><font color="Maroon" title="c10">f2</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c41">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="card_2.html#T13" title="CARD_2:th.13">CARD_2:13</a></i>
;<br/>
<b>then </b> <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">Pr</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c40">d</font></span>)</span> = 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c40">d</font></span>)</span>
<b>by </b><i><a class="txt" href="int_5.html#E5:59_33_8_2"><i><font color="Green" title="E142">A250</font></i></a>, <a class="ref" href="finseq_1.html#T76" title="FINSEQ_1:th.76">FINSEQ_1:76</a></i>
<br/>.= 
<font color="Maroon" title="c10">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c40">d</font>
<b>by </b><i><a class="ref" href="card_1.html#T66" title="CARD_1:th.66">CARD_1:66</a></i>
;<br/>
<b>hence </b><a NAME="E9:59_33_8_2"/>
<span class="p1">(<span class="default"><a href="int_5.html#K3" title="INT_5:func.3">Card</a> <font color="Maroon" title="c36">Pr</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c40">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c40">d</font>
 <b>by </b><i><a class="txt" href="int_5.html#E2:59_33_8_2"><i><font color="Green" title="E141">A249</font></i></a>, <a class="ref" href="card_3.html#D2" title="CARD_3:def.2">CARD_3:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:59_33_8"/>
 <a href="int_5.html#K3" title="INT_5:func.3">Card</a> <font color="Maroon" title="c36">Pr</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c10">f2</font>
 <b>by </b><i><a class="txt" href="int_5.html#E1:59_33_8"><i><font color="Green" title="E139">A247</font></i></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:59_33"/>
 <a href="int_5.html#K3" title="INT_5:func.3">Card</a> <font color="Maroon" title="c37">Pk</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c23">g2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_33_9"><b>proof </b></a><div class="add">

<i><font color="Green" title="E139">A268</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="int_5.html#K3" title="INT_5:func.3">Card</a> <font color="Maroon" title="c37">Pk</font></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">g2</font></span>)</span>
<b>by </b><i><a class="txt" href="int_5.html#E94:59"><i><font color="Green" title="E96">A124</font></i></a>, <a class="txt" href="int_5.html#E13:59_33"><i><font color="Green" title="E133">A222</font></i></a>, <a class="ref" href="card_3.html#D2" title="CARD_3:def.2">CARD_3:def 2</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">g2</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E2:59_33_9"/>
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"><a href="int_5.html#K3" title="INT_5:func.3">Card</a> <font color="Maroon" title="c37">Pk</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="int_5.html#K3" title="INT_5:func.3">Card</a> <font color="Maroon" title="c37">Pk</font></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="c23">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_33_9_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c40">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:59_33_9_2"/><i><font color="Green" title="E140">A269</font></i>: 
<font color="Maroon" title="c40">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"><a href="int_5.html#K3" title="INT_5:func.3">Card</a> <font color="Maroon" title="c37">Pk</font></span>)</span>
 ;<br/>

<a NAME="E2:59_33_9_2"/><b>then </b><i><font color="Green" title="E141">A270</font></i>: 
<font color="Maroon" title="c40">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="c37">Pk</font>
 
<b>by </b><i><a class="ref" href="card_3.html#D2" title="CARD_3:def.2">CARD_3:def 2</a></i>;<br/>
<a NAME="E3:59_33_9_2"/>
<font color="Maroon" title="c40">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="c4">q</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="txt" href="int_5.html#E94:59"><i><font color="Green" title="E96">A124</font></i></a>, <a class="txt" href="int_5.html#E1:59_33_9"><i><font color="Green" title="E139">A268</font></i></a>, <a class="txt" href="int_5.html#E1:59_33_9_2"><i><font color="Green" title="E140">A269</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c41">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c34">B</font><b> such that </b><br/><a NAME="E5:59_33_9_2"/><i><font color="Green" title="E142">A271</font></i>: 
( <font color="Maroon" title="c41">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c40">d</font> &amp; <font color="Maroon" title="c37">Pk</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c40">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Maroon" title="c41">n</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c33">A</font> : <font color="Maroon" title="c35">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c41">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span>  )
 <b>by </b><i><a class="txt" href="int_5.html#E13:59_33"><i><font color="Green" title="E133">A222</font></i></a></i>;<br/>
<a NAME="E6:59_33_9_2"/>
<font color="Maroon" title="c37">Pk</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c40">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">g2</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c41">n</font></span>)</span></span>)</span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c41">n</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_33_9_2_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c42">L</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">g2</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c41">n</font></span>)</span></span>)</span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c41">n</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>
<a NAME="E1:59_33_9_2_1"/><i><font color="Green" title="E143">A272</font></i>: 
<font color="Maroon" title="c37">Pk</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c40">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">g2</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c41">n</font></span>)</span></span>)</span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c41">n</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_33_9_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c43">l</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:59_33_9_2_1_1"/>
<font color="Maroon" title="c43">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c37">Pk</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c40">d</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c44">x</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c33">A</font><b> such that </b><br/><a NAME="E3:59_33_9_2_1_1"/><i><font color="Green" title="E144">A273</font></i>: 
( <font color="Maroon" title="c43">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c44">x</font>,<font color="Maroon" title="c41">n</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c35">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c44">x</font>,<font color="Maroon" title="c41">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="int_5.html#E5:59_33_9_2"><i><font color="Green" title="E142">A271</font></i></a></i>;<br/>
<a NAME="E4:59_33_9_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c44">x</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">p</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="c41">n</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="int_5.html#E4:59_33"><i><font color="Green" title="E127">A208</font></i></a>, <a class="txt" href="int_5.html#E3:59_33_9_2_1_1"><i><font color="Green" title="E144">A273</font></i></a></i>;<br/>
<a NAME="E5:59_33_9_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c44">x</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c41">n</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">n</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">n</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E6:59_33_9_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c44">x</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">n</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">p</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/>
<a NAME="E7:59_33_9_2_1_1"/><b>then </b>
<font color="Maroon" title="c44">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">q</font>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>;<br/>
<a NAME="E8:59_33_9_2_1_1"/><b>then </b><i><font color="Green" title="E145">A274</font></i>: 
<font color="Maroon" title="c44">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">q</font>
 
<b>by </b><i><a class="ref" href="int_1.html#T81" title="INT_1:th.81">INT_1:81</a></i>;<br/>
<a NAME="E9:59_33_9_2_1_1"/>
<font color="Maroon" title="c41">n</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="c4">q</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>
 
;<br/>
<a NAME="E10:59_33_9_2_1_1"/><b>then </b><i><font color="Green" title="E146">A275</font></i>: 
<font color="Maroon" title="c41">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g1</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E88:59"><i><font color="Green" title="E93">A112</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="E11:59_33_9_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c41">n</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">q</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c44">x</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E89:59"><i><font color="Green" title="E94">A113</font></i></a>, <a class="txt" href="int_5.html#E8:59_33_9_2_1_1"><i><font color="Green" title="E145">A274</font></i></a></i>;<br/>
<a NAME="E12:59_33_9_2_1_1"/><b>then </b><i><font color="Green" title="E147">A276</font></i>: 
<font color="Maroon" title="c44">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">g2</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c41">n</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E94:59"><i><font color="Green" title="E96">A124</font></i></a>, <a class="txt" href="int_5.html#E99:59"><i><font color="Green" title="E99">A127</font></i></a>, <a class="txt" href="int_5.html#E10:59_33_9_2_1_1"><i><font color="Green" title="E146">A275</font></i></a></i>;<br/>
<a NAME="E13:59_33_9_2_1_1"/>
<font color="Maroon" title="c44">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E14:59_33_9_2_1_1"/><b>then </b><i><font color="Green" title="E148">A277</font></i>: 
<font color="Maroon" title="c44">x</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"><font color="Maroon" title="c23">g2</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c41">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E12:59_33_9_2_1_1"><i><font color="Green" title="E147">A276</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E15:59_33_9_2_1_1"/>
<font color="Maroon" title="c41">n</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"><font color="Maroon" title="c41">n</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<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="E16:59_33_9_2_1_1"/>
<font color="Maroon" title="c43">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">g2</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c41">n</font></span>)</span></span>)</span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c41">n</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="int_5.html#E3:59_33_9_2_1_1"><i><font color="Green" title="E144">A273</font></i></a>, <a class="txt" href="int_5.html#E14:59_33_9_2_1_1"><i><font color="Green" title="E148">A277</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:59_33_9_2_1"/>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">g2</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c41">n</font></span>)</span></span>)</span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c41">n</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c37">Pk</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c40">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_33_9_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c43">l</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:59_33_9_2_1_2"/>
<font color="Maroon" title="c43">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">g2</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c41">n</font></span>)</span></span>)</span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c41">n</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c44">x</font>, <font color="Maroon" title="c45">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:59_33_9_2_1_2"/><i><font color="Green" title="E144">A278</font></i>: 
( <font color="Maroon" title="c44">x</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"><font color="Maroon" title="c23">g2</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c41">n</font></span>)</span> &amp; <font color="Maroon" title="c45">y</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"><font color="Maroon" title="c41">n</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c43">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c44">x</font>,<font color="Maroon" title="c45">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E4:59_33_9_2_1_2"/>
<font color="Maroon" title="c41">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="c4">q</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_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E5:59_33_9_2_1_2"/><b>then </b>
<font color="Maroon" title="c41">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">p</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="c4">q</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#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">p</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/>
<a NAME="E6:59_33_9_2_1_2"/><b>then </b><i><font color="Green" title="E145">A279</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c41">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">q</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"><span class="p3">(<span class="default"><font color="Maroon" title="c4">q</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#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">q</font>
 
<b>by </b><i><a class="ref" href="nat_2.html#T26" title="NAT_2:th.26">NAT_2:26</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_33_9_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_33_9_2_1_2_1"/>
<font color="Maroon" title="c3">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E2:59_33_9_2_1_2_1"/><b>then </b><i><font color="Green" title="E146">A280</font></i>: 
<font color="Maroon" title="c4">q</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">p</font>
 <b>by </b><i><a class="ref" href="pepin.html#T6" title="PEPIN:th.6">PEPIN:6</a></i>;<br/><a NAME="E3:59_33_9_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c4">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">p</font>
 <b>by </b><i><a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></i>;<br/><a NAME="E4:59_33_9_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c4">q</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">p</font>
 <b>by </b><i><a class="txt" href="int_5.html#E1:59"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E5:59_33_9_2_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="int_5.html#E9:59"><i><font color="Green" title="E49">A7</font></i></a>, <a class="txt" href="int_5.html#E2:59_33_9_2_1_2_1"><i><font color="Green" title="E146">A280</font></i></a>, <a class="ref" href="nat_4.html#T12" title="NAT_4:th.12">NAT_4:12</a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:59_33_9_2_1_2"/><b>then </b><i><font color="Green" title="E146">A281</font></i>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">q</font></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="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="wsierp_1.html#T49" title="WSIERP_1:th.49">WSIERP_1:49</a></i>;<br/>
<a NAME="E9:59_33_9_2_1_2"/>
2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">p</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E16:59"><i><font color="Green" title="E53">A11</font></i></a>, <a class="ref" href="nat_d.html#T9" title="NAT_D:th.9">NAT_D:9</a></i>;<br/>
<a NAME="E10:59_33_9_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <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="pepin.html#T6" title="PEPIN:th.6">PEPIN:6</a></i>;<br/>
<a NAME="E11:59_33_9_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 
<b>by </b><i><a class="ref" href="real_3.html#T4" title="REAL_3:th.4">REAL_3:4</a></i>;<br/>
<b>then </b><i><font color="Green" title="E147">A282</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">q</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#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">q</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</font></span>)</span>
<b>by </b><i><a class="txt" href="int_5.html#E10:59"><i><font color="Green" title="E50">A8</font></i></a>, <a class="txt" href="int_5.html#E17:59"><i><font color="Green" title="E54">A12</font></i></a>, <a class="ref" href="nat_2.html#T29" title="NAT_2:th.29">NAT_2:29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2
<b>by </b><i><a class="ref" href="pre_ff.html#T5" title="PRE_FF:th.5">PRE_FF:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2
<b>by </b><i><a class="txt" href="int_5.html#E8:59_33_9_2_1_2"><i><font color="Green" title="E146">A281</font></i></a>, <a class="ref" href="int_3.html#T8" title="INT_3:th.8">INT_3:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2
<b>by </b><i><a class="txt" href="int_5.html#E10:59"><i><font color="Green" title="E50">A8</font></i></a>, <a class="txt" href="int_5.html#E17:59"><i><font color="Green" title="E54">A12</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">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#K2" title="XCMPLX_0:func.2">+</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> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2</span>)</span>
<b>by </b><i><a class="ref" href="int_3.html#T8" title="INT_3:th.8">INT_3:8</a></i>
;<br/>
<a NAME="E13:59_33_9_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">q</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#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">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="59_33_9_2_1_2_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_33_9_2_1_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_33_9_2_1_2_3_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_33_9_2_1_2_3_1_1"><b>suppose </b></a><a NAME="E1:59_33_9_2_1_2_3_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2 = 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">q</font></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="wsierp_1.html#T50" title="WSIERP_1:th.50">WSIERP_1:50</a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="nat_2.html#T29" title="NAT_2:th.29">NAT_2:29</a></i>
;<br/><a NAME="E3:59_33_9_2_1_2_3_1_1"/><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">q</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#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="int_5.html#E12:59_33_9_2_1_2"><i><font color="Green" title="E147">A282</font></i></a></i>;<br/><b>hence </b><a NAME="E4:59_33_9_2_1_2_3_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">q</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#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">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="xreal_1.html#T45" title="XREAL_1:th.45">XREAL_1:45</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_33_9_2_1_2_3_1_2"><b>suppose </b></a><a NAME="E1:59_33_9_2_1_2_3_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:59_33_9_2_1_2_3_1_2"/><b>then </b>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</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="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="wsierp_1.html#T49" title="WSIERP_1:th.49">WSIERP_1:49</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2 = 
<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"><font color="Maroon" title="c3">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1

<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"><font color="Maroon" title="c3">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
<b>by </b><i><a class="ref" href="nat_2.html#T29" title="NAT_2:th.29">NAT_2:29</a></i>
;<br/><a NAME="E4:59_33_9_2_1_2_3_1_2"/><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">q</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#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="int_5.html#E12:59_33_9_2_1_2"><i><font color="Green" title="E147">A282</font></i></a></i>;<br/><b>hence </b><a NAME="E5:59_33_9_2_1_2_3_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">q</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#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">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="xreal_1.html#T45" title="XREAL_1:th.45">XREAL_1:45</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E14:59_33_9_2_1_2"/><b>then </b><i><font color="Green" title="E148">A283</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c41">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">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#E6:59_33_9_2_1_2"><i><font color="Green" title="E145">A279</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E15:59_33_9_2_1_2"/>
<font color="Maroon" title="c41">n</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="c4">q</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>
 
;<br/>
<a NAME="E16:59_33_9_2_1_2"/><b>then </b><i><font color="Green" title="E149">A284</font></i>: 
<font color="Maroon" title="c41">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g1</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E88:59"><i><font color="Green" title="E93">A112</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>then </b><i><font color="Green" title="E150">A285</font></i>: <font color="Maroon" title="c23">g2</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c41">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">g1</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c41">n</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">q</font>
<b>by </b><i><a class="txt" href="int_5.html#E94:59"><i><font color="Green" title="E96">A124</font></i></a>, <a class="txt" href="int_5.html#E99:59"><i><font color="Green" title="E99">A127</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c41">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">q</font>
<b>by </b><i><a class="txt" href="int_5.html#E89:59"><i><font color="Green" title="E94">A113</font></i></a>, <a class="txt" href="int_5.html#E16:59_33_9_2_1_2"><i><font color="Green" title="E149">A284</font></i></a></i>
;<br/>
<b>reconsider </b><font color="Maroon" title="c46">x</font> = <font color="Maroon" title="c44">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="int_5.html#E3:59_33_9_2_1_2"><i><font color="Green" title="E144">A278</font></i></a></i>;<br/>
<a NAME="E19:59_33_9_2_1_2"/><i><font color="Green" title="E151">A286</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c46">x</font> &amp; <font color="Maroon" title="c46">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">g2</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c41">n</font> )
 
<b>by </b><i><a class="txt" href="int_5.html#E3:59_33_9_2_1_2"><i><font color="Green" title="E144">A278</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E20:59_33_9_2_1_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c46">x</font> &amp; <font color="Maroon" title="c46">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">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#E14:59_33_9_2_1_2"><i><font color="Green" title="E148">A283</font></i></a>, <a class="txt" href="int_5.html#E17:59_33_9_2_1_2"><i><font color="Green" title="E150">A285</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c47">x</font> = <font color="Maroon" title="c46">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c33">A</font> <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_33_9_2_1_2_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_33_9_2_1_2_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c41">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">q</font> is <a href="int_1.html#V1" title="INT_1:attr.1">integer</a>
 ;<br/><a NAME="E2:59_33_9_2_1_2_5"/><b>then </b>
<font color="Maroon" title="c4">q</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c41">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">p</font>
 <b>by </b><i><a class="ref" href="wsierp_1.html#T22" title="WSIERP_1:th.22">WSIERP_1:22</a></i>;<br/><a NAME="E3:59_33_9_2_1_2_5"/><b>then </b><i><font color="Green" title="E152">A287</font></i>: 
<font color="Maroon" title="c4">q</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c41">n</font>
 <b>by </b><i><a class="txt" href="int_5.html#E3:59"><i><font color="Green" title="E45">A3</font></i></a>, <a class="ref" href="wsierp_1.html#T37" title="WSIERP_1:th.37">WSIERP_1:37</a></i>;<br/><a NAME="E4:59_33_9_2_1_2_5"/><i><font color="Green" title="E153">A288</font></i>: 
( <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c41">n</font> &amp; <font color="Maroon" title="c41">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="c4">q</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_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:59_33_9_2_1_2_5"/><b>then </b>
<font color="Maroon" title="c4">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c41">n</font>
 <b>by </b><i><a class="txt" href="int_5.html#E3:59_33_9_2_1_2_5"><i><font color="Green" title="E152">A287</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="E6:59_33_9_2_1_2_5"/>
contradiction
 <b>by </b><i><a class="txt" href="int_5.html#E22:59"><i><font color="Green" title="E58">A16</font></i></a>, <a class="txt" href="int_5.html#E4:59_33_9_2_1_2_5"><i><font color="Green" title="E153">A288</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="E23:59_33_9_2_1_2"/><b>then </b>
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c41">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">q</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">q</font>
 
<b>by </b><i><a class="ref" href="int_1.html#T48" title="INT_1:th.48">INT_1:48</a></i>;<br/>
<a NAME="E24:59_33_9_2_1_2"/><b>then </b>
<font color="Maroon" title="c47">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">q</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E17:59_33_9_2_1_2"><i><font color="Green" title="E150">A285</font></i></a>, <a class="txt" href="int_5.html#E19:59_33_9_2_1_2"><i><font color="Green" title="E151">A286</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E25:59_33_9_2_1_2"/><b>then </b>
<font color="Maroon" title="c47">x</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c41">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">q</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/>
<a NAME="E26:59_33_9_2_1_2"/><b>then </b>
<font color="Maroon" title="c47">x</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c41">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">p</font>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>;<br/>
<a NAME="E27:59_33_9_2_1_2"/><b>then </b>
<font color="Maroon" title="c47">x</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c41">n</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">q</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T108" title="XREAL_1:th.108">XREAL_1:108</a></i>;<br/>
<a NAME="E28:59_33_9_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c47">x</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">p</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="c41">n</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T51" title="XREAL_1:th.51">XREAL_1:51</a></i>;<br/>
<a NAME="E29:59_33_9_2_1_2"/><b>then </b>
<font color="Maroon" title="c35">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c47">x</font>,<font color="Maroon" title="c41">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="int_5.html#E4:59_33"><i><font color="Green" title="E127">A208</font></i></a></i>;<br/>
<a NAME="E30:59_33_9_2_1_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c47">x</font>,<font color="Maroon" title="c41">n</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c37">Pk</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c40">d</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E5:59_33_9_2"><i><font color="Green" title="E142">A271</font></i></a></i>;<br/>
<b>hence </b><a NAME="E31:59_33_9_2_1_2"/>
<font color="Maroon" title="c43">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c37">Pk</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c40">d</font>
 <b>by </b><i><a class="txt" href="int_5.html#E3:59_33_9_2_1_2"><i><font color="Green" title="E144">A278</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:59_33_9_2_1"/>
<font color="Maroon" title="c37">Pk</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c40">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">g2</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c41">n</font></span>)</span></span>)</span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c41">n</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="int_5.html#E1:59_33_9_2_1"><i><font color="Green" title="E143">A272</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:59_33_9_2"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c37">Pk</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c40">d</font></span>)</span> <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"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">g2</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c41">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="card_2.html#T13" title="CARD_2:th.13">CARD_2:13</a></i>;<br/>
<b>then </b> <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c37">Pk</font> <a href="prob_3.html#K7" title="PROB_3:func.7">.</a> <font color="Maroon" title="c40">d</font></span>)</span> = 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c40">d</font></span>)</span>
<b>by </b><i><a class="txt" href="int_5.html#E5:59_33_9_2"><i><font color="Green" title="E142">A271</font></i></a>, <a class="ref" href="finseq_1.html#T76" title="FINSEQ_1:th.76">FINSEQ_1:76</a></i>
<br/>.= 
<font color="Maroon" title="c23">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c40">d</font>
<b>by </b><i><a class="ref" href="card_1.html#T66" title="CARD_1:th.66">CARD_1:66</a></i>
;<br/>
<b>hence </b><a NAME="E9:59_33_9_2"/>
<span class="p1">(<span class="default"><a href="int_5.html#K3" title="INT_5:func.3">Card</a> <font color="Maroon" title="c37">Pk</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c40">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c40">d</font>
 <b>by </b><i><a class="txt" href="int_5.html#E2:59_33_9_2"><i><font color="Green" title="E141">A270</font></i></a>, <a class="ref" href="card_3.html#D2" title="CARD_3:def.2">CARD_3:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:59_33_9"/>
 <a href="int_5.html#K3" title="INT_5:func.3">Card</a> <font color="Maroon" title="c37">Pk</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c23">g2</font>
 <b>by </b><i><a class="txt" href="int_5.html#E1:59_33_9"><i><font color="Green" title="E139">A268</font></i></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E24:59_33"/>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c10">f2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c23">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">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#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</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="txt" href="int_5.html#E21:59_33"><i><font color="Green" title="E137">A245</font></i></a>, <a class="txt" href="int_5.html#E22:59_33"><i><font color="Green" title="E138">A246</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E152:59"/>
<span class="p1">(<span class="default"><a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c1">p</font></span>)</span> <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> <span class="p1">(<span class="default"><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> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">q</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#E85:59"><i><font color="Green" title="E92">A69</font></i></a>, <a class="txt" href="int_5.html#E150:59"><i><font color="Green" title="E125">A164</font></i></a>, <a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></i>;<br/>


</div>
