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<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> implies <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="nat_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:56"/><span class="lab"><font color="Green" title="E43">A1</font></span>: 
<font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2
 <span class="kw">and </span><br/><a NAME="E2:56"/><span class="lab"><font color="Green" title="E44">A2</font></span>: 
<font color="Maroon" title="c2">q</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2
 <span class="kw">and </span><br/><a NAME="E3:56"/><span class="lab"><font color="Green" title="E45">A3</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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="nat_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span></span><br/>

<a NAME="E4:56"/><span class="lab"><font color="Green" title="E46">A4</font></span>: 
<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c1">p</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="ref" href="int_2.html#T30" title="INT_2:th.30">INT_2:30</a></span>;<br/>
<a NAME="E5:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A5</font></span>: 
<font color="Maroon" title="c2">q</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D3" title="INT_2:def.3">INT_2:def 3</a></span>;<br/>
<span class="kw">reconsider </span><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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">p9</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2;<br/>
<a NAME="E7:56"/><span class="lab"><font color="Green" title="E48">A6</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D4" title="INT_2:def.4">INT_2:def 4</a></span>;<br/>
<a NAME="E8:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A7</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/>
<a NAME="E9:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A8</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:56"><span class="lab"><font color="Green" title="E48">A6</font></span></a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>;<br/>
<a NAME="E10:56"/>
 not <font color="Maroon" title="c3">p</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="ref" href="pepin.html#T17" title="PEPIN:th.17">PEPIN:17</a></span>;<br/>
<a NAME="E11:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A9</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:56"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="ref" href="hilbert3.html#T2" title="HILBERT3:th.2">HILBERT3:2</a></span>;<br/>
<a NAME="E12:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A10</font></span>: 
2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pepin.html#T22" title="PEPIN:th.22">PEPIN:22</a></span>;<br/>
<a NAME="E13:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A11</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T3" title="NAT_D:th.3">NAT_D:3</a></span>;<br/>
<a NAME="E14:56"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></span>;<br/>
<a NAME="E15:56"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:56"><span class="lab"><font color="Green" title="E50">A8</font></span></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></span>;<br/>
<a NAME="E16:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A12</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:56"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="ref" href="xreal_1.html#T146" title="XREAL_1:th.146">XREAL_1:146</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">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="nat_d.html#K7" title="NAT_D:func.7">-'</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="E17:56"/><span class="lab"><font color="Green" title="E55">A13</font></span>: 
 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="nat_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</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="56_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span></span>)</span> implies <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="nat_d.html#K7" title="NAT_D:func.7">-'</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="c7">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="c7">d</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_1"/><span class="lab"><font color="Green" title="E56">A14</font></span>: 
<font color="Maroon" title="c7">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="nat_d.html#K7" title="NAT_D:func.7">-'</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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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="nat_d.html#K7" title="NAT_D:func.7">-'</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="c7">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="c7">d</font></span><br/>

<a NAME="E2:56_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>;<br/>
<a NAME="E3:56_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">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="nat_d.html#K7" title="NAT_D:func.7">-'</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 class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E4:56_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A15</font></span>: 
<font color="Maroon" title="c7">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/>
<a NAME="E5:56_1"/>
<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="nat_d.html#K7" title="NAT_D:func.7">-'</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="c7">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="nat_d.html#K7" title="NAT_D:func.7">-'</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="c7">d</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1"><span class="lab"><font color="Green" title="E56">A14</font></span></a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:56_1"/>
<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="nat_d.html#K7" title="NAT_D:func.7">-'</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="c7">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="c7">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_1"><span class="lab"><font color="Green" title="E57">A15</font></span></a>, <a class="ref" href="finseq_2.html#T49" title="FINSEQ_2:th.49">FINSEQ_2:49</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E18:56"/><span class="lab"><font color="Green" title="E56">A16</font></span>: 
 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="nat_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</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="56_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span></span>)</span> implies <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="nat_d.html#K7" title="NAT_D:func.7">-'</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="c7">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_2"/>
<font color="Maroon" title="c7">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="nat_d.html#K7" title="NAT_D:func.7">-'</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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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="nat_d.html#K7" title="NAT_D:func.7">-'</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="c7">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><br/>

<a NAME="E2:56_2"/><span class="kw">then </span>
<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="nat_d.html#K7" title="NAT_D:func.7">-'</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="c7">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="c7">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:56"><span class="lab"><font color="Green" title="E55">A13</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:56_2"/>
<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="nat_d.html#K7" title="NAT_D:func.7">-'</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="c7">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E19:56"/>
 <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="nat_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>;<br/>
<a NAME="E20:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A17</font></span>: 
 <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="nat_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/>
<a NAME="E21:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A18</font></span>: 
 <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="nat_d.html#K7" title="NAT_D:func.7">-'</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_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">q9</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">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="nat_d.html#K7" title="NAT_D:func.7">-'</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="E22:56"/><span class="lab"><font color="Green" title="E59">A19</font></span>: 
 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="nat_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</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="56_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> <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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span></span>)</span> implies <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="nat_d.html#K7" title="NAT_D:func.7">-'</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="c9">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="c9">d</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_3"/><span class="lab"><font color="Green" title="E60">A20</font></span>: 
<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> <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="nat_d.html#K7" title="NAT_D:func.7">-'</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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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="nat_d.html#K7" title="NAT_D:func.7">-'</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="c9">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="c9">d</font></span><br/>

<a NAME="E2:56_3"/><span class="kw">then </span>
<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> <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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>;<br/>
<a NAME="E3:56_3"/><span class="kw">then </span>
<font color="Maroon" title="c9">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="nat_d.html#K7" title="NAT_D:func.7">-'</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 class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E4:56_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A21</font></span>: 
<font color="Maroon" title="c9">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/>
<a NAME="E5:56_3"/>
<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="nat_d.html#K7" title="NAT_D:func.7">-'</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="c9">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="nat_d.html#K7" title="NAT_D:func.7">-'</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="c9">d</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_3"><span class="lab"><font color="Green" title="E60">A20</font></span></a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:56_3"/>
<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="nat_d.html#K7" title="NAT_D:func.7">-'</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="c9">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="c9">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_3"><span class="lab"><font color="Green" title="E61">A21</font></span></a>, <a class="ref" href="finseq_2.html#T49" title="FINSEQ_2:th.49">FINSEQ_2:49</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E23:56"/><span class="lab"><font color="Green" title="E60">A22</font></span>: 
 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="nat_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</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="56_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> <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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span></span>)</span> implies <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="nat_d.html#K7" title="NAT_D:func.7">-'</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="c9">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_4"/>
<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> <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="nat_d.html#K7" title="NAT_D:func.7">-'</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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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="nat_d.html#K7" title="NAT_D:func.7">-'</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="c9">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><br/>

<a NAME="E2:56_4"/><span class="kw">then </span>
<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="nat_d.html#K7" title="NAT_D:func.7">-'</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="c9">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="c9">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:56"><span class="lab"><font color="Green" title="E59">A19</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:56_4"/>
<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="nat_d.html#K7" title="NAT_D:func.7">-'</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="c9">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E24:56"/>
 <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="nat_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>;<br/>
<a NAME="E25:56"/><span class="kw">then </span>
 <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="nat_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/>
<a NAME="E26:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A23</font></span>: 
 <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="nat_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">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="nat_d.html#K7" title="NAT_D:func.7">-'</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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:56"><span class="lab"><font color="Green" title="E60">A22</font></span></a>, <a class="ref" href="finseq_2.html#T12" title="FINSEQ_2:th.12">FINSEQ_2:12</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">g3</font> = <font color="Maroon" title="c9">g1</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c4">q</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">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#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">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/>
<a NAME="E28:56"/><span class="lab"><font color="Green" title="E62">A24</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">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="c9">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euler_2.html#D1" title="EULER_2:def.1">EULER_2:def 1</a></span>;<br/>
<a NAME="E29:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A25</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">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="c9">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 class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">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#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">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="E30:56"/>
 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#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">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#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">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="56_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">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#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">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>  implies <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">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><br/>

<span class="kw">assume </span><a NAME="E1:56_5"/>
<font color="Maroon" title="c13">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#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">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><br/>

<a NAME="E2:56_5"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">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>  st <br/>( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">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 )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:56_5"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E31:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A26</font></span>: 
<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#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">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#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">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 class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">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="nat_d.html#K7" title="NAT_D:func.7">-'</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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:56"><span class="lab"><font color="Green" title="E56">A16</font></span></a>, <a class="ref" href="finseq_2.html#T12" title="FINSEQ_2:th.12">FINSEQ_2:12</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <span class="kw">-&gt; </span>   <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="c13">f1</font> <a href="recdef_1.html#K1" title="RECDEF_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/>
<span class="kw">consider </span><font color="Maroon" title="c14">f2</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E34:56"/><span class="lab"><font color="Green" title="E65">A27</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">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="nat_d.html#K7" title="NAT_D:func.7">-'</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="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="c14">f2</font> holds <br/><font color="Maroon" title="c14">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>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_1.html#S2" title="FINSEQ_1:sch.2">FINSEQ_1:sch 2</a>();<br/></span>
<a NAME="E35:56"/><span class="lab"><font color="Green" title="E66">A28</font></span>: 
<font color="Maroon" title="c4">q</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D4" title="INT_2:def.4">INT_2:def 4</a></span>;<br/>
<a NAME="E36:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A29</font></span>: 
<font color="Maroon" title="c4">q</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/>
<a NAME="E37:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A30</font></span>: 
<font color="Maroon" title="c4">q</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:56"><span class="lab"><font color="Green" title="E66">A28</font></span></a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>;<br/>
<a NAME="E38:56"/>
<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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E39:56"/><span class="kw">then </span>
<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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E40:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A31</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:56"><span class="lab"><font color="Green" title="E67">A29</font></span></a>, <a class="ref" href="nat_2.html#T13" title="NAT_2:th.13">NAT_2:13</a></span>;<br/>
<a NAME="E41:56"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:56"><span class="lab"><font color="Green" title="E61">A23</font></span></a>, <a class="ref" href="euler_2.html#D1" title="EULER_2:def.1">EULER_2:def 1</a></span>;<br/>
<a NAME="E42:56"/><span class="kw">then </span>
<font color="Maroon" title="c9">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> 
 
;<br/>
<a NAME="E43:56"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">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> <span class="kw"> such that </span><br/><a NAME="E45:56"/><span class="lab"><font color="Green" title="E70">A32</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">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="c15">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <span class="kw">-&gt; </span>   <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="c9">g1</font> <a href="recdef_1.html#K1" title="RECDEF_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/>
<span class="kw">consider </span><font color="Maroon" title="c16">g2</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E47:56"/><span class="lab"><font color="Green" title="E71">A33</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">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="nat_d.html#K7" title="NAT_D:func.7">-'</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="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="c16">g2</font> holds <br/><font color="Maroon" title="c16">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>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_1.html#S2" title="FINSEQ_1:sch.2">FINSEQ_1:sch 2</a>();<br/></span>
<a NAME="E48:56"/>
 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="c16">g2</font> holds <br/><font color="Maroon" title="c16">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="56_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">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="c16">g2</font> implies <font color="Maroon" title="c16">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_6"/>
<font color="Maroon" title="c17">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="c16">g2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><br/>

<a NAME="E2:56_6"/><span class="kw">then </span>
<font color="Maroon" title="c16">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">d</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="kw">by</span> <span class="lab"><a class="txt" href="#E47:56"><span class="lab"><font color="Green" title="E71">A33</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:56_6"/>
<font color="Maroon" title="c16">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">g2</font> = <font color="Maroon" title="c16">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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T12" title="FINSEQ_2:th.12">FINSEQ_2:12</a></span>;<br/>
<a NAME="E50:56"/><span class="lab"><font color="Green" title="E72">A34</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">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="c17">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:56"><span class="lab"><font color="Green" title="E61">A23</font></span></a>, <a class="txt" href="#E47:56"><span class="lab"><font color="Green" title="E71">A33</font></span></a>, <a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/>
<a NAME="E51:56"/><span class="lab"><font color="Green" title="E73">A35</font></span>: 
 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">g1</font> holds <br/><font color="Maroon" title="c9">g1</font> <a href="recdef_1.html#K1" title="RECDEF_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="c17">g2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">d</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#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">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="56_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">g1</font> implies <font color="Maroon" title="c9">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c18">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">g2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c18">d</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#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">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="c18">d</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_7"/><span class="lab"><font color="Green" title="E74">A36</font></span>: 
<font color="Maroon" title="c18">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">g1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c18">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">g2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c18">d</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#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">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="c18">d</font></span>)</span></span><br/>

<a NAME="E2:56_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A37</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">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="c18">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c18">d</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euler_2.html#D1" title="EULER_2:def.1">EULER_2:def 1</a></span>;<br/>
<a NAME="E3:56_7"/>
<font color="Maroon" title="c17">g2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c18">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c18">d</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="kw">by</span> <span class="lab"><a class="txt" href="#E47:56"><span class="lab"><font color="Green" title="E71">A33</font></span></a>, <a class="txt" href="#E50:56"><span class="lab"><font color="Green" title="E72">A34</font></span></a>, <a class="txt" href="#E1:56_7"><span class="lab"><font color="Green" title="E74">A36</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:56_7"/>
<font color="Maroon" title="c9">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c18">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">g2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c18">d</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#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">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="c18">d</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_7"><span class="lab"><font color="Green" title="E75">A37</font></span></a>, <a class="ref" href="nat_d.html#T2" title="NAT_D:th.2">NAT_D:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E52:56"/>
 not <font color="Maroon" title="c4">q</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="ref" href="pepin.html#T17" title="PEPIN:th.17">PEPIN:17</a></span>;<br/>
<a NAME="E53:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A38</font></span>: 
<font color="Maroon" title="c4">q</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:56"><span class="lab"><font color="Green" title="E67">A29</font></span></a>, <a class="ref" href="hilbert3.html#T2" title="HILBERT3:th.2">HILBERT3:2</a></span>;<br/>
<a NAME="E54:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A39</font></span>: 
2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">q</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pepin.html#T22" title="PEPIN:th.22">PEPIN:22</a></span>;<br/>
<a NAME="E55:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A40</font></span>: 
<font color="Maroon" title="c4">q</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T3" title="NAT_D:th.3">NAT_D:3</a></span>;<br/>
<a NAME="E56:56"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></span>;<br/>
<a NAME="E57:56"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:56"><span class="lab"><font color="Green" title="E68">A30</font></span></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></span>;<br/>
<a NAME="E58:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A41</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:56"><span class="lab"><font color="Green" title="E67">A29</font></span></a>, <a class="ref" href="xreal_1.html#T146" title="XREAL_1:th.146">XREAL_1:146</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E59:56"/>
 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#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">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="56_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:56_8"/>
 <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#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">a</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:56_8"/><span class="lab"><font color="Green" title="E78">A42</font></span>: 
<font color="Maroon" title="c18">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="c9">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 class="kw">and </span><br/><a NAME="E4:56_8"/><span class="lab"><font color="Green" title="E79">A43</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">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="c18">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></span>;<br/>
<a NAME="E5:56_8"/>
<font color="Maroon" title="c18">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="c9">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:56"><span class="lab"><font color="Green" title="E62">A24</font></span></a>, <a class="txt" href="#E3:56_8"><span class="lab"><font color="Green" title="E78">A42</font></span></a>, <a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/>
<a NAME="E6:56_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A44</font></span>: 
<font color="Maroon" title="c9">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c18">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="c17">g2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c18">a</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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E51:56"><span class="lab"><font color="Green" title="E73">A35</font></span></a>, <a class="txt" href="#E4:56_8"><span class="lab"><font color="Green" title="E79">A43</font></span></a></span>;<br/>
<a NAME="E7:56_8"/>
<font color="Maroon" title="c18">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="c9">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:56"><span class="lab"><font color="Green" title="E62">A24</font></span></a>, <a class="txt" href="#E3:56_8"><span class="lab"><font color="Green" title="E78">A42</font></span></a>, <a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/>
<a NAME="E8:56_8"/><span class="kw">then </span>
<font color="Maroon" title="c3">p</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c18">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">g2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c18">a</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:56"><span class="lab"><font color="Green" title="E59">A19</font></span></a>, <a class="txt" href="#E6:56_8"><span class="lab"><font color="Green" title="E80">A44</font></span></a></span>;<br/>
<a NAME="E9:56_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A45</font></span>: 
<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="c18">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></span>;<br/>
<a NAME="E10:56_8"/>
<font color="Maroon" title="c18">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_8"><span class="lab"><font color="Green" title="E78">A42</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E11:56_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A46</font></span>: 
<font color="Maroon" title="c4">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="txt" href="#E9:56_8"><span class="lab"><font color="Green" title="E81">A45</font></span></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a>, <a class="ref" href="pepin.html#T3" title="PEPIN:th.3">PEPIN:3</a></span>;<br/>
<a NAME="E12:56_8"/>
<font color="Maroon" title="c18">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:56"><span class="lab"><font color="Green" title="E61">A23</font></span></a>, <a class="txt" href="#E28:56"><span class="lab"><font color="Green" title="E62">A24</font></span></a>, <a class="txt" href="#E3:56_8"><span class="lab"><font color="Green" title="E78">A42</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:56_8"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:56"><span class="lab"><font color="Green" title="E77">A41</font></span></a>, <a class="txt" href="#E11:56_8"><span class="lab"><font color="Green" title="E82">A46</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E60:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A47</font></span>: 
<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#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">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 class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T50" title="ZFMISC_1:th.50">ZFMISC_1:50</a></span>;<br/>
<a NAME="E61:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A48</font></span>: 
 <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:56"><span class="lab"><font color="Green" title="E70">A32</font></span></a>, <a class="ref" href="finseq_3.html#T92" title="FINSEQ_3:th.92">FINSEQ_3:92</a>, <a class="ref" href="xboole_1.html#T73" title="XBOOLE_1:th.73">XBOOLE_1:73</a></span>;<br/>
<a NAME="E62:56"/><span class="lab"><font color="Green" title="E80">A49</font></span>: 
 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="c9">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="c9">g1</font> &amp; <font color="Maroon" title="c4">q</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g1</font> <a href="recdef_1.html#K1" title="RECDEF_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="c9">g1</font> <a href="recdef_1.html#K1" title="RECDEF_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="56_9"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:56_9"/><span class="lab"><font color="Green" title="E81">A50</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="ref" href="int_2.html#T30" title="INT_2:th.30">INT_2:30</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c18">d</font>, <font color="Maroon" title="c19">e</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">g1</font> &amp; <font color="Maroon" title="c19">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="c9">g1</font> &amp; <font color="Maroon" title="c4">q</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c18">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="c9">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c19">e</font></span>)</span> implies <font color="Maroon" title="c18">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">e</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:56_9"/><span class="lab"><font color="Green" title="E82">A51</font></span>: 
<font color="Maroon" title="c18">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">g1</font>
 <span class="kw">and </span><br/><a NAME="E3:56_9"/><span class="lab"><font color="Green" title="E83">A52</font></span>: 
<font color="Maroon" title="c19">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="c9">g1</font>
 <span class="kw">and </span><br/><a NAME="E4:56_9"/><span class="lab"><font color="Green" title="E84">A53</font></span>: 
<font color="Maroon" title="c4">q</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c18">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="c9">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c19">e</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">e</font></span><br/>

<a NAME="E5:56_9"/><span class="lab"><font color="Green" title="E85">A54</font></span>: 
<font color="Maroon" title="c9">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c19">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="c19">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:56"><span class="lab"><font color="Green" title="E59">A19</font></span></a>, <a class="txt" href="#E3:56_9"><span class="lab"><font color="Green" title="E83">A52</font></span></a></span>;<br/>
<a NAME="E6:56_9"/>
<font color="Maroon" title="c9">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c18">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c18">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:56"><span class="lab"><font color="Green" title="E59">A19</font></span></a>, <a class="txt" href="#E2:56_9"><span class="lab"><font color="Green" title="E82">A51</font></span></a></span>;<br/>
<a NAME="E7:56_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A55</font></span>: 
<font color="Maroon" title="c4">q</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">d</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">e</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_9"><span class="lab"><font color="Green" title="E84">A53</font></span></a>, <a class="txt" href="#E5:56_9"><span class="lab"><font color="Green" title="E85">A54</font></span></a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_9_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:56_9_1"/>
<font color="Maroon" title="c18">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c19">e</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:56_9_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">d</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">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="E3:56_9_1"/><span class="kw">then </span>
 <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="c18">d</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">e</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:56_9"><span class="lab"><font color="Green" title="E86">A55</font></span></a>, <a class="txt" href="#E1:56_9"><span class="lab"><font color="Green" title="E81">A50</font></span></a>, <a class="ref" href="int_2.html#T25" title="INT_2:th.25">INT_2:25</a>, <a class="ref" href="int_4.html#T6" title="INT_4:th.6">INT_4:6</a></span>;<br/><a NAME="E4:56_9_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A56</font></span>: 
<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="c18">d</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">e</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><a NAME="E5:56_9_1"/><span class="lab"><font color="Green" title="E88">A57</font></span>: 
<font color="Maroon" title="c19">e</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_9"><span class="lab"><font color="Green" title="E83">A52</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E6:56_9_1"/><span class="lab"><font color="Green" title="E89">A58</font></span>: 
<font color="Maroon" title="c18">d</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_9"><span class="lab"><font color="Green" title="E82">A51</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E7:56_9_1"/>
<font color="Maroon" title="c19">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:56"><span class="lab"><font color="Green" title="E61">A23</font></span></a>, <a class="txt" href="#E3:56_9"><span class="lab"><font color="Green" title="E83">A52</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E8:56_9_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A59</font></span>: 
<font color="Maroon" title="c18">d</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:56_9_1"><span class="lab"><font color="Green" title="E89">A58</font></span></a>, <a class="ref" href="xreal_1.html#T13" title="XREAL_1:th.13">XREAL_1:13</a></span>;<br/><a NAME="E9:56_9_1"/><span class="lab"><font color="Green" title="E91">A60</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:56"><span class="lab"><font color="Green" title="E77">A41</font></span></a>, <a class="ref" href="xreal_1.html#T147" title="XREAL_1:th.147">XREAL_1:147</a></span>;<br/><a NAME="E10:56_9_1"/>
<font color="Maroon" title="c18">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:56"><span class="lab"><font color="Green" title="E61">A23</font></span></a>, <a class="txt" href="#E2:56_9"><span class="lab"><font color="Green" title="E82">A51</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E11:56_9_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">d</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">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="nat_d.html#K7" title="NAT_D:func.7">-'</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E5:56_9_1"><span class="lab"><font color="Green" title="E88">A57</font></span></a>, <a class="ref" href="xreal_1.html#T13" title="XREAL_1:th.13">XREAL_1:13</a></span>;<br/><a NAME="E12:56_9_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A61</font></span>: 
<font color="Maroon" title="c18">d</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">e</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:56_9_1"><span class="lab"><font color="Green" title="E91">A60</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E13:56_9_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="nat_d.html#K7" title="NAT_D:func.7">-'</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:56_9_1"><span class="lab"><font color="Green" title="E91">A60</font></span></a>, <a class="ref" href="xreal_1.html#T24" title="XREAL_1:th.24">XREAL_1:24</a></span>;<br/><a NAME="E14:56_9_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">d</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:56_9_1"><span class="lab"><font color="Green" title="E90">A59</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E15:56_9_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_9_1"><span class="lab"><font color="Green" title="E87">A56</font></span></a>, <a class="txt" href="#E12:56_9_1"><span class="lab"><font color="Green" title="E92">A61</font></span></a>, <a class="ref" href="seq_2.html#T1" title="SEQ_2:th.1">SEQ_2:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E9:56_9"/>
<font color="Maroon" title="c18">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">e</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E63:56"/>
 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="c9">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="c9">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="c9">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="c9">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="56_10"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">x</font>, <font color="Maroon" title="c19">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">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="c9">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="c19">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="c9">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="c9">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="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">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="c19">y</font> implies <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:56_10"/><span class="lab"><font color="Green" title="E81">A62</font></span>: 
<font color="Maroon" title="c18">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="c9">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 class="kw">and </span><br/><a NAME="E2:56_10"/><span class="lab"><font color="Green" title="E82">A63</font></span>: 
<font color="Maroon" title="c19">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="c9">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 class="kw">and </span><br/><a NAME="E3:56_10"/><span class="lab"><font color="Green" title="E83">A64</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">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="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">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="c19">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c20">x</font> = <font color="Maroon" title="c18">x</font>, <font color="Maroon" title="c21">y</font> = <font color="Maroon" title="c19">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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_10"><span class="lab"><font color="Green" title="E81">A62</font></span></a>, <a class="txt" href="#E2:56_10"><span class="lab"><font color="Green" title="E82">A63</font></span></a></span>;<br/>
<a NAME="E5:56_10"/><span class="lab"><font color="Green" title="E84">A65</font></span>: 
<font color="Maroon" title="c9">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">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="c17">g2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">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="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="c9">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="c21">y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:56"><span class="lab"><font color="Green" title="E63">A25</font></span></a>, <a class="txt" href="#E51:56"><span class="lab"><font color="Green" title="E73">A35</font></span></a>, <a class="txt" href="#E2:56_10"><span class="lab"><font color="Green" title="E82">A63</font></span></a></span>;<br/>
<a NAME="E6:56_10"/>
<font color="Maroon" title="c9">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">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="c17">g2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">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="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="c9">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="c20">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:56"><span class="lab"><font color="Green" title="E63">A25</font></span></a>, <a class="txt" href="#E51:56"><span class="lab"><font color="Green" title="E73">A35</font></span></a>, <a class="txt" href="#E1:56_10"><span class="lab"><font color="Green" title="E81">A62</font></span></a></span>;<br/>
<a NAME="E7:56_10"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">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="c9">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">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="c17">g2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">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="c17">g2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_10"><span class="lab"><font color="Green" title="E83">A64</font></span></a>, <a class="txt" href="#E5:56_10"><span class="lab"><font color="Green" title="E84">A65</font></span></a></span>;<br/>
<a NAME="E8:56_10"/><span class="kw">then </span>
<font color="Maroon" title="c4">q</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">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="c9">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D3" title="INT_1:def.3">INT_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:56_10"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E62:56"><span class="lab"><font color="Green" title="E80">A49</font></span></a>, <a class="txt" href="#E29:56"><span class="lab"><font color="Green" title="E63">A25</font></span></a>, <a class="txt" href="#E1:56_10"><span class="lab"><font color="Green" title="E81">A62</font></span></a>, <a class="txt" href="#E2:56_10"><span class="lab"><font color="Green" title="E82">A63</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E64:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A66</font></span>: 
<font color="Maroon" title="c9">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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>;<br/>
<a NAME="E65:56"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">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#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">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="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T62" title="FINSEQ_4:th.62">FINSEQ_4:62</a></span>;<br/>
<a NAME="E66:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A67</font></span>: 
 <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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:56"><span class="lab"><font color="Green" title="E61">A23</font></span></a>, <a class="txt" href="#E28:56"><span class="lab"><font color="Green" title="E62">A24</font></span></a>, <a class="txt" href="#E45:56"><span class="lab"><font color="Green" title="E70">A32</font></span></a>, <a class="txt" href="#E60:56"><span class="lab"><font color="Green" title="E78">A47</font></span></a>, <a class="ref" href="finseq_3.html#T39" title="FINSEQ_3:th.39">FINSEQ_3:39</a>, <a class="ref" href="xboole_1.html#T73" title="XBOOLE_1:th.73">XBOOLE_1:73</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">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#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:56"><span class="lab"><font color="Green" title="E64">A26</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c19">mm</font> =  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c18">XX</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c20">YY</font> = <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">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="c18">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="E69:56"/><span class="lab"><font color="Green" title="E83">A68</font></span>: 
<font color="Maroon" title="c9">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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/>
<a NAME="E70:56"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:56"><span class="lab"><font color="Green" title="E61">A23</font></span></a>, <a class="ref" href="euler_2.html#D1" title="EULER_2:def.1">EULER_2:def 1</a></span>;<br/>
<a NAME="E71:56"/><span class="kw">then </span>
<font color="Maroon" title="c9">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="nat_d.html#K7" title="NAT_D:func.7">-'</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E69:56"><span class="lab"><font color="Green" title="E83">A68</font></span></a></span>;<br/>
<a NAME="E72:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A69</font></span>: 
<font color="Maroon" title="c9">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="nat_d.html#K7" title="NAT_D:func.7">-'</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></span>;<br/>
<a NAME="E73:56"/>
 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="nat_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E74:56"/><span class="kw">then </span>
 <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="nat_d.html#K7" title="NAT_D:func.7">-'</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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T12" title="FINSEQ_2:th.12">FINSEQ_2:12</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c21">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="nat_d.html#K7" title="NAT_D:func.7">-'</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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="int_5.html#E55"><span class="lab"><font color="Green" title="E41">Lm4</font></span></a></span>;<br/>
<a NAME="E76:56"/><span class="lab"><font color="Green" title="E85">A70</font></span>: 
2,<font color="Maroon" title="c4">q</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="ref" href="euler_1.html#T2" title="EULER_1:th.2">EULER_1:2</a></span>;<br/>
<a NAME="E77:56"/>
 <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="c17">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="c17">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>;<br/>
<a NAME="E78:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A71</font></span>: 
 <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="c17">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:56"><span class="lab"><font color="Green" title="E71">A33</font></span></a>, <a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/>
<a NAME="E79:56"/>
<font color="Maroon" title="c4">q</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c17">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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/>
<a NAME="E80:56"/><span class="kw">then </span>
<font color="Maroon" title="c4">q</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c17">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="nat_d.html#K7" title="NAT_D:func.7">-'</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:56"><span class="lab"><font color="Green" title="E86">A71</font></span></a></span>;<br/>
<a NAME="E81:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A72</font></span>: 
<font color="Maroon" title="c4">q</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c17">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="nat_d.html#K7" title="NAT_D:func.7">-'</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></span>;<br/>
<span class="lab"><font color="Green" title="E88">A73</font></span>:  <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="c17">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="c9">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="c17">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="c9">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="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c17">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="c9">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="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">g1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:56"><span class="lab"><font color="Green" title="E63">A25</font></span></a>, <a class="txt" href="#E50:56"><span class="lab"><font color="Green" title="E72">A34</font></span></a></span>
;<br/>
<a NAME="E83:56"/>
 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">g1</font> holds <br/><font color="Maroon" title="c9">g1</font> <a href="recdef_1.html#K1" title="RECDEF_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="c17">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="c9">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="56_12"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c22">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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="c9">g1</font> implies <font color="Maroon" title="c9">g1</font> <a href="recdef_1.html#K1" title="RECDEF_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"><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="c17">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="c9">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="c22">d</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_12"/><span class="lab"><font color="Green" title="E89">A74</font></span>: 
<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="c9">g1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">g1</font> <a href="recdef_1.html#K1" title="RECDEF_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"><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="c17">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="c9">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="c22">d</font></span><br/>

<a NAME="E2:56_12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A75</font></span>: 
<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"><font color="Maroon" title="c4">q</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c17">g2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:56"><span class="lab"><font color="Green" title="E72">A34</font></span></a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>;<br/>
<a NAME="E3:56_12"/>
<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="c17">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="c9">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="c22">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="c17">g2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">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="c9">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="c22">d</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E82:56"><span class="lab"><font color="Green" title="E88">A73</font></span></a>, <a class="txt" href="#E1:56_12"><span class="lab"><font color="Green" title="E89">A74</font></span></a>, <a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></span>;<br/>
<span class="kw">hence </span><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="c17">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="c9">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="c22">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="c17">g2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">d</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"><font color="Maroon" title="c9">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="c22">d</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_12"><span class="lab"><font color="Green" title="E90">A75</font></span></a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>
<br/>.= 
<font color="Maroon" title="c9">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">d</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E51:56"><span class="lab"><font color="Green" title="E73">A35</font></span></a>, <a class="txt" href="#E1:56_12"><span class="lab"><font color="Green" title="E89">A74</font></span></a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E84:56"/><span class="kw">then </span>
<font color="Maroon" title="c9">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="c17">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="c9">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E82:56"><span class="lab"><font color="Green" title="E88">A73</font></span></a>, <a class="ref" href="finseq_1.html#T13" title="FINSEQ_1:th.13">FINSEQ_1:13</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E89">A76</font></span>:  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c9">g1</font> = 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">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="c17">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#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">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="kw">by</span> <span class="lab"><a class="txt" href="#E81:56"><span class="lab"><font color="Green" title="E87">A72</font></span></a>, <a class="txt" href="#E72:56"><span class="lab"><font color="Green" title="E84">A69</font></span></a>, <a class="ref" href="rvsum_1.html#T89" title="RVSUM_1:th.89">RVSUM_1:89</a></span>
<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#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c17">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#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">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="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T87" title="RVSUM_1:th.87">RVSUM_1:87</a></span>
;<br/>
<a NAME="E86:56"/><span class="lab"><font color="Green" title="E90">A77</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">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="c15">n2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:56"><span class="lab"><font color="Green" title="E70">A32</font></span></a>, <a class="txt" href="#E60:56"><span class="lab"><font color="Green" title="E78">A47</font></span></a>, <a class="ref" href="xboole_1.html#T73" title="XBOOLE_1:th.73">XBOOLE_1:73</a></span>;<br/>
<a NAME="E87:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A78</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">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#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">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 class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></span>;<br/>
<a NAME="E88:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A79</font></span>: 
<font color="Maroon" title="c18">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="c15">n2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E86:56"><span class="lab"><font color="Green" title="E90">A77</font></span></a>, <a class="txt" href="#E31:56"><span class="lab"><font color="Green" title="E64">A26</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E89:56"/><span class="lab"><font color="Green" title="E93">A80</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">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#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E64:56"><span class="lab"><font color="Green" title="E81">A66</font></span></a>, <a class="txt" href="#E87:56"><span class="lab"><font color="Green" title="E91">A78</font></span></a>, <a class="ref" href="finseq_4.html#T62" title="FINSEQ_4:th.62">FINSEQ_4:62</a></span>;<br/>
<a NAME="E90:56"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c18">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#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:56"><span class="lab"><font color="Green" title="E64">A26</font></span></a>, <a class="ref" href="nat_1.html#T43" title="NAT_1:th.43">NAT_1:43</a></span>;<br/>
<a NAME="E91:56"/><span class="kw">then </span>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c18">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:56"><span class="lab"><font color="Green" title="E61">A23</font></span></a>, <a class="txt" href="#E89:56"><span class="lab"><font color="Green" title="E93">A80</font></span></a>, <a class="ref" href="euler_2.html#D1" title="EULER_2:def.1">EULER_2:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c22">nn</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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="c18">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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a></span>;<br/>
<a NAME="E93:56"/><span class="lab"><font color="Green" title="E94">A81</font></span>: 
 <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfinseq.html#T8" title="RFINSEQ:th.8">RFINSEQ:8</a></span>;<br/>
<a NAME="E94:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A82</font></span>: 
<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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E61:56"><span class="lab"><font color="Green" title="E79">A48</font></span></a>, <a class="ref" href="finseq_3.html#T91" title="FINSEQ_3:th.91">FINSEQ_3:91</a></span>;<br/>
<span class="lab"><font color="Green" title="E96">A83</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</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
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E53:56"><span class="lab"><font color="Green" title="E74">A38</font></span></a>, <a class="ref" href="nat_2.html#T26" title="NAT_2:th.26">NAT_2:26</a></span>
<br/>.= 
<font color="Maroon" title="c4">q</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:56"><span class="lab"><font color="Green" title="E66">A28</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>
;<br/>
<a NAME="E96:56"/>
 <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T24" title="FINSEQ_2:th.24">FINSEQ_2:24</a></span>;<br/>
<a NAME="E97:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A84</font></span>: 
 <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">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#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E64:56"><span class="lab"><font color="Green" title="E81">A66</font></span></a>, <a class="txt" href="#E87:56"><span class="lab"><font color="Green" title="E91">A78</font></span></a>, <a class="txt" href="#E61:56"><span class="lab"><font color="Green" title="E79">A48</font></span></a>, <a class="ref" href="rfinseq.html#T9" title="RFINSEQ:th.9">RFINSEQ:9</a>, <a class="ref" href="rfinseq.html#T26" title="RFINSEQ:th.26">RFINSEQ:26</a></span>;<br/>
<a NAME="E98:56"/><span class="lab"><font color="Green" title="E98">A85</font></span>: 
<span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">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="c18">XX</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">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 class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/>
<a NAME="E99:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A86</font></span>: 
<font color="Maroon" title="c20">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="c15">n2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E86:56"><span class="lab"><font color="Green" title="E90">A77</font></span></a>, <a class="txt" href="#E87:56"><span class="lab"><font color="Green" title="E91">A78</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E100:56"/>
 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="c20">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="c18">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="56_15"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">k</font>, <font color="Maroon" title="c24">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">YY</font> &amp; <font color="Maroon" title="c24">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">XX</font> implies <font color="Maroon" title="c23">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">l</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:56_15"/><span class="lab"><font color="Green" title="E100">A87</font></span>: 
<font color="Maroon" title="c23">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">YY</font>
 <span class="kw">and </span><br/><a NAME="E2:56_15"/><span class="lab"><font color="Green" title="E101">A88</font></span>: 
<font color="Maroon" title="c24">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">XX</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">l</font></span><br/>

<a NAME="E3:56_15"/><span class="lab"><font color="Green" title="E102">A89</font></span>: 
 not <font color="Maroon" title="c23">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">XX</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_15"><span class="lab"><font color="Green" title="E100">A87</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E4:56_15"/><span class="lab"><font color="Green" title="E103">A90</font></span>: 
 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="c24">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#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">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 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_15"><span class="lab"><font color="Green" title="E101">A88</font></span></a></span>;<br/>
<a NAME="E5:56_15"/>
<font color="Maroon" title="c23">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_15"><span class="lab"><font color="Green" title="E100">A87</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E6:56_15"/><span class="kw">then </span>
<font color="Maroon" title="c23">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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_15"><span class="lab"><font color="Green" title="E102">A89</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:56_15"/>
<font color="Maroon" title="c23">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_15"><span class="lab"><font color="Green" title="E103">A90</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E101:56"/><span class="kw">then </span>
 <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">YY</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c18">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="c20">YY</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.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="c18">XX</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E99:56"><span class="lab"><font color="Green" title="E99">A86</font></span></a>, <a class="txt" href="#E88:56"><span class="lab"><font color="Green" title="E92">A79</font></span></a>, <a class="ref" href="finseq_3.html#T42" title="FINSEQ_3:th.42">FINSEQ_3:42</a></span>;<br/>
<a NAME="E102:56"/><span class="kw">then </span>
 <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#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c18">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="c20">YY</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.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="c18">XX</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T39" title="XBOOLE_1:th.39">XBOOLE_1:39</a></span>;<br/>
<a NAME="E103:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A91</font></span>: 
 <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">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="c20">YY</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.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="c18">XX</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E87:56"><span class="lab"><font color="Green" title="E91">A78</font></span></a>, <a class="txt" href="#E31:56"><span class="lab"><font color="Green" title="E64">A26</font></span></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></span>;<br/>
<a NAME="E104:56"/><span class="kw">then </span>
 <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">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#K3" title="WSIERP_1:func.3">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="c20">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#K3" title="WSIERP_1:func.3">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="c18">XX</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T75" title="RVSUM_1:th.75">RVSUM_1:75</a></span>;<br/>
<a NAME="E105:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A92</font></span>: 
<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="nat_d.html#K7" title="NAT_D:func.7">-'</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#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c17">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#K3" title="WSIERP_1:func.3">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="c20">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#K3" title="WSIERP_1:func.3">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="c18">XX</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E85:56"><span class="lab"><font color="Green" title="E89">A76</font></span></a>, <a class="txt" href="#E97:56"><span class="lab"><font color="Green" title="E97">A84</font></span></a>, <a class="ref" href="rvsum_1.html#T87" title="RVSUM_1:th.87">RVSUM_1:87</a></span>;<br/>
<span class="lab"><font color="Green" title="E102">A93</font></span>:  <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="c20">YY</font></span>)</span> = 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c20">YY</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E86:56"><span class="lab"><font color="Green" title="E90">A77</font></span></a>, <a class="txt" href="#E87:56"><span class="lab"><font color="Green" title="E91">A78</font></span></a>, <a class="txt" href="#E98:56"><span class="lab"><font color="Green" title="E98">A85</font></span></a>, <a class="ref" href="finseq_3.html#T39" title="FINSEQ_3:th.39">FINSEQ_3:39</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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="c18">XX</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:56"><span class="lab"><font color="Green" title="E61">A23</font></span></a>, <a class="txt" href="#E28:56"><span class="lab"><font color="Green" title="E62">A24</font></span></a>, <a class="txt" href="#E31:56"><span class="lab"><font color="Green" title="E64">A26</font></span></a>, <a class="txt" href="#E89:56"><span class="lab"><font color="Green" title="E93">A80</font></span></a>, <a class="ref" href="card_2.html#T44" title="CARD_2:th.44">CARD_2:44</a></span>
;<br/>
<a NAME="E107:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A94</font></span>: 
<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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c18">XX</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E103:56"><span class="lab"><font color="Green" title="E100">A91</font></span></a>, <a class="ref" href="finseq_5.html#T37" title="FINSEQ_5:th.37">FINSEQ_5:37</a></span>;<br/>
<a NAME="E108:56"/>
 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="c14">f2</font> holds <br/><font color="Maroon" title="c14">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="56_17"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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="c14">f2</font> implies <font color="Maroon" title="c14">f2</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>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_17"/>
<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="c14">f2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">f2</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> </span><br/>

<a NAME="E2:56_17"/><span class="kw">then </span>
<font color="Maroon" title="c14">f2</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="c13">f1</font> <a href="recdef_1.html#K1" title="RECDEF_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="c3">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:56"><span class="lab"><font color="Green" title="E65">A27</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:56_17"/>
<font color="Maroon" title="c14">f2</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c23">f2</font> = <font color="Maroon" title="c14">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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T12" title="FINSEQ_2:th.12">FINSEQ_2:12</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c24">f3</font> = <font color="Maroon" title="c13">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="E110:56"/><span class="lab"><font color="Green" title="E104">A95</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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="c13">f1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euler_2.html#D1" title="EULER_2:def.1">EULER_2:def 1</a></span>;<br/>
<a NAME="E111:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A96</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">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="c13">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 class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c25">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#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">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/>
<a NAME="E112:56"/>
<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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E113:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A97</font></span>: 
<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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E114:56"/><span class="kw">then </span>
<font color="Maroon" title="c13">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:56"><span class="lab"><font color="Green" title="E58">A18</font></span></a>, <a class="txt" href="#E8:56"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E110:56"><span class="lab"><font color="Green" title="E104">A95</font></span></a>, <a class="ref" href="nat_2.html#T13" title="NAT_2:th.13">NAT_2:13</a></span>;<br/>
<a NAME="E115:56"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c26">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> <span class="kw"> such that </span><br/><a NAME="E117:56"/><span class="lab"><font color="Green" title="E107">A98</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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="c26">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a></span>;<br/>
<a NAME="E118:56"/><span class="lab"><font color="Green" title="E108">A99</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">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="c23">f2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:56"><span class="lab"><font color="Green" title="E58">A18</font></span></a>, <a class="txt" href="#E34:56"><span class="lab"><font color="Green" title="E65">A27</font></span></a>, <a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/>
<a NAME="E119:56"/><span class="lab"><font color="Green" title="E109">A100</font></span>: 
 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="c13">f1</font> holds <br/><font color="Maroon" title="c13">f1</font> <a href="recdef_1.html#K1" title="RECDEF_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">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">d</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#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">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="56_18"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c27">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c27">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="c13">f1</font> implies <font color="Maroon" title="c13">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">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">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">d</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#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">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">d</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_18"/><span class="lab"><font color="Green" title="E110">A101</font></span>: 
<font color="Maroon" title="c27">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="c13">f1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">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">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">d</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#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">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">d</font></span>)</span></span><br/>

<a NAME="E2:56_18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E111">A102</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">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">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">d</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euler_2.html#D1" title="EULER_2:def.1">EULER_2:def 1</a></span>;<br/>
<a NAME="E3:56_18"/>
<font color="Maroon" title="c23">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">d</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="kw">by</span> <span class="lab"><a class="txt" href="#E34:56"><span class="lab"><font color="Green" title="E65">A27</font></span></a>, <a class="txt" href="#E118:56"><span class="lab"><font color="Green" title="E108">A99</font></span></a>, <a class="txt" href="#E1:56_18"><span class="lab"><font color="Green" title="E110">A101</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:56_18"/>
<font color="Maroon" title="c13">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">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">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">d</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#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">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">d</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_18"><span class="lab"><font color="Green" title="E111">A102</font></span></a>, <a class="ref" href="nat_d.html#T2" title="NAT_D:th.2">NAT_D:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E120:56"/>
 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#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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="56_19"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:56_19"/>
 <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#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c27">a</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:56_19"/><span class="lab"><font color="Green" title="E110">A103</font></span>: 
<font color="Maroon" title="c27">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="c13">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 class="kw">and </span><br/><a NAME="E4:56_19"/><span class="lab"><font color="Green" title="E111">A104</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">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">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></span>;<br/>
<a NAME="E5:56_19"/>
<font color="Maroon" title="c13">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">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">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">a</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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E111:56"><span class="lab"><font color="Green" title="E105">A96</font></span></a>, <a class="txt" href="#E119:56"><span class="lab"><font color="Green" title="E109">A100</font></span></a>, <a class="txt" href="#E3:56_19"><span class="lab"><font color="Green" title="E110">A103</font></span></a>, <a class="txt" href="#E4:56_19"><span class="lab"><font color="Green" title="E111">A104</font></span></a></span>;<br/>
<a NAME="E6:56_19"/><span class="kw">then </span>
<font color="Maroon" title="c4">q</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c27">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">a</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:56"><span class="lab"><font color="Green" title="E55">A13</font></span></a>, <a class="txt" href="#E111:56"><span class="lab"><font color="Green" title="E105">A96</font></span></a>, <a class="txt" href="#E3:56_19"><span class="lab"><font color="Green" title="E110">A103</font></span></a></span>;<br/>
<a NAME="E7:56_19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E112">A105</font></span>: 
<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="c27">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></span>;<br/>
<a NAME="E8:56_19"/>
<font color="Maroon" title="c27">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_19"><span class="lab"><font color="Green" title="E110">A103</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E9:56_19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E113">A106</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="txt" href="#E7:56_19"><span class="lab"><font color="Green" title="E112">A105</font></span></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a>, <a class="ref" href="pepin.html#T3" title="PEPIN:th.3">PEPIN:3</a></span>;<br/>
<a NAME="E10:56_19"/>
<font color="Maroon" title="c27">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:56"><span class="lab"><font color="Green" title="E58">A18</font></span></a>, <a class="txt" href="#E110:56"><span class="lab"><font color="Green" title="E104">A95</font></span></a>, <a class="txt" href="#E3:56_19"><span class="lab"><font color="Green" title="E110">A103</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:56_19"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:56"><span class="lab"><font color="Green" title="E54">A12</font></span></a>, <a class="txt" href="#E9:56_19"><span class="lab"><font color="Green" title="E113">A106</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E121:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E110">A107</font></span>: 
<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#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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 class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T50" title="ZFMISC_1:th.50">ZFMISC_1:50</a></span>;<br/>
<a NAME="E122:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E111">A108</font></span>: 
 <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E117:56"><span class="lab"><font color="Green" title="E107">A98</font></span></a>, <a class="ref" href="finseq_3.html#T92" title="FINSEQ_3:th.92">FINSEQ_3:92</a>, <a class="ref" href="xboole_1.html#T73" title="XBOOLE_1:th.73">XBOOLE_1:73</a></span>;<br/>
<a NAME="E123:56"/><span class="lab"><font color="Green" title="E112">A109</font></span>: 
 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="c13">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="c13">f1</font> &amp; <font color="Maroon" title="c3">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">f1</font> <a href="recdef_1.html#K1" title="RECDEF_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="c13">f1</font> <a href="recdef_1.html#K1" title="RECDEF_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="56_20"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:56_20"/><span class="lab"><font color="Green" title="E113">A110</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="ref" href="int_2.html#T30" title="INT_2:th.30">INT_2:30</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c27">d</font>, <font color="Maroon" title="c28">e</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c27">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="c13">f1</font> &amp; <font color="Maroon" title="c28">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="c13">f1</font> &amp; <font color="Maroon" title="c3">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">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="c13">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c28">e</font></span>)</span> implies <font color="Maroon" title="c27">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">e</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:56_20"/><span class="lab"><font color="Green" title="E114">A111</font></span>: 
<font color="Maroon" title="c27">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="c13">f1</font>
 <span class="kw">and </span><br/><a NAME="E3:56_20"/><span class="lab"><font color="Green" title="E115">A112</font></span>: 
<font color="Maroon" title="c28">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="c13">f1</font>
 <span class="kw">and </span><br/><a NAME="E4:56_20"/><span class="lab"><font color="Green" title="E116">A113</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">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="c13">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c28">e</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c27">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">e</font></span><br/>

<a NAME="E5:56_20"/><span class="lab"><font color="Green" title="E117">A114</font></span>: 
<font color="Maroon" title="c13">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c28">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="c28">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:56"><span class="lab"><font color="Green" title="E55">A13</font></span></a>, <a class="txt" href="#E3:56_20"><span class="lab"><font color="Green" title="E115">A112</font></span></a></span>;<br/>
<a NAME="E6:56_20"/>
<font color="Maroon" title="c13">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">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="c27">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:56"><span class="lab"><font color="Green" title="E55">A13</font></span></a>, <a class="txt" href="#E2:56_20"><span class="lab"><font color="Green" title="E114">A111</font></span></a></span>;<br/>
<a NAME="E7:56_20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E118">A115</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">d</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c28">e</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_20"><span class="lab"><font color="Green" title="E116">A113</font></span></a>, <a class="txt" href="#E5:56_20"><span class="lab"><font color="Green" title="E117">A114</font></span></a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_20_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:56_20_1"/>
<font color="Maroon" title="c27">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c28">e</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:56_20_1"/><span class="kw">then </span>
<font color="Maroon" title="c27">d</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c28">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="E3:56_20_1"/><span class="kw">then </span>
 <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="c27">d</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c28">e</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:56_20"><span class="lab"><font color="Green" title="E118">A115</font></span></a>, <a class="txt" href="#E1:56_20"><span class="lab"><font color="Green" title="E113">A110</font></span></a>, <a class="ref" href="int_2.html#T25" title="INT_2:th.25">INT_2:25</a>, <a class="ref" href="int_4.html#T6" title="INT_4:th.6">INT_4:6</a></span>;<br/><a NAME="E4:56_20_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E119">A116</font></span>: 
<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="c27">d</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c28">e</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><a NAME="E5:56_20_1"/><span class="lab"><font color="Green" title="E120">A117</font></span>: 
<font color="Maroon" title="c28">e</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_20"><span class="lab"><font color="Green" title="E115">A112</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E6:56_20_1"/><span class="lab"><font color="Green" title="E121">A118</font></span>: 
<font color="Maroon" title="c27">d</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_20"><span class="lab"><font color="Green" title="E114">A111</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E7:56_20_1"/>
<font color="Maroon" title="c28">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:56"><span class="lab"><font color="Green" title="E58">A18</font></span></a>, <a class="txt" href="#E3:56_20"><span class="lab"><font color="Green" title="E115">A112</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E8:56_20_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E122">A119</font></span>: 
<font color="Maroon" title="c27">d</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c28">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:56_20_1"><span class="lab"><font color="Green" title="E121">A118</font></span></a>, <a class="ref" href="xreal_1.html#T13" title="XREAL_1:th.13">XREAL_1:13</a></span>;<br/><a NAME="E9:56_20_1"/><span class="lab"><font color="Green" title="E123">A120</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:56"><span class="lab"><font color="Green" title="E54">A12</font></span></a>, <a class="ref" href="xreal_1.html#T147" title="XREAL_1:th.147">XREAL_1:147</a></span>;<br/><a NAME="E10:56_20_1"/>
<font color="Maroon" title="c27">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:56"><span class="lab"><font color="Green" title="E58">A18</font></span></a>, <a class="txt" href="#E2:56_20"><span class="lab"><font color="Green" title="E114">A111</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E11:56_20_1"/><span class="kw">then </span>
<font color="Maroon" title="c27">d</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c28">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="nat_d.html#K7" title="NAT_D:func.7">-'</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E5:56_20_1"><span class="lab"><font color="Green" title="E120">A117</font></span></a>, <a class="ref" href="xreal_1.html#T13" title="XREAL_1:th.13">XREAL_1:13</a></span>;<br/><a NAME="E12:56_20_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E124">A121</font></span>: 
<font color="Maroon" title="c27">d</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c28">e</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:56_20_1"><span class="lab"><font color="Green" title="E123">A120</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E13:56_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="c3">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:56_20_1"><span class="lab"><font color="Green" title="E123">A120</font></span></a>, <a class="ref" href="xreal_1.html#T24" title="XREAL_1:th.24">XREAL_1:24</a></span>;<br/><a NAME="E14:56_20_1"/><span class="kw">then </span>
<font color="Maroon" title="c27">d</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c28">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:56_20_1"><span class="lab"><font color="Green" title="E122">A119</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E15:56_20_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_20_1"><span class="lab"><font color="Green" title="E119">A116</font></span></a>, <a class="txt" href="#E12:56_20_1"><span class="lab"><font color="Green" title="E124">A121</font></span></a>, <a class="ref" href="seq_2.html#T1" title="SEQ_2:th.1">SEQ_2:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E9:56_20"/>
<font color="Maroon" title="c27">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">e</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E124:56"/>
 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="c13">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="c13">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="c13">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="c13">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="56_21"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c27">x</font>, <font color="Maroon" title="c28">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c27">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="c13">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="c28">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="c13">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="c13">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">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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="c28">y</font> implies <font color="Maroon" title="c27">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">y</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:56_21"/><span class="lab"><font color="Green" title="E113">A122</font></span>: 
<font color="Maroon" title="c27">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="c13">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 class="kw">and </span><br/><a NAME="E2:56_21"/><span class="lab"><font color="Green" title="E114">A123</font></span>: 
<font color="Maroon" title="c28">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="c13">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 class="kw">and </span><br/><a NAME="E3:56_21"/><span class="lab"><font color="Green" title="E115">A124</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">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">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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="c28">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c27">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">y</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c29">x</font> = <font color="Maroon" title="c27">x</font>, <font color="Maroon" title="c30">y</font> = <font color="Maroon" title="c28">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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_21"><span class="lab"><font color="Green" title="E113">A122</font></span></a>, <a class="txt" href="#E2:56_21"><span class="lab"><font color="Green" title="E114">A123</font></span></a></span>;<br/>
<a NAME="E5:56_21"/><span class="lab"><font color="Green" title="E116">A125</font></span>: 
<font color="Maroon" title="c13">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c30">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">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c30">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="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="c13">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="c30">y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E111:56"><span class="lab"><font color="Green" title="E105">A96</font></span></a>, <a class="txt" href="#E119:56"><span class="lab"><font color="Green" title="E109">A100</font></span></a>, <a class="txt" href="#E2:56_21"><span class="lab"><font color="Green" title="E114">A123</font></span></a></span>;<br/>
<a NAME="E6:56_21"/>
<font color="Maroon" title="c13">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">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">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">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#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">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="c29">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E111:56"><span class="lab"><font color="Green" title="E105">A96</font></span></a>, <a class="txt" href="#E119:56"><span class="lab"><font color="Green" title="E109">A100</font></span></a>, <a class="txt" href="#E1:56_21"><span class="lab"><font color="Green" title="E113">A122</font></span></a></span>;<br/>
<a NAME="E7:56_21"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">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="c13">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c30">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">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">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">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c30">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_21"><span class="lab"><font color="Green" title="E115">A124</font></span></a>, <a class="txt" href="#E5:56_21"><span class="lab"><font color="Green" title="E116">A125</font></span></a></span>;<br/>
<a NAME="E8:56_21"/><span class="kw">then </span>
<font color="Maroon" title="c3">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">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="c13">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c30">y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D3" title="INT_1:def.3">INT_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:56_21"/>
<font color="Maroon" title="c27">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E123:56"><span class="lab"><font color="Green" title="E112">A109</font></span></a>, <a class="txt" href="#E111:56"><span class="lab"><font color="Green" title="E105">A96</font></span></a>, <a class="txt" href="#E1:56_21"><span class="lab"><font color="Green" title="E113">A122</font></span></a>, <a class="txt" href="#E2:56_21"><span class="lab"><font color="Green" title="E114">A123</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E125:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E113">A126</font></span>: 
<font color="Maroon" title="c13">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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>;<br/>
<a NAME="E126:56"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">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="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T62" title="FINSEQ_4:th.62">FINSEQ_4:62</a></span>;<br/>
<a NAME="E127:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E114">A127</font></span>: 
 <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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:56"><span class="lab"><font color="Green" title="E58">A18</font></span></a>, <a class="txt" href="#E110:56"><span class="lab"><font color="Green" title="E104">A95</font></span></a>, <a class="txt" href="#E117:56"><span class="lab"><font color="Green" title="E107">A98</font></span></a>, <a class="txt" href="#E121:56"><span class="lab"><font color="Green" title="E110">A107</font></span></a>, <a class="ref" href="finseq_3.html#T39" title="FINSEQ_3:th.39">FINSEQ_3:39</a>, <a class="ref" href="xboole_1.html#T73" title="XBOOLE_1:th.73">XBOOLE_1:73</a></span>;<br/>
<a NAME="E128:56"/><span class="lab"><font color="Green" title="E115">A128</font></span>: 
<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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c20">YY</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E103:56"><span class="lab"><font color="Green" title="E100">A91</font></span></a>, <a class="txt" href="#E106:56"><span class="lab"><font color="Green" title="E102">A93</font></span></a>, <a class="ref" href="finseq_3.html#T113" title="FINSEQ_3:th.113">FINSEQ_3:113</a>, <a class="ref" href="finseq_6.html#T10" title="FINSEQ_6:th.10">FINSEQ_6:10</a></span>;<br/>
<a NAME="E129:56"/><span class="lab"><font color="Green" title="E116">A129</font></span>: 
<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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E61:56"><span class="lab"><font color="Green" title="E79">A48</font></span></a>, <a class="txt" href="#E93:56"><span class="lab"><font color="Green" title="E94">A81</font></span></a>, <a class="ref" href="finseq_3.html#T91" title="FINSEQ_3:th.91">FINSEQ_3:91</a></span>;<br/>
<a NAME="E130:56"/><span class="lab"><font color="Green" title="E117">A130</font></span>: 
 <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#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c17">g2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_22"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c27">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="c18">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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c28">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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">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="c22">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="c18">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span></span>)</span>;<br/>
<a NAME="E1:56_22"/><span class="lab"><font color="Green" title="E118">A131</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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="nat_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T45" title="RELAT_1:th.45">RELAT_1:45</a></span>;<br/>
<a NAME="E2:56_22"/><span class="lab"><font color="Green" title="E119">A132</font></span>: 
<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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">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="c22">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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T24" title="FINSEQ_2:th.24">FINSEQ_2:24</a></span>;<br/>
<a NAME="E3:56_22"/><span class="lab"><font color="Green" title="E120">A133</font></span>: 
 <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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">nn</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E66:56"><span class="lab"><font color="Green" title="E82">A67</font></span></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a>, <a class="ref" href="xreal_1.html#T43" title="XREAL_1:th.43">XREAL_1:43</a></span>;<br/>
<span class="lab"><font color="Green" title="E121">A134</font></span>:  <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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">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="c22">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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c22">nn</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfinseq.html#T29" title="RFINSEQ:th.29">RFINSEQ:29</a></span>
<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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E66:56"><span class="lab"><font color="Green" title="E82">A67</font></span></a>, <a class="ref" href="xreal_1.html#T43" title="XREAL_1:th.43">XREAL_1:43</a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>
<br/>.= 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c18">XX</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E66:56"><span class="lab"><font color="Green" title="E82">A67</font></span></a></span>
;<br/>
<span class="lab"><font color="Green" title="E122">A135</font></span>:  <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="c18">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">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="c18">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T12" title="VALUED_1:th.12">VALUED_1:12</a></span>
<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="c18">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<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#K2" title="RELSET_1:func.2">rng</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">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="c22">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_22"><span class="lab"><font color="Green" title="E121">A134</font></span></a>, <a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>
<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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span>

;<br/>
<a NAME="E6:56_22"/><span class="kw">then </span><span class="lab"><font color="Green" title="E123">A136</font></span>: 
 <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="c18">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/>
<a NAME="E7:56_22"/><span class="lab"><font color="Green" title="E124">A137</font></span>: 
 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="c18">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">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="c18">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">d</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_22_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c29">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c29">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="c18">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span></span>)</span> implies <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="c18">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">d</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_22_3"/><span class="lab"><font color="Green" title="E125">A138</font></span>: 
<font color="Maroon" title="c29">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="c18">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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="c18">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">d</font></span>)</span></span><br/>

<a NAME="E2:56_22_3"/><span class="kw">then </span>
<font color="Maroon" title="c29">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="c18">XX</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_22"><span class="lab"><font color="Green" title="E121">A134</font></span></a>, <a class="txt" href="#E5:56_22"><span class="lab"><font color="Green" title="E122">A135</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E3:56_22_3"/><span class="kw">then </span>
<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="c18">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="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T57" title="FINSEQ_2:th.57">FINSEQ_2:57</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:56_22_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="c18">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">d</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_22_3"><span class="lab"><font color="Green" title="E125">A138</font></span></a>, <a class="ref" href="valued_1.html#T13" title="VALUED_1:th.13">VALUED_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:56_22"/><span class="lab"><font color="Green" title="E125">A139</font></span>: 
 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="c18">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">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="c18">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">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="c18">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">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="nat_d.html#K7" title="NAT_D:func.7">-'</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="56_22_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c29">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c29">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="c18">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span></span>)</span> implies ( <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="c18">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">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="c18">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c30">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="c18">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:56_22_4"/><span class="lab"><font color="Green" title="E126">A140</font></span>: 
<font color="Maroon" title="c29">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="c18">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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="c18">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">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="c18">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 )</span><br/>

<a NAME="E3:56_22_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c18">XX</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">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="c18">XX</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E107:56"><span class="lab"><font color="Green" title="E103">A94</font></span></a>, <a class="txt" href="#E5:56_22"><span class="lab"><font color="Green" title="E122">A135</font></span></a>, <a class="ref" href="funct_1.html#T3" title="FUNCT_1:th.3">FUNCT_1:3</a></span>;<br/>
<a NAME="E4:56_22_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c18">XX</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">XX</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E88:56"><span class="lab"><font color="Green" title="E92">A79</font></span></a>, <a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></span>;<br/>
<a NAME="E5:56_22_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E127">A141</font></span>: 
 ex <font color="Olive" title="b1">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>  st <br/>( <font color="Olive" title="b1">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="c18">XX</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">d</font> &amp; <font color="Olive" title="b1">ll</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">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="b1">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 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E87:56"><span class="lab"><font color="Green" title="E91">A78</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c31">e</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E7:56_22_4"/><span class="lab"><font color="Green" title="E128">A142</font></span>: 
<font color="Maroon" title="c31">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="c9">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 class="kw">and </span><br/><a NAME="E8:56_22_4"/><span class="lab"><font color="Green" title="E129">A143</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">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="c31">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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E107:56"><span class="lab"><font color="Green" title="E103">A94</font></span></a>, <a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></span>;<br/>
<a NAME="E9:56_22_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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c31">e</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:56"><span class="lab"><font color="Green" title="E63">A25</font></span></a>, <a class="txt" href="#E7:56_22_4"><span class="lab"><font color="Green" title="E128">A142</font></span></a>, <a class="txt" href="#E8:56_22_4"><span class="lab"><font color="Green" title="E129">A143</font></span></a>, <a class="ref" href="euler_2.html#D1" title="EULER_2:def.1">EULER_2:def 1</a></span>;<br/>
<a NAME="E10:56_22_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E130">A144</font></span>: 
<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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T1" title="NAT_D:th.1">NAT_D:1</a></span>;<br/>
<a NAME="E11:56_22_4"/><span class="lab"><font color="Green" title="E131">A145</font></span>: 
<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="c18">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">d</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:56_22"><span class="lab"><font color="Green" title="E124">A137</font></span></a>, <a class="txt" href="#E2:56_22_4"><span class="lab"><font color="Green" title="E126">A140</font></span></a></span>;<br/>
<a NAME="E12:56_22_4"/><span class="kw">then </span>
<font color="Maroon" title="c30">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E107:56"><span class="lab"><font color="Green" title="E103">A94</font></span></a>, <a class="txt" href="#E5:56_22_4"><span class="lab"><font color="Green" title="E127">A141</font></span></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:56_22_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="c18">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">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="c18">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E95:56"><span class="lab"><font color="Green" title="E96">A83</font></span></a>, <a class="txt" href="#E11:56_22_4"><span class="lab"><font color="Green" title="E131">A145</font></span></a>, <a class="txt" href="#E10:56_22_4"><span class="lab"><font color="Green" title="E130">A144</font></span></a>, <a class="ref" href="int_1.html#T54" title="INT_1:th.54">INT_1:54</a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E9:56_22"/>
 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="c18">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">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="c18">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">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="56_22_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c29">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c29">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="c18">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span></span>)</span> implies <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="c18">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_22_5"/>
<font color="Maroon" title="c29">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="c18">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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="c18">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><br/>

<a NAME="E2:56_22_5"/><span class="kw">then </span>
<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="c18">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:56_22"><span class="lab"><font color="Green" title="E125">A139</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:56_22_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="c18">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T3" title="INT_1:th.3">INT_1:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c29">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="c18">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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">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="c22">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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T12" title="FINSEQ_2:th.12">FINSEQ_2:12</a></span>;<br/>
<a NAME="E11:56_22"/>
<font color="Maroon" title="c29">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/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c30">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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">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="c22">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="c18">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T75" title="FINSEQ_1:th.75">FINSEQ_1:75</a></span>;<br/>
<a NAME="E13:56_22"/><span class="lab"><font color="Green" title="E126">A146</font></span>: 
<font color="Maroon" title="c30">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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T24" title="FINSEQ_2:th.24">FINSEQ_2:24</a></span>;<br/>
<a NAME="E14:56_22"/><span class="lab"><font color="Green" title="E127">A147</font></span>: 
<font color="Maroon" title="c22">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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E66:56"><span class="lab"><font color="Green" title="E82">A67</font></span></a>, <a class="ref" href="xreal_1.html#T43" title="XREAL_1:th.43">XREAL_1:43</a></span>;<br/>
<a NAME="E15:56_22"/><span class="lab"><font color="Green" title="E128">A148</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c29">g5</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_22_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:56_22_6"/>
 not  <a href="relset_1.html#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c29">g5</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c31">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:56_22_6"/><span class="lab"><font color="Green" title="E129">A149</font></span>: 
<font color="Maroon" title="c31">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:56_22_6"/><span class="lab"><font color="Green" title="E130">A150</font></span>: 
<font color="Maroon" title="c31">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c29">g5</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c32">e</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E6:56_22_6"/><span class="lab"><font color="Green" title="E131">A151</font></span>: 
<font color="Maroon" title="c32">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="c29">g5</font>
 <span class="kw">and </span><br/><a NAME="E7:56_22_6"/><span class="lab"><font color="Green" title="E132">A152</font></span>: 
<font color="Maroon" title="c29">g5</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c32">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_22_6"><span class="lab"><font color="Green" title="E130">A150</font></span></a>, <a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></span>;<br/>
<a NAME="E8:56_22_6"/>
<font color="Maroon" title="c31">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c32">e</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:56_22"><span class="lab"><font color="Green" title="E124">A137</font></span></a>, <a class="txt" href="#E6:56_22_6"><span class="lab"><font color="Green" title="E131">A151</font></span></a>, <a class="txt" href="#E7:56_22_6"><span class="lab"><font color="Green" title="E132">A152</font></span></a></span>;<br/>
<a NAME="E9:56_22_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E133">A153</font></span>: 
<font color="Maroon" title="c31">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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">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="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c32">e</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c22">nn</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:56_22"><span class="lab"><font color="Green" title="E127">A147</font></span></a>, <a class="txt" href="#E5:56_22"><span class="lab"><font color="Green" title="E122">A135</font></span></a>, <a class="txt" href="#E6:56_22_6"><span class="lab"><font color="Green" title="E131">A151</font></span></a>, <a class="ref" href="rfinseq.html#D1" title="RFINSEQ:def.1">RFINSEQ:def 1</a></span>;<br/>
<a NAME="E10:56_22_6"/>
<font color="Maroon" title="c32">e</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c22">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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E5:56_22"><span class="lab"><font color="Green" title="E122">A135</font></span></a>, <a class="txt" href="#E6:56_22_6"><span class="lab"><font color="Green" title="E131">A151</font></span></a>, <a class="ref" href="finseq_5.html#T26" title="FINSEQ_5:th.26">FINSEQ_5:26</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c33">e1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E12:56_22_6"/><span class="lab"><font color="Green" title="E134">A154</font></span>: 
<font color="Maroon" title="c33">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="c9">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 class="kw">and </span><br/><a NAME="E13:56_22_6"/><span class="lab"><font color="Green" title="E135">A155</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">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="c33">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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">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="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">e</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c22">nn</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E87:56"><span class="lab"><font color="Green" title="E91">A78</font></span></a>, <a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a>, <a class="ref" href="funct_1.html#T3" title="FUNCT_1:th.3">FUNCT_1:3</a></span>;<br/>
<a NAME="E14:56_22_6"/><span class="lab"><font color="Green" title="E136">A156</font></span>: 
<font color="Maroon" title="c33">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:56"><span class="lab"><font color="Green" title="E61">A23</font></span></a>, <a class="txt" href="#E28:56"><span class="lab"><font color="Green" title="E62">A24</font></span></a>, <a class="txt" href="#E12:56_22_6"><span class="lab"><font color="Green" title="E134">A154</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E15:56_22_6"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">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 class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T19" title="FINSEQ_5:th.19">FINSEQ_5:19</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c34">d1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E17:56_22_6"/><span class="lab"><font color="Green" title="E137">A157</font></span>: 
<font color="Maroon" title="c34">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="c9">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 class="kw">and </span><br/><a NAME="E18:56_22_6"/><span class="lab"><font color="Green" title="E138">A158</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">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="c34">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E87:56"><span class="lab"><font color="Green" title="E91">A78</font></span></a>, <a class="txt" href="#E3:56_22_6"><span class="lab"><font color="Green" title="E129">A149</font></span></a>, <a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></span>;<br/>
<a NAME="E19:56_22_6"/>
<font color="Maroon" title="c34">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:56"><span class="lab"><font color="Green" title="E61">A23</font></span></a>, <a class="txt" href="#E28:56"><span class="lab"><font color="Green" title="E62">A24</font></span></a>, <a class="txt" href="#E17:56_22_6"><span class="lab"><font color="Green" title="E137">A157</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E20:56_22_6"/><span class="kw">then </span>
<font color="Maroon" title="c34">d1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c33">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="nat_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:56_22_6"><span class="lab"><font color="Green" title="E136">A156</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/>
<a NAME="E21:56_22_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E139">A159</font></span>: 
<font color="Maroon" title="c34">d1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c33">e1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:56"><span class="lab"><font color="Green" title="E67">A29</font></span></a>, <a class="txt" href="#E55:56"><span class="lab"><font color="Green" title="E76">A40</font></span></a>, <a class="ref" href="xreal_1.html#T146" title="XREAL_1:th.146">XREAL_1:146</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E22:56_22_6"/><span class="lab"><font color="Green" title="E140">A160</font></span>: 
<font color="Maroon" title="c33">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="c9">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:56"><span class="lab"><font color="Green" title="E62">A24</font></span></a>, <a class="txt" href="#E12:56_22_6"><span class="lab"><font color="Green" title="E134">A154</font></span></a>, <a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/>
<a NAME="E23:56_22_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E141">A161</font></span>: 
<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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">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="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">e</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c22">nn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c33">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E13:56_22_6"><span class="lab"><font color="Green" title="E135">A155</font></span></a>, <a class="ref" href="euler_2.html#D1" title="EULER_2:def.1">EULER_2:def 1</a></span>;<br/>
<a NAME="E24:56_22_6"/><span class="lab"><font color="Green" title="E142">A162</font></span>: 
<font color="Maroon" title="c34">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="c9">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:56"><span class="lab"><font color="Green" title="E62">A24</font></span></a>, <a class="txt" href="#E17:56_22_6"><span class="lab"><font color="Green" title="E137">A157</font></span></a>, <a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/>
<a NAME="E25:56_22_6"/><span class="kw">then </span>
<font color="Maroon" title="c31">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c34">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E18:56_22_6"><span class="lab"><font color="Green" title="E138">A158</font></span></a>, <a class="ref" href="euler_2.html#D1" title="EULER_2:def.1">EULER_2:def 1</a></span>;<br/>
<a NAME="E26:56_22_6"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c34">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="c9">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c33">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:56_22_6"><span class="lab"><font color="Green" title="E133">A153</font></span></a>, <a class="txt" href="#E23:56_22_6"><span class="lab"><font color="Green" title="E141">A161</font></span></a>, <a class="ref" href="nat_d.html#T25" title="NAT_D:th.25">NAT_D:25</a></span>;<br/>
<a NAME="E27:56_22_6"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c34">d1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c33">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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euler_2.html#T6" title="EULER_2:th.6">EULER_2:6</a></span>;<br/>
<a NAME="E28:56_22_6"/><span class="kw">then </span>
<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="c9">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c34">d1</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="c9">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c33">e1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pepin.html#T6" title="PEPIN:th.6">PEPIN:6</a></span>;<br/>
<a NAME="E29:56_22_6"/><span class="kw">then </span>
<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="c34">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="c9">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c33">e1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:56"><span class="lab"><font color="Green" title="E59">A19</font></span></a>, <a class="txt" href="#E24:56_22_6"><span class="lab"><font color="Green" title="E142">A162</font></span></a></span>;<br/>
<a NAME="E30:56_22_6"/><span class="kw">then </span>
<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="c34">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="c33">e1</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c3">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:56"><span class="lab"><font color="Green" title="E59">A19</font></span></a>, <a class="txt" href="#E22:56_22_6"><span class="lab"><font color="Green" title="E140">A160</font></span></a></span>;<br/>
<a NAME="E31:56_22_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E143">A163</font></span>: 
<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="c34">d1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c33">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="E32:56_22_6"/>
<font color="Maroon" title="c34">d1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:56_22_6"><span class="lab"><font color="Green" title="E137">A157</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E33:56_22_6"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="txt" href="#E31:56_22_6"><span class="lab"><font color="Green" title="E143">A163</font></span></a>, <a class="txt" href="#E21:56_22_6"><span class="lab"><font color="Green" title="E139">A159</font></span></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a>, <a class="ref" href="pepin.html#T3" title="PEPIN:th.3">PEPIN:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E16:56_22"/>
 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="c29">g5</font> holds <br/><font color="Maroon" title="c29">g5</font> <a href="recdef_1.html#K1" title="RECDEF_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="c29">g5</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">e</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_22_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c31">d</font>, <font color="Maroon" title="c32">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">d</font> &amp; <font color="Maroon" title="c31">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c32">e</font> &amp; <font color="Maroon" title="c32">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="c29">g5</font> implies <font color="Maroon" title="c29">g5</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c31">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c29">g5</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c32">e</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:56_22_7"/><span class="lab"><font color="Green" title="E129">A164</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">d</font>
 <span class="kw">and </span><br/><a NAME="E2:56_22_7"/><span class="lab"><font color="Green" title="E130">A165</font></span>: 
<font color="Maroon" title="c31">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c32">e</font>
 <span class="kw">and </span><br/><a NAME="E3:56_22_7"/><span class="lab"><font color="Green" title="E131">A166</font></span>: 
<font color="Maroon" title="c32">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="c29">g5</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c29">g5</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c31">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c29">g5</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c32">e</font></span><br/>

<a NAME="E4:56_22_7"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c32">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_22_7"><span class="lab"><font color="Green" title="E129">A164</font></span></a>, <a class="txt" href="#E2:56_22_7"><span class="lab"><font color="Green" title="E130">A165</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E5:56_22_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E132">A167</font></span>: 
<font color="Maroon" title="c32">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="c29">g5</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_22_7"><span class="lab"><font color="Green" title="E131">A166</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E6:56_22_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E133">A168</font></span>: 
<font color="Maroon" title="c29">g5</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c32">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c32">e</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:56_22"><span class="lab"><font color="Green" title="E124">A137</font></span></a></span>;<br/>
<a NAME="E7:56_22_7"/>
<font color="Maroon" title="c31">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="c29">g5</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_22_7"><span class="lab"><font color="Green" title="E130">A165</font></span></a>, <a class="txt" href="#E3:56_22_7"><span class="lab"><font color="Green" title="E131">A166</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E8:56_22_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E134">A169</font></span>: 
<font color="Maroon" title="c31">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="c29">g5</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_22_7"><span class="lab"><font color="Green" title="E129">A164</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E9:56_22_7"/><span class="kw">then </span>
<font color="Maroon" title="c29">g5</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c31">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c31">d</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:56_22"><span class="lab"><font color="Green" title="E124">A137</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:56_22_7"/>
<font color="Maroon" title="c29">g5</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c31">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c29">g5</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c32">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E94:56"><span class="lab"><font color="Green" title="E95">A82</font></span></a>, <a class="txt" href="#E5:56_22"><span class="lab"><font color="Green" title="E122">A135</font></span></a>, <a class="txt" href="#E2:56_22_7"><span class="lab"><font color="Green" title="E130">A165</font></span></a>, <a class="txt" href="#E8:56_22_7"><span class="lab"><font color="Green" title="E134">A169</font></span></a>, <a class="txt" href="#E5:56_22_7"><span class="lab"><font color="Green" title="E132">A167</font></span></a>, <a class="txt" href="#E6:56_22_7"><span class="lab"><font color="Green" title="E133">A168</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E17:56_22"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c29">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#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c29">g5</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="graph_5.html#T7" title="GRAPH_5:th.7">GRAPH_5:7</a></span>;<br/>
<a NAME="E18:56_22"/><span class="kw">then </span>
<font color="Maroon" title="c29">g5</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T62" title="FINSEQ_4:th.62">FINSEQ_4:62</a></span>;<br/>
<a NAME="E19:56_22"/><span class="kw">then </span><span class="lab"><font color="Green" title="E129">A170</font></span>: 
<font color="Maroon" title="c30">g6</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E129:56"><span class="lab"><font color="Green" title="E116">A129</font></span></a>, <a class="txt" href="#E15:56_22"><span class="lab"><font color="Green" title="E128">A148</font></span></a>, <a class="ref" href="finseq_3.html#T91" title="FINSEQ_3:th.91">FINSEQ_3:91</a></span>;<br/>
<a NAME="E20:56_22"/><span class="lab"><font color="Green" title="E130">A171</font></span>: 
 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="c30">g6</font> holds <br/>( <font color="Maroon" title="c30">g6</font> <a href="recdef_1.html#K1" title="RECDEF_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="c30">g6</font> <a href="recdef_1.html#K1" title="RECDEF_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="nat_d.html#K7" title="NAT_D:func.7">-'</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="56_22_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c31">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c31">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="c30">g6</font> implies ( <font color="Maroon" title="c30">g6</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c31">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="c30">g6</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c31">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_22_8"/><span class="lab"><font color="Green" title="E131">A172</font></span>: 
<font color="Maroon" title="c31">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="c30">g6</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c30">g6</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c31">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="c30">g6</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c31">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_22_8_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:56_22_8_1"/>
( <font color="Maroon" title="c31">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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">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="c22">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="c29">g5</font> &amp; <font color="Maroon" title="c31">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_22_8"><span class="lab"><font color="Green" title="E131">A172</font></span></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_22_8_1_1"><span class="kw">suppose </span></a><a NAME="E1:56_22_8_1_1"/><span class="lab"><font color="Green" title="E132">A173</font></span>: 
<font color="Maroon" title="c31">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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c30">g6</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c31">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="c30">g6</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c31">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 )</span><br/><div class="add"><a NAME="E2:56_22_8_1_1"/><span class="kw">then </span>
<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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c31">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">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="c20">YY</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E128:56"><span class="lab"><font color="Green" title="E115">A128</font></span></a>, <a class="ref" href="funct_1.html#T3" title="FUNCT_1:th.3">FUNCT_1:3</a></span>;<br/><a NAME="E3:56_22_8_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E133">A174</font></span>: 
<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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c31">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">YY</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E99:56"><span class="lab"><font color="Green" title="E99">A86</font></span></a>, <a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></span>;<br/><a NAME="E4:56_22_8_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E134">A175</font></span>: 
<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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c31">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">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 class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E5:56_22_8_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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c31">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">XX</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_22_8_1_1"><span class="lab"><font color="Green" title="E133">A174</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E6:56_22_8_1_1"/><span class="kw">then </span>
<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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c31">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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_22_8_1_1"><span class="lab"><font color="Green" title="E134">A175</font></span></a></span>;<br/><a NAME="E7:56_22_8_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E135">A176</font></span>: 
<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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c31">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E95:56"><span class="lab"><font color="Green" title="E96">A83</font></span></a>, <a class="ref" href="int_1.html#T54" title="INT_1:th.54">INT_1:54</a></span>;<br/><a NAME="E8:56_22_8_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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c31">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E60:56"><span class="lab"><font color="Green" title="E78">A47</font></span></a>, <a class="txt" href="#E87:56"><span class="lab"><font color="Green" title="E91">A78</font></span></a>, <a class="txt" href="#E4:56_22_8_1_1"><span class="lab"><font color="Green" title="E134">A175</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><a NAME="E9:56_22_8_1_1"/><span class="kw">then </span>
<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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c31">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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E10:56_22_8_1_1"/>
( <font color="Maroon" title="c30">g6</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c31">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="c30">g6</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c31">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_22_8_1_1"><span class="lab"><font color="Green" title="E132">A173</font></span></a>, <a class="txt" href="#E7:56_22_8_1_1"><span class="lab"><font color="Green" title="E135">A176</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_22_8_1_2"><span class="kw">suppose </span></a><a NAME="E1:56_22_8_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="c29">g5</font> &amp; <font color="Maroon" title="c31">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c30">g6</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c31">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="c30">g6</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c31">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 )</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c32">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> <span class="kw"> such that </span><br/><a NAME="E3:56_22_8_1_2"/><span class="lab"><font color="Green" title="E132">A177</font></span>: 
<font color="Maroon" title="c32">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="c29">g5</font>
 <span class="kw">and </span><br/><a NAME="E4:56_22_8_1_2"/><span class="lab"><font color="Green" title="E133">A178</font></span>: 
<font color="Maroon" title="c31">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c32">l</font>
 ;<br/><a NAME="E5:56_22_8_1_2"/>
<font color="Maroon" title="c30">g6</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c31">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">g5</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c32">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_22_8_1_2"><span class="lab"><font color="Green" title="E132">A177</font></span></a>, <a class="txt" href="#E4:56_22_8_1_2"><span class="lab"><font color="Green" title="E133">A178</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E6:56_22_8_1_2"/>
( <font color="Maroon" title="c30">g6</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c31">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="c30">g6</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c31">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:56_22"><span class="lab"><font color="Green" title="E125">A139</font></span></a>, <a class="txt" href="#E3:56_22_8_1_2"><span class="lab"><font color="Green" title="E132">A177</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c30">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">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="c29">g5</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" title="FINSEQ_1:th.22">FINSEQ_1:22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_22"><span class="lab"><font color="Green" title="E120">A133</font></span></a>, <a class="txt" href="#E4:56_22"><span class="lab"><font color="Green" title="E121">A134</font></span></a>, <a class="txt" href="#E6:56_22"><span class="lab"><font color="Green" title="E123">A136</font></span></a></span>
;<br/>
<a NAME="E22:56_22"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c30">g6</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_22"><span class="lab"><font color="Green" title="E118">A131</font></span></a>, <a class="txt" href="#E19:56_22"><span class="lab"><font color="Green" title="E129">A170</font></span></a>, <a class="txt" href="#E20:56_22"><span class="lab"><font color="Green" title="E130">A171</font></span></a>, <a class="ref" href="int_5.html#T40" target="_self" title="INT_5:th.40">Th40</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c21">N</font> = 
 <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c30">g6</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:56_22"><span class="lab"><font color="Green" title="E129">A170</font></span></a>, <a class="txt" href="#E13:56_22"><span class="lab"><font color="Green" title="E126">A146</font></span></a>, <a class="ref" href="rfinseq.html#T9" title="RFINSEQ:th.9">RFINSEQ:9</a>, <a class="ref" href="rfinseq.html#T26" title="RFINSEQ:th.26">RFINSEQ:26</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">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#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c29">g5</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T75" title="RVSUM_1:th.75">RVSUM_1:75</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">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="c18">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#K3" title="WSIERP_1:func.3">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#K2" title="RELSET_1:func.2">rng</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_22"><span class="lab"><font color="Green" title="E121">A134</font></span></a>, <a class="txt" href="#E2:56_22"><span class="lab"><font color="Green" title="E119">A132</font></span></a>, <a class="ref" href="int_5.html#T47" target="_self" title="INT_5:th.47">Th47</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">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#K2" title="RELSET_1:func.2">rng</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">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="c22">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="c18">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#K3" title="WSIERP_1:func.3">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">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="c22">nn</font></span>)</span></span>)</span>

;<br/>
<a NAME="E24:56_22"/><span class="kw">then </span>
<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="c21">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#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c17">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#K3" title="WSIERP_1:func.3">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="c18">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="c18">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E105:56"><span class="lab"><font color="Green" title="E101">A92</font></span></a>, <a class="txt" href="#E107:56"><span class="lab"><font color="Green" title="E103">A94</font></span></a>, <a class="txt" href="#E128:56"><span class="lab"><font color="Green" title="E115">A128</font></span></a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E131">A179</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c21">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#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c17">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="c18">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#K3" title="WSIERP_1:func.3">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="c18">XX</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:56"><span class="lab"><font color="Green" title="E48">A6</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>
<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#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c17">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="c18">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
<span class="kw">by</span> <span class="lab"><a class="ref" href="euler_1.html#T12" title="EULER_1:th.12">EULER_1:12</a></span>
;<br/>
<a NAME="E26:56_22"/>
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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c21">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:56"><span class="lab"><font color="Green" title="E52">A10</font></span></a>, <a class="ref" href="nat_d.html#T9" title="NAT_D:th.9">NAT_D:9</a></span>;<br/>
<a NAME="E27:56_22"/><span class="kw">then </span>
<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#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c17">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="c18">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:56_22"><span class="lab"><font color="Green" title="E131">A179</font></span></a>, <a class="ref" href="pepin.html#T6" title="PEPIN:th.6">PEPIN:6</a></span>;<br/>
<a NAME="E28:56_22"/><span class="kw">then </span>
2 <a href="int_1.html#R1" title="INT_1:pred.1">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#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c17">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="c18">XX</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="int_5.html#E3"><span class="lab"><font color="Green" title="E3">Lm1</font></span></a></span>;<br/>
<a NAME="E29:56_22"/><span class="kw">then </span>
2 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c17">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="c18">XX</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E76:56"><span class="lab"><font color="Green" title="E85">A70</font></span></a>, <a class="ref" href="int_2.html#T25" title="INT_2:th.25">INT_2:25</a></span>;<br/>
<a NAME="E30:56_22"/><span class="kw">then </span>
 <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c17">g2</font>, <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c18">XX</font> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T15" title="INT_2:th.15">INT_2:15</a></span>;<br/>
<a NAME="E31:56_22"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c17">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="c18">XX</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T64" title="NAT_D:th.64">NAT_D:64</a></span>;<br/>
<a NAME="E32:56_22"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c17">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="c18">XX</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_5.html#T45" target="_self" title="INT_5:th.45">Th45</a></span>;<br/>
<span class="kw">hence </span><a NAME="E33:56_22"/>
 <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#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c17">g2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="txt" href="#E5:56"><span class="lab"><font color="Green" title="E47">A5</font></span></a>, <a class="txt" href="#E87:56"><span class="lab"><font color="Green" title="E91">A78</font></span></a>, <a class="ref" href="int_5.html#T41" target="_self" title="INT_5:th.41">Th41</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E131:56"/>
 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="nat_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</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="56_23"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c27">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c27">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span> implies <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="nat_d.html#K7" title="NAT_D:func.7">-'</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="c27">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_23"/><span class="lab"><font color="Green" title="E118">A180</font></span>: 
<font color="Maroon" title="c27">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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="nat_d.html#K7" title="NAT_D:func.7">-'</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="c27">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><br/>

<a NAME="E2:56_23"/><span class="kw">then </span>
<font color="Maroon" title="c27">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="nat_d.html#K7" title="NAT_D:func.7">-'</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 class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E3:56_23"/><span class="kw">then </span>
<font color="Maroon" title="c27">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/>
<a NAME="E4:56_23"/><span class="kw">then </span>
<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="nat_d.html#K7" title="NAT_D:func.7">-'</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="c27">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T49" title="FINSEQ_2:th.49">FINSEQ_2:49</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:56_23"/>
<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="nat_d.html#K7" title="NAT_D:func.7">-'</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="c27">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_23"><span class="lab"><font color="Green" title="E118">A180</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E132:56"/><span class="kw">then </span>
 <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="nat_d.html#K7" title="NAT_D:func.7">-'</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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T12" title="FINSEQ_2:th.12">FINSEQ_2:12</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c27">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="nat_d.html#K7" title="NAT_D:func.7">-'</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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="int_5.html#E55"><span class="lab"><font color="Green" title="E41">Lm4</font></span></a></span>;<br/>
<a NAME="E134:56"/><span class="lab"><font color="Green" title="E118">A181</font></span>: 
2,<font color="Maroon" title="c3">p</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="ref" href="euler_1.html#T2" title="EULER_1:th.2">EULER_1:2</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c28">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#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">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="E135:56"/>
 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#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">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#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">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="56_24"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c29">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c29">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#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">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>  implies <font color="Maroon" title="c29">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">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><br/>

<span class="kw">assume </span><a NAME="E1:56_24"/>
<font color="Maroon" title="c29">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#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c29">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">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><br/>

<a NAME="E2:56_24"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">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>  st <br/>( <font color="Maroon" title="c29">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">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 )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:56_24"/>
<font color="Maroon" title="c29">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E136:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E119">A182</font></span>: 
<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#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">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#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">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 class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E137:56"/><span class="lab"><font color="Green" title="E120">A183</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:56"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E113:56"><span class="lab"><font color="Green" title="E106">A97</font></span></a>, <a class="ref" href="nat_2.html#T13" title="NAT_2:th.13">NAT_2:13</a></span>;<br/>
<a NAME="E138:56"/><span class="lab"><font color="Green" title="E121">A184</font></span>: 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c23">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#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c17">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="nat_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</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="56_25"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c29">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span>, <font color="Maroon" title="c30">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="nat_d.html#K7" title="NAT_D:func.7">-'</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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E137:56"><span class="lab"><font color="Green" title="E120">A183</font></span></a>, <a class="txt" href="#E40:56"><span class="lab"><font color="Green" title="E69">A31</font></span></a></span>;<br/>
<span class="kw">deffunc </span><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="c29">A</font>,   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c30">B</font>) <span class="kw">-&gt; </span>   <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="E2:56_25"/><span class="lab"><font color="Green" title="E122">A185</font></span>: 
 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="c29">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="c30">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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c31">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="c29">A</font>,<font color="Maroon" title="c30">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><span class="kw"> such that </span><br/><a NAME="E4:56_25"/><span class="lab"><font color="Green" title="E123">A186</font></span>: 
 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="c29">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="c30">B</font> holds  <font color="Maroon" title="c31">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>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_7.html#S1" title="FUNCT_7:sch.1">FUNCT_7:sch 1</a>(<a class="txt" href="#E2:56_25"><span class="lab"><font color="Green" title="E122">A185</font></span></a>);<br/></span>
<span class="kw">defpred </span><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> ] <span class="kw">means </span> 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="c29">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="c30">B</font> : <font color="Maroon" title="c31">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="E5:56_25"/><span class="lab"><font color="Green" title="E124">A187</font></span>: 
 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="nat_d.html#K7" title="NAT_D:func.7">-'</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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c31">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="56_25_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c32">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span> implies  ex <font color="Olive" title="b1">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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c31">z</font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c32">d</font>,<font color="Olive" title="b1">x1</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_25_1"/>
<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"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c31">z</font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c32">d</font>,<font color="Olive" title="b1">x1</font>]</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c33">d</font> = <font color="Maroon" title="c32">d</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c29">A</font> ;<br/>
<span class="kw">take </span><font color="Maroon" title="c34">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="c33">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="c30">B</font> : <font color="Maroon" title="c31">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c33">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c34">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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c31">z</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c32">d</font>,<font color="Maroon" title="c34">x1</font>] )</span><br/>

<a NAME="E3:56_25_1"/>
<font color="Maroon" title="c34">x1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c31">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_25_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c35">l</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c35">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c34">x1</font> or <font color="Maroon" title="c35">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c31">z</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_25_1_1"/>
<font color="Maroon" title="c35">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c34">x1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c35">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c31">z</font></span><br/>

<a NAME="E2:56_25_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">yy</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c30">B</font> st <br/>( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c33">d</font>,<font color="Olive" title="b1">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="c35">l</font> &amp; <font color="Maroon" title="c31">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c33">d</font>,<font color="Olive" title="b1">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="E3:56_25_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c35">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="c29">A</font>,<font color="Maroon" title="c30">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
;<br/>
<span class="kw">hence </span><a NAME="E4:56_25_1_1"/>
<font color="Maroon" title="c35">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c31">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:56_25_1"/>
( <font color="Maroon" title="c34">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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c31">z</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c32">d</font>,<font color="Maroon" title="c34">x1</font>] )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c32">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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c31">z</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E7:56_25"/><span class="lab"><font color="Green" title="E125">A188</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c32">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="nat_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</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="c32">Pr</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Olive" title="b1">d</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_1.html#S5" title="FINSEQ_1:sch.5">FINSEQ_1:sch 5</a>(<a class="txt" href="#E5:56_25"><span class="lab"><font color="Green" title="E124">A187</font></span></a>);<br/></span>
<span class="lab"><font color="Green" title="E126">A189</font></span>:  <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="c32">Pr</font></span>)</span> = 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c32">Pr</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D2" title="CARD_3:def.2">CARD_3:def 2</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">f2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:56"><span class="lab"><font color="Green" title="E65">A27</font></span></a>, <a class="txt" href="#E7:56_25"><span class="lab"><font color="Green" title="E125">A188</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
;<br/>
<a NAME="E9:56_25"/>
 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="c32">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="c32">Pr</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_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">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_25_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c33">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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="int_5.html#K3" title="INT_5:func.3">Card</a> <font color="Maroon" title="c32">Pr</font></span>)</span> implies <span class="p1">(<span class="default"><a href="int_5.html#K3" title="INT_5:func.3">Card</a> <font color="Maroon" title="c32">Pr</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c33">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c33">d</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_25_3"/><span class="lab"><font color="Green" title="E127">A190</font></span>: 
<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="int_5.html#K3" title="INT_5:func.3">Card</a> <font color="Maroon" title="c32">Pr</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="int_5.html#K3" title="INT_5:func.3">Card</a> <font color="Maroon" title="c32">Pr</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c33">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c33">d</font></span><br/>

<a NAME="E2:56_25_3"/><span class="kw">then </span>
<font color="Maroon" title="c33">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:56"><span class="lab"><font color="Green" title="E65">A27</font></span></a>, <a class="txt" href="#E8:56_25"><span class="lab"><font color="Green" title="E126">A189</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c34">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c29">A</font><span class="kw"> such that </span><br/><a NAME="E4:56_25_3"/><span class="lab"><font color="Green" title="E128">A191</font></span>: 
<font color="Maroon" title="c34">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">d</font>
 <span class="kw">and </span><br/><a NAME="E5:56_25_3"/><span class="lab"><font color="Green" title="E129">A192</font></span>: 
<font color="Maroon" title="c32">Pr</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c33">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="c34">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="c30">B</font> : <font color="Maroon" title="c31">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c34">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:56_25"><span class="lab"><font color="Green" title="E125">A188</font></span></a></span>;<br/>
<a NAME="E6:56_25_3"/>
<font color="Maroon" title="c32">Pr</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c33">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="c34">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="c23">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c34">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="56_25_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c35">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="c34">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="c23">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c34">m</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>
<a NAME="E1:56_25_3_1"/><span class="lab"><font color="Green" title="E130">A193</font></span>: 
<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="c34">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="c23">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c34">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="c32">Pr</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c33">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_25_3_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_25_3_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:56_25_3_1_1_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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:56_25_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E131">A194</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pepin.html#T6" title="PEPIN:th.6">PEPIN:6</a></span>;<br/><a NAME="E3:56_25_3_1_1_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></span>;<br/><a NAME="E4:56_25_3_1_1_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E5:56_25_3_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:56"><span class="lab"><font color="Green" title="E48">A6</font></span></a>, <a class="txt" href="#E2:56_25_3_1_1_1"><span class="lab"><font color="Green" title="E131">A194</font></span></a>, <a class="ref" href="nat_4.html#T12" title="NAT_4:th.12">NAT_4:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E2:56_25_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E131">A195</font></span>: 
 <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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="wsierp_1.html#T41" title="WSIERP_1:th.41">WSIERP_1:41</a></span>;<br/>
<a NAME="E3:56_25_3_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="c3">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:56"><span class="lab"><font color="Green" title="E52">A10</font></span></a>, <a class="ref" href="nat_d.html#T9" title="NAT_D:th.9">NAT_D:9</a></span>;<br/>
<a NAME="E4:56_25_3_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pepin.html#T6" title="PEPIN:th.6">PEPIN:6</a></span>;<br/>
<a NAME="E5:56_25_3_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_3.html#T4" title="REAL_3:th.4">REAL_3:4</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E132">A196</font></span>: <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_d.html#K7" title="NAT_D:func.7">-'</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:56"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E13:56"><span class="lab"><font color="Green" title="E53">A11</font></span></a>, <a class="ref" href="nat_2.html#T27" title="NAT_2:th.27">NAT_2:27</a></span>
<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
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_ff.html#T5" title="PRE_FF:th.5">PRE_FF:5</a></span>
<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
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_25_3_1_1"><span class="lab"><font color="Green" title="E131">A195</font></span></a>, <a class="ref" href="nat_d.html#T61" title="NAT_D:th.61">NAT_D:61</a></span>
<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="nat_d.html#K7" title="NAT_D:func.7">-'</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
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:56"><span class="lab"><font color="Green" title="E67">A29</font></span></a>, <a class="txt" href="#E55:56"><span class="lab"><font color="Green" title="E76">A40</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T61" title="NAT_D:th.61">NAT_D:61</a></span>
;<br/>
<a NAME="E7:56_25_3_1_1"/><span class="lab"><font color="Green" title="E133">A197</font></span>: 
<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_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</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="56_25_3_1_1_3"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_25_3_1_1_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:56_25_3_1_1_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="56_25_3_1_1_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:56_25_3_1_1_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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span><br/><div class="add"><span class="kw">then </span><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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="wsierp_1.html#T42" title="WSIERP_1:th.42">WSIERP_1:42</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_2.html#T27" title="NAT_2:th.27">NAT_2:27</a></span>
;<br/><a NAME="E3:56_25_3_1_1_3_1_1"/><span class="kw">then </span>
<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_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:56_25_3_1_1"><span class="lab"><font color="Green" title="E132">A196</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:56_25_3_1_1_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="nat_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T43" title="XREAL_1:th.43">XREAL_1:43</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_25_3_1_1_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:56_25_3_1_1_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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span><br/><div class="add"><a NAME="E2:56_25_3_1_1_3_1_2"/><span class="kw">then </span>
 <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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="wsierp_1.html#T41" title="WSIERP_1:th.41">WSIERP_1:41</a></span>;<br/><span class="kw">then </span><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
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_2.html#T27" title="NAT_2:th.27">NAT_2:27</a></span>
;<br/><a NAME="E4:56_25_3_1_1_3_1_2"/><span class="kw">then </span>
<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_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:56_25_3_1_1"><span class="lab"><font color="Green" title="E132">A196</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:56_25_3_1_1_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="nat_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T43" title="XREAL_1:th.43">XREAL_1:43</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E8:56_25_3_1_1"/>
<font color="Maroon" title="c34">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E9:56_25_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c34">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="nat_d.html#K7" title="NAT_D:func.7">-'</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 class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/>
<a NAME="E10:56_25_3_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c34">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="nat_d.html#K7" title="NAT_D:func.7">-'</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="kw">by</span> <span class="lab"><a class="ref" href="nat_2.html#T24" title="NAT_2:th.24">NAT_2:24</a></span>;<br/>
<a NAME="E11:56_25_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E134">A198</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c34">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:56_25_3_1_1"><span class="lab"><font color="Green" title="E133">A197</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E12:56_25_3_1_1"/>
<font color="Maroon" title="c34">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span>
 
;<br/>
<a NAME="E13:56_25_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E135">A199</font></span>: 
<font color="Maroon" title="c34">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="c13">f1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:56"><span class="lab"><font color="Green" title="E58">A18</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E136">A200</font></span>: <font color="Maroon" title="c23">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c34">m</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c34">m</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="kw">by</span> <span class="lab"><a class="txt" href="#E34:56"><span class="lab"><font color="Green" title="E65">A27</font></span></a>, <a class="txt" href="#E118:56"><span class="lab"><font color="Green" title="E108">A99</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c34">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:56"><span class="lab"><font color="Green" title="E55">A13</font></span></a>, <a class="txt" href="#E13:56_25_3_1_1"><span class="lab"><font color="Green" title="E135">A199</font></span></a></span>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_25_3_1_1_5"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:56_25_3_1_1_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c34">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:56_25_3_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E137">A201</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c34">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="wsierp_1.html#T17" title="WSIERP_1:th.17">WSIERP_1:17</a></span>;<br/><a NAME="E3:56_25_3_1_1_5"/><span class="lab"><font color="Green" title="E138">A202</font></span>: 
<font color="Maroon" title="c34">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E4:56_25_3_1_1_5"/>
<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="c34">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E5:56_25_3_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c34">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:56"><span class="lab"><font color="Green" title="E47">A5</font></span></a>, <a class="txt" href="#E2:56_25_3_1_1_5"><span class="lab"><font color="Green" title="E137">A201</font></span></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a>, <a class="ref" href="wsierp_1.html#T30" title="WSIERP_1:th.30">WSIERP_1:30</a></span>;<br/><span class="kw">hence </span><a NAME="E6:56_25_3_1_1_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:56"><span class="lab"><font color="Green" title="E54">A12</font></span></a>, <a class="txt" href="#E3:56_25_3_1_1_5"><span class="lab"><font color="Green" title="E138">A202</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E16:56_25_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E137">A203</font></span>: 
<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="c34">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="c34">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 class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T26" title="INT_1:th.26">INT_1:26</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c36">l</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c36">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="c34">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="c23">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c34">m</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> or <font color="Maroon" title="c36">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c32">Pr</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c33">d</font> )</span><br/>

<span class="kw">assume </span><a NAME="E17:56_25_3_1_1"/>
<font color="Maroon" title="c36">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="c34">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="c23">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c34">m</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c36">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c32">Pr</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c33">d</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c37">x</font>, <font color="Maroon" title="c38">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E19:56_25_3_1_1"/><span class="lab"><font color="Green" title="E138">A204</font></span>: 
<font color="Maroon" title="c37">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="c34">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">and </span><br/><a NAME="E20:56_25_3_1_1"/><span class="lab"><font color="Green" title="E139">A205</font></span>: 
<font color="Maroon" title="c38">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="c23">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c34">m</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E21:56_25_3_1_1"/><span class="lab"><font color="Green" title="E140">A206</font></span>: 
<font color="Maroon" title="c36">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="c37">x</font>,<font color="Maroon" title="c38">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c39">y</font> = <font color="Maroon" title="c38">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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:56_25_3_1_1"><span class="lab"><font color="Green" title="E139">A205</font></span></a></span>;<br/>
<a NAME="E23:56_25_3_1_1"/><span class="lab"><font color="Green" title="E141">A207</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c39">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:56_25_3_1_1"><span class="lab"><font color="Green" title="E139">A205</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E24:56_25_3_1_1"/>
<font color="Maroon" title="c39">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c34">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:56_25_3_1_1"><span class="lab"><font color="Green" title="E139">A205</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E25:56_25_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c39">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:56_25_3_1_1"><span class="lab"><font color="Green" title="E134">A198</font></span></a>, <a class="txt" href="#E14:56_25_3_1_1"><span class="lab"><font color="Green" title="E136">A200</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c40">y</font> = <font color="Maroon" title="c39">y</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c30">B</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:56_25_3_1_1"><span class="lab"><font color="Green" title="E141">A207</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E27:56_25_3_1_1"/>
<font color="Maroon" title="c40">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="c34">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:56_25_3_1_1"><span class="lab"><font color="Green" title="E139">A205</font></span></a>, <a class="txt" href="#E14:56_25_3_1_1"><span class="lab"><font color="Green" title="E136">A200</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E28:56_25_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c40">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="c34">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E16:56_25_3_1_1"><span class="lab"><font color="Green" title="E137">A203</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E29:56_25_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c40">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="c34">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T68" title="XREAL_1:th.68">XREAL_1:68</a></span>;<br/>
<a NAME="E30:56_25_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c40">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="c34">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T87" title="XCMPLX_1:th.87">XCMPLX_1:87</a></span>;<br/>
<a NAME="E31:56_25_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c40">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="c34">m</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T106" title="XREAL_1:th.106">XREAL_1:106</a></span>;<br/>
<a NAME="E32:56_25_3_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c34">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="c40">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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>;<br/>
<a NAME="E33:56_25_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c31">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c34">m</font>,<font color="Maroon" title="c40">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_25"><span class="lab"><font color="Green" title="E123">A186</font></span></a></span>;<br/>
<a NAME="E34:56_25_3_1_1"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c34">m</font>,<font color="Maroon" title="c40">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="c32">Pr</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c33">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:56_25_3"><span class="lab"><font color="Green" title="E129">A192</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E35:56_25_3_1_1"/>
<font color="Maroon" title="c36">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c32">Pr</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c33">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:56_25_3_1_1"><span class="lab"><font color="Green" title="E138">A204</font></span></a>, <a class="txt" href="#E21:56_25_3_1_1"><span class="lab"><font color="Green" title="E140">A206</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:56_25_3_1"/>
<font color="Maroon" title="c32">Pr</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c33">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="c34">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="c23">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c34">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="56_25_3_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c36">l</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c36">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c32">Pr</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c33">d</font> or <font color="Maroon" title="c36">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="c34">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="c23">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c34">m</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )</span><br/>

<a NAME="E1:56_25_3_1_2"/><span class="lab"><font color="Green" title="E131">A208</font></span>: 
<font color="Maroon" title="c34">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="c34">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E2:56_25_3_1_2"/>
<font color="Maroon" title="c34">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span>
 
;<br/>
<a NAME="E3:56_25_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E132">A209</font></span>: 
<font color="Maroon" title="c34">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="c13">f1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:56"><span class="lab"><font color="Green" title="E58">A18</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:56_25_3_1_2"/>
<font color="Maroon" title="c36">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c32">Pr</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c33">d</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c36">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="c34">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="c23">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c34">m</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c37">y1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c30">B</font><span class="kw"> such that </span><br/><a NAME="E6:56_25_3_1_2"/><span class="lab"><font color="Green" title="E133">A210</font></span>: 
<font color="Maroon" title="c36">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="c34">m</font>,<font color="Maroon" title="c37">y1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <span class="kw">and </span><br/><a NAME="E7:56_25_3_1_2"/><span class="lab"><font color="Green" title="E134">A211</font></span>: 
<font color="Maroon" title="c31">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c34">m</font>,<font color="Maroon" title="c37">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:56_25_3"><span class="lab"><font color="Green" title="E129">A192</font></span></a></span>;<br/>
<a NAME="E8:56_25_3_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c34">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="c37">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_25"><span class="lab"><font color="Green" title="E123">A186</font></span></a>, <a class="txt" href="#E7:56_25_3_1_2"><span class="lab"><font color="Green" title="E134">A211</font></span></a></span>;<br/>
<a NAME="E9:56_25_3_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c34">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="c37">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="c37">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="c37">y1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">q</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/>
<a NAME="E10:56_25_3_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c34">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="c37">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T68" title="XREAL_1:th.68">XREAL_1:68</a></span>;<br/>
<a NAME="E11:56_25_3_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c34">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="c37">y1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T87" title="XCMPLX_1:th.87">XCMPLX_1:87</a></span>;<br/>
<a NAME="E12:56_25_3_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c34">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="c37">y1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T54" title="INT_1:th.54">INT_1:54</a></span>;<br/>
<a NAME="E13:56_25_3_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c34">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="c37">y1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:56"><span class="lab"><font color="Green" title="E55">A13</font></span></a>, <a class="txt" href="#E3:56_25_3_1_2"><span class="lab"><font color="Green" title="E132">A209</font></span></a></span>;<br/>
<a NAME="E14:56_25_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E135">A212</font></span>: 
<font color="Maroon" title="c37">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c34">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:56"><span class="lab"><font color="Green" title="E65">A27</font></span></a>, <a class="txt" href="#E118:56"><span class="lab"><font color="Green" title="E108">A99</font></span></a>, <a class="txt" href="#E3:56_25_3_1_2"><span class="lab"><font color="Green" title="E132">A209</font></span></a></span>;<br/>
<a NAME="E15:56_25_3_1_2"/>
<font color="Maroon" title="c37">y1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E16:56_25_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c37">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="c23">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c34">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:56_25_3_1_2"><span class="lab"><font color="Green" title="E135">A212</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:56_25_3_1_2"/>
<font color="Maroon" title="c36">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="c34">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="c23">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c34">m</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:56_25_3_1_2"><span class="lab"><font color="Green" title="E133">A210</font></span></a>, <a class="txt" href="#E1:56_25_3_1_2"><span class="lab"><font color="Green" title="E131">A208</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:56_25_3_1"/>
<font color="Maroon" title="c32">Pr</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c33">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="c34">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="c23">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c34">m</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_25_3_1"><span class="lab"><font color="Green" title="E130">A193</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span> <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">Pr</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c33">d</font></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"><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">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c34">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="c34">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T4" title="CARD_2:th.4">CARD_2:4</a></span>
<br/>.= 
 <a href="card_1.html#K5" title="CARD_1:func.5">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">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c34">m</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T69" title="CARD_1:th.69">CARD_1:69</a></span>
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E130">A213</font></span>:  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">Pr</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c33">d</font></span>)</span> = 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c33">d</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_25_3"><span class="lab"><font color="Green" title="E128">A191</font></span></a>, <a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a></span>
<br/>.= 
<font color="Maroon" title="c23">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c33">d</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D2" title="CARD_1:def.2">CARD_1:def 2</a></span>
;<br/>
<a NAME="E9:56_25_3"/>
<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> <font color="Maroon" title="c32">Pr</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_25_3"><span class="lab"><font color="Green" title="E127">A190</font></span></a>, <a class="ref" href="card_3.html#D2" title="CARD_3:def.2">CARD_3:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:56_25_3"/>
<span class="p1">(<span class="default"><a href="int_5.html#K3" title="INT_5:func.3">Card</a> <font color="Maroon" title="c32">Pr</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c33">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c33">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:56_25_3"><span class="lab"><font color="Green" title="E130">A213</font></span></a>, <a class="ref" href="card_3.html#D2" title="CARD_3:def.2">CARD_3:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E10:56_25"/><span class="kw">then </span><span class="lab"><font color="Green" title="E127">A214</font></span>: 
 <a href="int_5.html#K3" title="INT_5:func.3">Card</a> <font color="Maroon" title="c32">Pr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">f2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:56_25"><span class="lab"><font color="Green" title="E126">A189</font></span></a>, <a class="ref" href="finseq_1.html#T13" title="FINSEQ_1:th.13">FINSEQ_1:13</a></span>;<br/>
<span class="kw">defpred </span><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> ] <span class="kw">means </span> 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="c30">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="c29">A</font> : <font color="Maroon" title="c31">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="E11:56_25"/><span class="lab"><font color="Green" title="E128">A215</font></span>: 
 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="nat_d.html#K7" title="NAT_D:func.7">-'</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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c31">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="56_25_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c33">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c33">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span> implies  ex <font color="Olive" title="b1">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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c31">z</font></span>)</span> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c33">d</font>,<font color="Olive" title="b1">x1</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_25_4"/>
<font color="Maroon" title="c33">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c31">z</font></span>)</span> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c33">d</font>,<font color="Olive" title="b1">x1</font>]</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c34">d</font> = <font color="Maroon" title="c33">d</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c30">B</font> ;<br/>
<span class="kw">take </span><font color="Maroon" title="c35">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="c34">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="c29">A</font> : <font color="Maroon" title="c31">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="c34">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c35">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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c31">z</font></span>)</span> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c33">d</font>,<font color="Maroon" title="c35">x1</font>] )</span><br/>

<a NAME="E3:56_25_4"/>
<font color="Maroon" title="c35">x1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c31">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_25_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c36">l</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c36">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c35">x1</font> or <font color="Maroon" title="c36">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c31">z</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_25_4_1"/>
<font color="Maroon" title="c36">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c35">x1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c36">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c31">z</font></span><br/>

<a NAME="E2:56_25_4_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">xx</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c29">A</font> st <br/>( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">xx</font>,<font color="Maroon" title="c34">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="c36">l</font> &amp; <font color="Maroon" title="c31">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b1">xx</font>,<font color="Maroon" title="c34">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="E3:56_25_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c36">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="c29">A</font>,<font color="Maroon" title="c30">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
;<br/>
<span class="kw">hence </span><a NAME="E4:56_25_4_1"/>
<font color="Maroon" title="c36">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c31">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:56_25_4"/>
( <font color="Maroon" title="c35">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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c31">z</font></span>)</span> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c33">d</font>,<font color="Maroon" title="c35">x1</font>] )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c33">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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c31">z</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E13:56_25"/><span class="lab"><font color="Green" title="E129">A216</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c33">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="nat_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</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="c33">Pk</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Olive" title="b1">d</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_1.html#S5" title="FINSEQ_1:sch.5">FINSEQ_1:sch 5</a>(<a class="txt" href="#E11:56_25"><span class="lab"><font color="Green" title="E128">A215</font></span></a>);<br/></span>
<span class="lab"><font color="Green" title="E130">A217</font></span>:  <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="c33">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="c17">g2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:56"><span class="lab"><font color="Green" title="E71">A33</font></span></a>, <a class="txt" href="#E13:56_25"><span class="lab"><font color="Green" title="E129">A216</font></span></a>, <a class="ref" href="card_3.html#D2" title="CARD_3:def.2">CARD_3:def 2</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c17">g2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
;<br/>
<a NAME="E15:56_25"/><span class="lab"><font color="Green" title="E131">A218</font></span>: 
 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="c33">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="c33">Pk</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_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="c17">g2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_25_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c34">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c34">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="c33">Pk</font></span>)</span> implies <span class="p1">(<span class="default"><a href="int_5.html#K3" title="INT_5:func.3">Card</a> <font color="Maroon" title="c33">Pk</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c34">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c34">d</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_25_6"/><span class="lab"><font color="Green" title="E132">A219</font></span>: 
<font color="Maroon" title="c34">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="c33">Pk</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="int_5.html#K3" title="INT_5:func.3">Card</a> <font color="Maroon" title="c33">Pk</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c34">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c34">d</font></span><br/>

<a NAME="E2:56_25_6"/><span class="kw">then </span>
<font color="Maroon" title="c34">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:56"><span class="lab"><font color="Green" title="E71">A33</font></span></a>, <a class="txt" href="#E14:56_25"><span class="lab"><font color="Green" title="E130">A217</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c35">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c30">B</font><span class="kw"> such that </span><br/><a NAME="E4:56_25_6"/><span class="lab"><font color="Green" title="E133">A220</font></span>: 
<font color="Maroon" title="c35">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">d</font>
 <span class="kw">and </span><br/><a NAME="E5:56_25_6"/><span class="lab"><font color="Green" title="E134">A221</font></span>: 
<font color="Maroon" title="c33">Pk</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c34">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="c35">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="c29">A</font> : <font color="Maroon" title="c31">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="c35">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:56_25"><span class="lab"><font color="Green" title="E129">A216</font></span></a></span>;<br/>
<a NAME="E6:56_25_6"/>
<font color="Maroon" title="c33">Pk</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c34">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="c17">g2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">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="c35">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="56_25_6_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c36">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="c17">g2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">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="c35">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="E1:56_25_6_1"/><span class="lab"><font color="Green" title="E135">A222</font></span>: 
<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="c17">g2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">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="c35">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="c33">Pk</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c34">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_25_6_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_25_6_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:56_25_6_1_1_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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:56_25_6_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E136">A223</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pepin.html#T6" title="PEPIN:th.6">PEPIN:6</a></span>;<br/><a NAME="E3:56_25_6_1_1_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></span>;<br/><a NAME="E4:56_25_6_1_1_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E5:56_25_6_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:56"><span class="lab"><font color="Green" title="E66">A28</font></span></a>, <a class="txt" href="#E2:56_25_6_1_1_1"><span class="lab"><font color="Green" title="E136">A223</font></span></a>, <a class="ref" href="nat_4.html#T12" title="NAT_4:th.12">NAT_4:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E2:56_25_6_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E136">A224</font></span>: 
 <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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="wsierp_1.html#T41" title="WSIERP_1:th.41">WSIERP_1:41</a></span>;<br/>
<a NAME="E3:56_25_6_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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:56"><span class="lab"><font color="Green" title="E75">A39</font></span></a>, <a class="ref" href="nat_d.html#T9" title="NAT_D:th.9">NAT_D:9</a></span>;<br/>
<a NAME="E4:56_25_6_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pepin.html#T6" title="PEPIN:th.6">PEPIN:6</a></span>;<br/>
<a NAME="E5:56_25_6_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_3.html#T4" title="REAL_3:th.4">REAL_3:4</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E137">A225</font></span>: <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_d.html#K7" title="NAT_D:func.7">-'</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:56"><span class="lab"><font color="Green" title="E67">A29</font></span></a>, <a class="txt" href="#E55:56"><span class="lab"><font color="Green" title="E76">A40</font></span></a>, <a class="ref" href="nat_2.html#T27" title="NAT_2:th.27">NAT_2:27</a></span>
<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
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_ff.html#T5" title="PRE_FF:th.5">PRE_FF:5</a></span>
<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
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_25_6_1_1"><span class="lab"><font color="Green" title="E136">A224</font></span></a>, <a class="ref" href="nat_d.html#T61" title="NAT_D:th.61">NAT_D:61</a></span>
<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="nat_d.html#K7" title="NAT_D:func.7">-'</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
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:56"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E13:56"><span class="lab"><font color="Green" title="E53">A11</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T61" title="NAT_D:th.61">NAT_D:61</a></span>
;<br/>
<a NAME="E7:56_25_6_1_1"/><span class="lab"><font color="Green" title="E138">A226</font></span>: 
<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_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</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="56_25_6_1_1_3"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_25_6_1_1_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:56_25_6_1_1_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="56_25_6_1_1_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:56_25_6_1_1_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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span><br/><div class="add"><span class="kw">then </span><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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="wsierp_1.html#T42" title="WSIERP_1:th.42">WSIERP_1:42</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_2.html#T27" title="NAT_2:th.27">NAT_2:27</a></span>
;<br/><a NAME="E3:56_25_6_1_1_3_1_1"/><span class="kw">then </span>
<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_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:56_25_6_1_1"><span class="lab"><font color="Green" title="E137">A225</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:56_25_6_1_1_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="nat_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T43" title="XREAL_1:th.43">XREAL_1:43</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_25_6_1_1_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:56_25_6_1_1_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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span><br/><div class="add"><a NAME="E2:56_25_6_1_1_3_1_2"/><span class="kw">then </span>
 <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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="wsierp_1.html#T41" title="WSIERP_1:th.41">WSIERP_1:41</a></span>;<br/><span class="kw">then </span><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
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_2.html#T27" title="NAT_2:th.27">NAT_2:27</a></span>
;<br/><a NAME="E4:56_25_6_1_1_3_1_2"/><span class="kw">then </span>
<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_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:56_25_6_1_1"><span class="lab"><font color="Green" title="E137">A225</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:56_25_6_1_1_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="nat_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T43" title="XREAL_1:th.43">XREAL_1:43</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E8:56_25_6_1_1"/>
<font color="Maroon" title="c35">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span>
 
;<br/>
<a NAME="E9:56_25_6_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E139">A227</font></span>: 
<font color="Maroon" title="c35">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="c9">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:56"><span class="lab"><font color="Green" title="E61">A23</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E140">A228</font></span>: <font color="Maroon" title="c17">g2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">n</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="kw">by</span> <span class="lab"><a class="txt" href="#E47:56"><span class="lab"><font color="Green" title="E71">A33</font></span></a>, <a class="txt" href="#E50:56"><span class="lab"><font color="Green" title="E72">A34</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c35">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:56"><span class="lab"><font color="Green" title="E59">A19</font></span></a>, <a class="txt" href="#E9:56_25_6_1_1"><span class="lab"><font color="Green" title="E139">A227</font></span></a></span>
;<br/>
<span class="kw">let </span><font color="Maroon" title="c37">l</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c37">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="c17">g2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">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="c35">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> or <font color="Maroon" title="c37">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c33">Pk</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c34">d</font> )</span><br/>

<span class="kw">assume </span><a NAME="E11:56_25_6_1_1"/>
<font color="Maroon" title="c37">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="c17">g2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">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="c35">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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c37">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c33">Pk</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c34">d</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c38">x</font>, <font color="Maroon" title="c39">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E13:56_25_6_1_1"/><span class="lab"><font color="Green" title="E141">A229</font></span>: 
<font color="Maroon" title="c38">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="c17">g2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E14:56_25_6_1_1"/><span class="lab"><font color="Green" title="E142">A230</font></span>: 
<font color="Maroon" title="c39">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="c35">n</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">and </span><br/><a NAME="E15:56_25_6_1_1"/><span class="lab"><font color="Green" title="E143">A231</font></span>: 
<font color="Maroon" title="c37">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="c38">x</font>,<font color="Maroon" title="c39">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c40">x</font> = <font color="Maroon" title="c38">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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:56_25_6_1_1"><span class="lab"><font color="Green" title="E141">A229</font></span></a></span>;<br/>
<a NAME="E17:56_25_6_1_1"/><span class="lab"><font color="Green" title="E144">A232</font></span>: 
<font color="Maroon" title="c40">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">g2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:56_25_6_1_1"><span class="lab"><font color="Green" title="E141">A229</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E18:56_25_6_1_1"/>
<font color="Maroon" title="c35">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E19:56_25_6_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c35">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="nat_d.html#K7" title="NAT_D:func.7">-'</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 class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/>
<a NAME="E20:56_25_6_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c35">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="nat_d.html#K7" title="NAT_D:func.7">-'</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="kw">by</span> <span class="lab"><a class="ref" href="nat_2.html#T24" title="NAT_2:th.24">NAT_2:24</a></span>;<br/>
<a NAME="E21:56_25_6_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c35">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:56_25_6_1_1"><span class="lab"><font color="Green" title="E138">A226</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E22:56_25_6_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E145">A233</font></span>: 
<font color="Maroon" title="c40">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:56_25_6_1_1"><span class="lab"><font color="Green" title="E140">A228</font></span></a>, <a class="txt" href="#E17:56_25_6_1_1"><span class="lab"><font color="Green" title="E144">A232</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E23:56_25_6_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c40">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:56_25_6_1_1"><span class="lab"><font color="Green" title="E141">A229</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c41">x</font> = <font color="Maroon" title="c40">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c29">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:56_25_6_1_1"><span class="lab"><font color="Green" title="E145">A233</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_25_6_1_1_5"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:56_25_6_1_1_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c35">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:56_25_6_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E146">A234</font></span>: 
<font color="Maroon" title="c4">q</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c35">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="wsierp_1.html#T17" title="WSIERP_1:th.17">WSIERP_1:17</a></span>;<br/><a NAME="E3:56_25_6_1_1_5"/><span class="lab"><font color="Green" title="E147">A235</font></span>: 
<font color="Maroon" title="c35">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E4:56_25_6_1_1_5"/>
<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="c35">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E5:56_25_6_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c4">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c35">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:56"><span class="lab"><font color="Green" title="E47">A5</font></span></a>, <a class="txt" href="#E2:56_25_6_1_1_5"><span class="lab"><font color="Green" title="E146">A234</font></span></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a>, <a class="ref" href="wsierp_1.html#T30" title="WSIERP_1:th.30">WSIERP_1:30</a></span>;<br/><span class="kw">hence </span><a NAME="E6:56_25_6_1_1_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:56"><span class="lab"><font color="Green" title="E77">A41</font></span></a>, <a class="txt" href="#E3:56_25_6_1_1_5"><span class="lab"><font color="Green" title="E147">A235</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E26:56_25_6_1_1"/><span class="kw">then </span>
<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="c35">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="c35">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 class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T26" title="INT_1:th.26">INT_1:26</a></span>;<br/>
<a NAME="E27:56_25_6_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c41">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="c35">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E10:56_25_6_1_1"><span class="lab"><font color="Green" title="E140">A228</font></span></a>, <a class="txt" href="#E17:56_25_6_1_1"><span class="lab"><font color="Green" title="E144">A232</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E28:56_25_6_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c41">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="c35">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T68" title="XREAL_1:th.68">XREAL_1:68</a></span>;<br/>
<a NAME="E29:56_25_6_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c41">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="c35">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T87" title="XCMPLX_1:th.87">XCMPLX_1:87</a></span>;<br/>
<a NAME="E30:56_25_6_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c41">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="c35">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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T49" title="XREAL_1:th.49">XREAL_1:49</a>, <a class="ref" href="xreal_1.html#T106" title="XREAL_1:th.106">XREAL_1:106</a></span>;<br/>
<a NAME="E31:56_25_6_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c31">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c41">x</font>,<font color="Maroon" title="c35">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_25"><span class="lab"><font color="Green" title="E123">A186</font></span></a></span>;<br/>
<a NAME="E32:56_25_6_1_1"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c41">x</font>,<font color="Maroon" title="c35">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="c33">Pk</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c34">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:56_25_6"><span class="lab"><font color="Green" title="E134">A221</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E33:56_25_6_1_1"/>
<font color="Maroon" title="c37">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c33">Pk</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c34">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:56_25_6_1_1"><span class="lab"><font color="Green" title="E142">A230</font></span></a>, <a class="txt" href="#E15:56_25_6_1_1"><span class="lab"><font color="Green" title="E143">A231</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:56_25_6_1"/>
<font color="Maroon" title="c33">Pk</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c34">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="c17">g2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">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="c35">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="56_25_6_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c37">l</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c37">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c33">Pk</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c34">d</font> or <font color="Maroon" title="c37">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="c17">g2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">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="c35">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> )</span><br/>

<a NAME="E1:56_25_6_1_2"/><span class="lab"><font color="Green" title="E136">A236</font></span>: 
<font color="Maroon" title="c35">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="c35">n</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E2:56_25_6_1_2"/>
<font color="Maroon" title="c35">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span>
 
;<br/>
<a NAME="E3:56_25_6_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E137">A237</font></span>: 
<font color="Maroon" title="c35">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="c9">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:56"><span class="lab"><font color="Green" title="E61">A23</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:56_25_6_1_2"/>
<font color="Maroon" title="c37">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c33">Pk</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c34">d</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c37">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="c17">g2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">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="c35">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></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c38">x</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c29">A</font><span class="kw"> such that </span><br/><a NAME="E6:56_25_6_1_2"/><span class="lab"><font color="Green" title="E138">A238</font></span>: 
<font color="Maroon" title="c37">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="c38">x</font>,<font color="Maroon" title="c35">n</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <span class="kw">and </span><br/><a NAME="E7:56_25_6_1_2"/><span class="lab"><font color="Green" title="E139">A239</font></span>: 
<font color="Maroon" title="c31">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c38">x</font>,<font color="Maroon" title="c35">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E5:56_25_6"><span class="lab"><font color="Green" title="E134">A221</font></span></a></span>;<br/>
<a NAME="E8:56_25_6_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c38">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="c35">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_25"><span class="lab"><font color="Green" title="E123">A186</font></span></a>, <a class="txt" href="#E7:56_25_6_1_2"><span class="lab"><font color="Green" title="E139">A239</font></span></a></span>;<br/>
<a NAME="E9:56_25_6_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c38">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="c35">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="c35">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="c35">n</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">q</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/>
<a NAME="E10:56_25_6_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c38">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="c35">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T68" title="XREAL_1:th.68">XREAL_1:68</a></span>;<br/>
<a NAME="E11:56_25_6_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c38">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="c35">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 class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T87" title="XCMPLX_1:th.87">XCMPLX_1:87</a></span>;<br/>
<a NAME="E12:56_25_6_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c38">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="c35">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T54" title="INT_1:th.54">INT_1:54</a></span>;<br/>
<a NAME="E13:56_25_6_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">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="c38">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:56"><span class="lab"><font color="Green" title="E59">A19</font></span></a>, <a class="txt" href="#E3:56_25_6_1_2"><span class="lab"><font color="Green" title="E137">A237</font></span></a></span>;<br/>
<a NAME="E14:56_25_6_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E140">A240</font></span>: 
<font color="Maroon" title="c38">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">g2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:56"><span class="lab"><font color="Green" title="E71">A33</font></span></a>, <a class="txt" href="#E50:56"><span class="lab"><font color="Green" title="E72">A34</font></span></a>, <a class="txt" href="#E3:56_25_6_1_2"><span class="lab"><font color="Green" title="E137">A237</font></span></a></span>;<br/>
<a NAME="E15:56_25_6_1_2"/>
<font color="Maroon" title="c38">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E16:56_25_6_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c38">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="c17">g2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:56_25_6_1_2"><span class="lab"><font color="Green" title="E140">A240</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:56_25_6_1_2"/>
<font color="Maroon" title="c37">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="c17">g2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">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="c35">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:56_25_6_1_2"><span class="lab"><font color="Green" title="E138">A238</font></span></a>, <a class="txt" href="#E1:56_25_6_1_2"><span class="lab"><font color="Green" title="E136">A236</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:56_25_6_1"/>
<font color="Maroon" title="c33">Pk</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c34">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="c17">g2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">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="c35">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_25_6_1"><span class="lab"><font color="Green" title="E135">A222</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:56_25_6"/><span class="kw">then </span>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">Pk</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c34">d</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="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">g2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">n</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T69" title="CARD_1:th.69">CARD_1:69</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E135">A241</font></span>:  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">Pk</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c34">d</font></span>)</span> = 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">g2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c34">d</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_25_6"><span class="lab"><font color="Green" title="E133">A220</font></span></a>, <a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a></span>
<br/>.= 
<font color="Maroon" title="c17">g2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c34">d</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D2" title="CARD_1:def.2">CARD_1:def 2</a></span>
;<br/>
<a NAME="E9:56_25_6"/>
<font color="Maroon" title="c34">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="c33">Pk</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_25_6"><span class="lab"><font color="Green" title="E132">A219</font></span></a>, <a class="ref" href="card_3.html#D2" title="CARD_3:def.2">CARD_3:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:56_25_6"/>
<span class="p1">(<span class="default"><a href="int_5.html#K3" title="INT_5:func.3">Card</a> <font color="Maroon" title="c33">Pk</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c34">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c34">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:56_25_6"><span class="lab"><font color="Green" title="E135">A241</font></span></a>, <a class="ref" href="card_3.html#D2" title="CARD_3:def.2">CARD_3:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">reconsider </span><font color="Maroon" title="c34">U1</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c32">Pr</font></span>)</span>, <font color="Maroon" title="c35">U2</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c33">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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c31">z</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_3.html#T48" title="PROB_3:th.48">PROB_3:48</a></span>;<br/>
<a NAME="E17:56_25"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c31">z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c34">U1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c35">U2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_25_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c36">l</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c36">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c31">z</font> or <font color="Maroon" title="c36">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c34">U1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c35">U2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_25_7"/>
<font color="Maroon" title="c36">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c31">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c36">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c34">U1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c35">U2</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c37">x</font>, <font color="Maroon" title="c38">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:56_25_7"/><span class="lab"><font color="Green" title="E132">A242</font></span>: 
<font color="Maroon" title="c37">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c29">A</font>
 <span class="kw">and </span><br/><a NAME="E4:56_25_7"/><span class="lab"><font color="Green" title="E133">A243</font></span>: 
<font color="Maroon" title="c38">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c30">B</font>
 <span class="kw">and </span><br/><a NAME="E5:56_25_7"/><span class="lab"><font color="Green" title="E134">A244</font></span>: 
<font color="Maroon" title="c36">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="c37">x</font>,<font color="Maroon" title="c38">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c39">y</font> = <font color="Maroon" title="c38">y</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c30">B</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_25_7"><span class="lab"><font color="Green" title="E133">A243</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c40">x</font> = <font color="Maroon" title="c37">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c29">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_25_7"><span class="lab"><font color="Green" title="E132">A242</font></span></a></span>;<br/>
<a NAME="E8:56_25_7"/><span class="lab"><font color="Green" title="E135">A245</font></span>: 
<font color="Maroon" title="c31">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c40">x</font>,<font color="Maroon" title="c39">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="56_25_7_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:56_25_7_1"/>
<font color="Maroon" title="c31">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c40">x</font>,<font color="Maroon" title="c39">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:56_25_7_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c40">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="c39">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_25"><span class="lab"><font color="Green" title="E123">A186</font></span></a></span>;<br/>
<a NAME="E3:56_25_7_1"/><span class="kw">then </span>
<font color="Maroon" title="c40">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="c39">y</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T95" title="XCMPLX_1:th.95">XCMPLX_1:95</a></span>;<br/>
<a NAME="E4:56_25_7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E136">A246</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c40">x</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></span>;<br/>
<a NAME="E5:56_25_7_1"/><span class="lab"><font color="Green" title="E137">A247</font></span>: 
<font color="Maroon" title="c40">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E6:56_25_7_1"/>
<font color="Maroon" title="c40">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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E7:56_25_7_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c40">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:56"><span class="lab"><font color="Green" title="E47">A5</font></span></a>, <a class="txt" href="#E4:56_25_7_1"><span class="lab"><font color="Green" title="E136">A246</font></span></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a>, <a class="ref" href="wsierp_1.html#T30" title="WSIERP_1:th.30">WSIERP_1:30</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:56_25_7_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:56"><span class="lab"><font color="Green" title="E54">A12</font></span></a>, <a class="txt" href="#E5:56_25_7_1"><span class="lab"><font color="Green" title="E137">A247</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_25_7_2"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:56_25_7_2"/>
( <font color="Maroon" title="c31">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c40">x</font>,<font color="Maroon" title="c39">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="c31">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c40">x</font>,<font color="Maroon" title="c39">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E8:56_25_7"><span class="lab"><font color="Green" title="E135">A245</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_25_7_2_1"><span class="kw">suppose </span></a><a NAME="E1:56_25_7_2_1"/><span class="lab"><font color="Green" title="E136">A248</font></span>: 
<font color="Maroon" title="c31">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c40">x</font>,<font color="Maroon" title="c39">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c36">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c34">U1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c35">U2</font></span><br/><div class="add"><a NAME="E2:56_25_7_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c40">x</font>,<font color="Maroon" title="c32">Pr</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c40">x</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:56_25"><span class="lab"><font color="Green" title="E125">A188</font></span></a></span>;<br/><a NAME="E3:56_25_7_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c36">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c32">Pr</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c40">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:56_25_7"><span class="lab"><font color="Green" title="E134">A244</font></span></a>, <a class="txt" href="#E1:56_25_7_2_1"><span class="lab"><font color="Green" title="E136">A248</font></span></a></span>;<br/><a NAME="E4:56_25_7_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c36">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_3.html#K5" title="PROB_3:func.5">Union</a> <font color="Maroon" title="c32">Pr</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:56_25"><span class="lab"><font color="Green" title="E125">A188</font></span></a>, <a class="ref" href="prob_3.html#T49" title="PROB_3:th.49">PROB_3:49</a></span>;<br/><span class="kw">hence </span><a NAME="E5:56_25_7_2_1"/>
<font color="Maroon" title="c36">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c34">U1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c35">U2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_25_7_2_2"><span class="kw">suppose </span></a><a NAME="E1:56_25_7_2_2"/><span class="lab"><font color="Green" title="E136">A249</font></span>: 
<font color="Maroon" title="c31">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c40">x</font>,<font color="Maroon" title="c39">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c36">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c34">U1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c35">U2</font></span><br/><div class="add"><a NAME="E2:56_25_7_2_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c39">y</font>,<font color="Maroon" title="c33">Pk</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c39">y</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:56_25"><span class="lab"><font color="Green" title="E129">A216</font></span></a></span>;<br/><a NAME="E3:56_25_7_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c36">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c33">Pk</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c39">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:56_25_7"><span class="lab"><font color="Green" title="E134">A244</font></span></a>, <a class="txt" href="#E1:56_25_7_2_2"><span class="lab"><font color="Green" title="E136">A249</font></span></a></span>;<br/><a NAME="E4:56_25_7_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c36">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_3.html#K5" title="PROB_3:func.5">Union</a> <font color="Maroon" title="c33">Pk</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:56_25"><span class="lab"><font color="Green" title="E129">A216</font></span></a>, <a class="ref" href="prob_3.html#T49" title="PROB_3:th.49">PROB_3:49</a></span>;<br/><span class="kw">hence </span><a NAME="E5:56_25_7_2_2"/>
<font color="Maroon" title="c36">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c34">U1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c35">U2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E18:56_25"/><span class="kw">then </span><span class="lab"><font color="Green" title="E132">A250</font></span>: 
<font color="Maroon" title="c34">U1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c35">U2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c31">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E19:56_25"/><span class="lab"><font color="Green" title="E133">A251</font></span>: 
<font color="Maroon" title="c34">U1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c35">U2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_25_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:56_25_8"/>
<font color="Maroon" title="c34">U1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c35">U2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c36">l</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:56_25_8"/><span class="lab"><font color="Green" title="E134">A252</font></span>: 
<font color="Maroon" title="c36">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c34">U1</font>
 <span class="kw">and </span><br/><a NAME="E4:56_25_8"/><span class="lab"><font color="Green" title="E135">A253</font></span>: 
<font color="Maroon" title="c36">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c35">U2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<a NAME="E5:56_25_8"/>
<font color="Maroon" title="c36">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_3.html#K5" title="PROB_3:func.5">Union</a> <font color="Maroon" title="c33">Pk</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_25_8"><span class="lab"><font color="Green" title="E135">A253</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c37">k2</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E7:56_25_8"/><span class="lab"><font color="Green" title="E136">A254</font></span>: 
<font color="Maroon" title="c37">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="c33">Pk</font>
 <span class="kw">and </span><br/><a NAME="E8:56_25_8"/><span class="lab"><font color="Green" title="E137">A255</font></span>: 
<font color="Maroon" title="c36">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c33">Pk</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c37">k2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_3.html#T49" title="PROB_3:th.49">PROB_3:49</a></span>;<br/>
<a NAME="E9:56_25_8"/>
<font color="Maroon" title="c36">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_3.html#K5" title="PROB_3:func.5">Union</a> <font color="Maroon" title="c32">Pr</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_25_8"><span class="lab"><font color="Green" title="E134">A252</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c38">k1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E11:56_25_8"/><span class="lab"><font color="Green" title="E138">A256</font></span>: 
<font color="Maroon" title="c38">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="c32">Pr</font>
 <span class="kw">and </span><br/><a NAME="E12:56_25_8"/><span class="lab"><font color="Green" title="E139">A257</font></span>: 
<font color="Maroon" title="c36">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c32">Pr</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c38">k1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_3.html#T49" title="PROB_3:th.49">PROB_3:49</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c39">k1</font> = <font color="Maroon" title="c38">k1</font>, <font color="Maroon" title="c40">k2</font> = <font color="Maroon" title="c37">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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c41">n1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c30">B</font><span class="kw"> such that </span><br/><a NAME="E15:56_25_8"/>
<font color="Maroon" title="c41">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c40">k2</font>
 <span class="kw">and </span><br/><a NAME="E16:56_25_8"/><span class="lab"><font color="Green" title="E140">A258</font></span>: 
<font color="Maroon" title="c33">Pk</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c40">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="c41">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="c29">A</font> : <font color="Maroon" title="c31">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">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:56_25"><span class="lab"><font color="Green" title="E129">A216</font></span></a>, <a class="txt" href="#E7:56_25_8"><span class="lab"><font color="Green" title="E136">A254</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c42">n2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c29">A</font><span class="kw"> such that </span><br/><a NAME="E18:56_25_8"/><span class="lab"><font color="Green" title="E141">A259</font></span>: 
<font color="Maroon" title="c36">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="c42">n2</font>,<font color="Maroon" title="c41">n1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <span class="kw">and </span><br/><a NAME="E19:56_25_8"/><span class="lab"><font color="Green" title="E142">A260</font></span>: 
<font color="Maroon" title="c31">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c42">n2</font>,<font color="Maroon" title="c41">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E8:56_25_8"><span class="lab"><font color="Green" title="E137">A255</font></span></a>, <a class="txt" href="#E16:56_25_8"><span class="lab"><font color="Green" title="E140">A258</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c43">m1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c29">A</font><span class="kw"> such that </span><br/><a NAME="E21:56_25_8"/>
<font color="Maroon" title="c43">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">k1</font>
 <span class="kw">and </span><br/><a NAME="E22:56_25_8"/><span class="lab"><font color="Green" title="E143">A261</font></span>: 
<font color="Maroon" title="c32">Pr</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c39">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="c43">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="c30">B</font> : <font color="Maroon" title="c31">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c43">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:56_25"><span class="lab"><font color="Green" title="E125">A188</font></span></a>, <a class="txt" href="#E11:56_25_8"><span class="lab"><font color="Green" title="E138">A256</font></span></a></span>;<br/>
<a NAME="E23:56_25_8"/><span class="lab"><font color="Green" title="E144">A262</font></span>: 
 ex <font color="Olive" title="b1">m2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c30">B</font> st <br/>( <font color="Maroon" title="c36">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">m1</font>,<font color="Olive" title="b1">m2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c31">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c43">m1</font>,<font color="Olive" title="b1">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>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:56_25_8"><span class="lab"><font color="Green" title="E139">A257</font></span></a>, <a class="txt" href="#E22:56_25_8"><span class="lab"><font color="Green" title="E143">A261</font></span></a></span>;<br/>
<a NAME="E24:56_25_8"/><span class="kw">then </span>
<font color="Maroon" title="c43">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">n2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:56_25_8"><span class="lab"><font color="Green" title="E141">A259</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E25:56_25_8"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:56_25_8"><span class="lab"><font color="Green" title="E144">A262</font></span></a>, <a class="txt" href="#E18:56_25_8"><span class="lab"><font color="Green" title="E141">A259</font></span></a>, <a class="txt" href="#E19:56_25_8"><span class="lab"><font color="Green" title="E142">A260</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E20:56_25"/><span class="lab"><font color="Green" title="E134">A263</font></span>: 
 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="c33">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="c33">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="c33">Pk</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</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="c33">Pk</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Olive" title="b2">e</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_25_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c36">d</font>, <font color="Maroon" title="c37">e</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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="c33">Pk</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="c33">Pk</font> &amp; <font color="Maroon" title="c36">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c37">e</font> implies <font color="Maroon" title="c33">Pk</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c36">d</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c33">Pk</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c37">e</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:56_25_9"/><span class="lab"><font color="Green" title="E135">A264</font></span>: 
<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="c33">Pk</font>
 <span class="kw">and </span><br/><a NAME="E2:56_25_9"/><span class="lab"><font color="Green" title="E136">A265</font></span>: 
<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="c33">Pk</font>
 <span class="kw">and </span><br/><a NAME="E3:56_25_9"/><span class="lab"><font color="Green" title="E137">A266</font></span>: 
<font color="Maroon" title="c36">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c37">e</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c33">Pk</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c36">d</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c33">Pk</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c37">e</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c38">y2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c30">B</font><span class="kw"> such that </span><br/><a NAME="E5:56_25_9"/><span class="lab"><font color="Green" title="E138">A267</font></span>: 
<font color="Maroon" title="c38">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">e</font>
 <span class="kw">and </span><br/><a NAME="E6:56_25_9"/><span class="lab"><font color="Green" title="E139">A268</font></span>: 
<font color="Maroon" title="c33">Pk</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c37">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="c38">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="c29">A</font> : <font color="Maroon" title="c31">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">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:56_25"><span class="lab"><font color="Green" title="E129">A216</font></span></a>, <a class="txt" href="#E2:56_25_9"><span class="lab"><font color="Green" title="E136">A265</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c39">y1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c30">B</font><span class="kw"> such that </span><br/><a NAME="E8:56_25_9"/><span class="lab"><font color="Green" title="E140">A269</font></span>: 
<font color="Maroon" title="c39">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c36">d</font>
 <span class="kw">and </span><br/><a NAME="E9:56_25_9"/><span class="lab"><font color="Green" title="E141">A270</font></span>: 
<font color="Maroon" title="c33">Pk</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c36">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="c39">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="c29">A</font> : <font color="Maroon" title="c31">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="c39">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:56_25"><span class="lab"><font color="Green" title="E129">A216</font></span></a>, <a class="txt" href="#E1:56_25_9"><span class="lab"><font color="Green" title="E135">A264</font></span></a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_25_9_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:56_25_9_1"/>
 not <font color="Maroon" title="c33">Pk</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c36">d</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c33">Pk</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c37">e</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c40">l</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:56_25_9_1"/><span class="lab"><font color="Green" title="E142">A271</font></span>: 
<font color="Maroon" title="c40">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c33">Pk</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c36">d</font>
 <span class="kw">and </span><br/><a NAME="E4:56_25_9_1"/><span class="lab"><font color="Green" title="E143">A272</font></span>: 
<font color="Maroon" title="c40">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c33">Pk</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c37">e</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><a NAME="E5:56_25_9_1"/><span class="lab"><font color="Green" title="E144">A273</font></span>: 
 ex <font color="Olive" title="b1">x2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c29">A</font> st <br/>( <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="Olive" title="b1">x2</font>,<font color="Maroon" title="c38">y2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c31">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b1">x2</font>,<font color="Maroon" title="c38">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E6:56_25_9"><span class="lab"><font color="Green" title="E139">A268</font></span></a>, <a class="txt" href="#E4:56_25_9_1"><span class="lab"><font color="Green" title="E143">A272</font></span></a></span>;<br/><a NAME="E6:56_25_9_1"/>
 ex <font color="Olive" title="b1">x1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c29">A</font> st <br/>( <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="Olive" title="b1">x1</font>,<font color="Maroon" title="c39">y1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c31">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b1">x1</font>,<font color="Maroon" title="c39">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E9:56_25_9"><span class="lab"><font color="Green" title="E141">A270</font></span></a>, <a class="txt" href="#E3:56_25_9_1"><span class="lab"><font color="Green" title="E142">A271</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:56_25_9_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_25_9"><span class="lab"><font color="Green" title="E137">A266</font></span></a>, <a class="txt" href="#E8:56_25_9"><span class="lab"><font color="Green" title="E140">A269</font></span></a>, <a class="txt" href="#E5:56_25_9"><span class="lab"><font color="Green" title="E138">A267</font></span></a>, <a class="txt" href="#E5:56_25_9_1"><span class="lab"><font color="Green" title="E144">A273</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E11:56_25_9"/>
<font color="Maroon" title="c33">Pk</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c36">d</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c33">Pk</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c37">e</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E21:56_25"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c33">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:56_25"><span class="lab"><font color="Green" title="E129">A216</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E22:56_25"/><span class="kw">then </span><span class="lab"><font color="Green" title="E135">A274</font></span>: 
 <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#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c33">Pk</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">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="c33">Pk</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:56_25"><span class="lab"><font color="Green" title="E134">A263</font></span></a>, <a class="ref" href="int_5.html#T48" target="_self" title="INT_5:th.48">Th48</a></span>;<br/>
<a NAME="E23:56_25"/><span class="lab"><font color="Green" title="E136">A275</font></span>: 
 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="c32">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="c32">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="c32">Pr</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</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="c32">Pr</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Olive" title="b2">e</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_25_10"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c36">d</font>, <font color="Maroon" title="c37">e</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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="c32">Pr</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="c32">Pr</font> &amp; <font color="Maroon" title="c36">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c37">e</font> implies <font color="Maroon" title="c32">Pr</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c36">d</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c32">Pr</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c37">e</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:56_25_10"/><span class="lab"><font color="Green" title="E137">A276</font></span>: 
<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="c32">Pr</font>
 <span class="kw">and </span><br/><a NAME="E2:56_25_10"/><span class="lab"><font color="Green" title="E138">A277</font></span>: 
<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="c32">Pr</font>
 <span class="kw">and </span><br/><a NAME="E3:56_25_10"/><span class="lab"><font color="Green" title="E139">A278</font></span>: 
<font color="Maroon" title="c36">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c37">e</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c32">Pr</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c36">d</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c32">Pr</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c37">e</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c38">x2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c29">A</font><span class="kw"> such that </span><br/><a NAME="E5:56_25_10"/><span class="lab"><font color="Green" title="E140">A279</font></span>: 
<font color="Maroon" title="c38">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">e</font>
 <span class="kw">and </span><br/><a NAME="E6:56_25_10"/><span class="lab"><font color="Green" title="E141">A280</font></span>: 
<font color="Maroon" title="c32">Pr</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c37">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="c38">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="c30">B</font> : <font color="Maroon" title="c31">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c38">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:56_25"><span class="lab"><font color="Green" title="E125">A188</font></span></a>, <a class="txt" href="#E2:56_25_10"><span class="lab"><font color="Green" title="E138">A277</font></span></a></span>;<br/>
<span class="kw">consider </span><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="c29">A</font><span class="kw"> such that </span><br/><a NAME="E8:56_25_10"/><span class="lab"><font color="Green" title="E142">A281</font></span>: 
<font color="Maroon" title="c39">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c36">d</font>
 <span class="kw">and </span><br/><a NAME="E9:56_25_10"/><span class="lab"><font color="Green" title="E143">A282</font></span>: 
<font color="Maroon" title="c32">Pr</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c36">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="c30">B</font> : <font color="Maroon" title="c31">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:56_25"><span class="lab"><font color="Green" title="E125">A188</font></span></a>, <a class="txt" href="#E1:56_25_10"><span class="lab"><font color="Green" title="E137">A276</font></span></a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_25_10_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:56_25_10_1"/>
 not <font color="Maroon" title="c32">Pr</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c36">d</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c32">Pr</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c37">e</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c40">l</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:56_25_10_1"/><span class="lab"><font color="Green" title="E144">A283</font></span>: 
<font color="Maroon" title="c40">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c32">Pr</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c36">d</font>
 <span class="kw">and </span><br/><a NAME="E4:56_25_10_1"/><span class="lab"><font color="Green" title="E145">A284</font></span>: 
<font color="Maroon" title="c40">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c32">Pr</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c37">e</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><a NAME="E5:56_25_10_1"/><span class="lab"><font color="Green" title="E146">A285</font></span>: 
 ex <font color="Olive" title="b1">y2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c30">B</font> st <br/>( <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="c38">x2</font>,<font color="Olive" title="b1">y2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c31">z</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c38">x2</font>,<font color="Olive" title="b1">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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:56_25_10"><span class="lab"><font color="Green" title="E141">A280</font></span></a>, <a class="txt" href="#E4:56_25_10_1"><span class="lab"><font color="Green" title="E145">A284</font></span></a></span>;<br/><a NAME="E6:56_25_10_1"/>
 ex <font color="Olive" title="b1">y1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c30">B</font> st <br/>( <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="c39">x1</font>,<font color="Olive" title="b1">y1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c31">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">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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:56_25_10"><span class="lab"><font color="Green" title="E143">A282</font></span></a>, <a class="txt" href="#E3:56_25_10_1"><span class="lab"><font color="Green" title="E144">A283</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:56_25_10_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_25_10"><span class="lab"><font color="Green" title="E139">A278</font></span></a>, <a class="txt" href="#E8:56_25_10"><span class="lab"><font color="Green" title="E142">A281</font></span></a>, <a class="txt" href="#E5:56_25_10"><span class="lab"><font color="Green" title="E140">A279</font></span></a>, <a class="txt" href="#E5:56_25_10_1"><span class="lab"><font color="Green" title="E146">A285</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E11:56_25_10"/>
<font color="Maroon" title="c32">Pr</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c36">d</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c32">Pr</font> <a href="prob_3.html#K4" title="PROB_3:func.4">.</a> <font color="Maroon" title="c37">e</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E24:56_25"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:56_25"><span class="lab"><font color="Green" title="E125">A188</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E25:56_25"/><span class="kw">then </span>
 <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#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c32">Pr</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">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="c32">Pr</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:56_25"><span class="lab"><font color="Green" title="E136">A275</font></span></a>, <a class="ref" href="int_5.html#T48" target="_self" title="INT_5:th.48">Th48</a></span>;<br/>
<a NAME="E26:56_25"/><span class="kw">then </span>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">U1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c35">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#K3" title="WSIERP_1:func.3">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="c32">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#K3" title="WSIERP_1:func.3">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="c33">Pk</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:56_25"><span class="lab"><font color="Green" title="E135">A274</font></span></a>, <a class="txt" href="#E19:56_25"><span class="lab"><font color="Green" title="E133">A251</font></span></a>, <a class="ref" href="card_2.html#T40" title="CARD_2:th.40">CARD_2:40</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">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="c32">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#K3" title="WSIERP_1:func.3">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="c33">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="c29">A</font>,<font color="Maroon" title="c30">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:56_25"><span class="lab"><font color="Green" title="E132">A250</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c29">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="c30">B</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T46" title="CARD_2:th.46">CARD_2:46</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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="c30">B</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E28:56_25"/>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c23">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#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c17">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="nat_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:56_25"><span class="lab"><font color="Green" title="E127">A214</font></span></a>, <a class="txt" href="#E14:56_25"><span class="lab"><font color="Green" title="E130">A217</font></span></a>, <a class="txt" href="#E15:56_25"><span class="lab"><font color="Green" title="E131">A218</font></span></a>, <a class="ref" href="finseq_1.html#T13" title="FINSEQ_1:th.13">FINSEQ_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E139:56"/>
 <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="c23">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="c23">f2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>;<br/>
<a NAME="E140:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E122">A286</font></span>: 
 <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="c23">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:56"><span class="lab"><font color="Green" title="E65">A27</font></span></a>, <a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/>
<a NAME="E141:56"/>
<font color="Maroon" title="c3">p</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c23">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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/>
<a NAME="E142:56"/><span class="kw">then </span>
<font color="Maroon" title="c3">p</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c23">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="nat_d.html#K7" title="NAT_D:func.7">-'</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E140:56"><span class="lab"><font color="Green" title="E122">A286</font></span></a></span>;<br/>
<a NAME="E143:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E123">A287</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c23">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="nat_d.html#K7" title="NAT_D:func.7">-'</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></span>;<br/>
<span class="lab"><font color="Green" title="E124">A288</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</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
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:56"><span class="lab"><font color="Green" title="E51">A9</font></span></a>, <a class="ref" href="nat_2.html#T26" title="NAT_2:th.26">NAT_2:26</a></span>
<br/>.= 
<font color="Maroon" title="c3">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:56"><span class="lab"><font color="Green" title="E48">A6</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c29">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#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E136:56"><span class="lab"><font color="Green" title="E119">A182</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c30">m</font> =  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c29">X</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c31">Y</font> = <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">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="c29">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="E147:56"/><span class="lab"><font color="Green" title="E125">A289</font></span>: 
<font color="Maroon" title="c13">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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/>
<a NAME="E148:56"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:56"><span class="lab"><font color="Green" title="E57">A17</font></span></a>, <a class="txt" href="#E110:56"><span class="lab"><font color="Green" title="E104">A95</font></span></a>, <a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/>
<a NAME="E149:56"/><span class="kw">then </span>
<font color="Maroon" title="c13">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="nat_d.html#K7" title="NAT_D:func.7">-'</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E147:56"><span class="lab"><font color="Green" title="E125">A289</font></span></a></span>;<br/>
<a NAME="E150:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E126">A290</font></span>: 
<font color="Maroon" title="c13">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="nat_d.html#K7" title="NAT_D:func.7">-'</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></span>;<br/>
<a NAME="E151:56"/><span class="lab"><font color="Green" title="E127">A291</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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="c26">n1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E117:56"><span class="lab"><font color="Green" title="E107">A98</font></span></a>, <a class="txt" href="#E121:56"><span class="lab"><font color="Green" title="E110">A107</font></span></a>, <a class="ref" href="xboole_1.html#T73" title="XBOOLE_1:th.73">XBOOLE_1:73</a></span>;<br/>
<a NAME="E152:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E128">A292</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">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#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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 class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></span>;<br/>
<a NAME="E153:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E129">A293</font></span>: 
<font color="Maroon" title="c29">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="c26">n1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E151:56"><span class="lab"><font color="Green" title="E127">A291</font></span></a>, <a class="txt" href="#E136:56"><span class="lab"><font color="Green" title="E119">A182</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="lab"><font color="Green" title="E130">A294</font></span>:  <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="c23">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="c13">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="c23">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="c13">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="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></span>
<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">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="c13">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="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">f1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E111:56"><span class="lab"><font color="Green" title="E105">A96</font></span></a>, <a class="txt" href="#E118:56"><span class="lab"><font color="Green" title="E108">A99</font></span></a></span>
;<br/>
<a NAME="E155:56"/>
 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="c13">f1</font> holds <br/><font color="Maroon" title="c13">f1</font> <a href="recdef_1.html#K1" title="RECDEF_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="c23">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="c13">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="56_28"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c32">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> <font color="Maroon" title="c13">f1</font> implies <font color="Maroon" title="c13">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c32">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="c23">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="c13">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="c32">d</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_28"/><span class="lab"><font color="Green" title="E131">A295</font></span>: 
<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> <font color="Maroon" title="c13">f1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c32">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="c23">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="c13">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="c32">d</font></span><br/>

<a NAME="E2:56_28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E132">A296</font></span>: 
<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"><font color="Maroon" title="c3">p</font> <a href="euler_2.html#K1" title="EULER_2:func.1">*</a> <font color="Maroon" title="c23">f2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E118:56"><span class="lab"><font color="Green" title="E108">A99</font></span></a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>;<br/>
<a NAME="E3:56_28"/>
<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="c23">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="c13">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="c32">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="c23">f2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c32">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="c13">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="c32">d</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E154:56"><span class="lab"><font color="Green" title="E130">A294</font></span></a>, <a class="txt" href="#E1:56_28"><span class="lab"><font color="Green" title="E131">A295</font></span></a>, <a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></span>;<br/>
<span class="kw">hence </span><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="c23">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="c13">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="c32">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="c23">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c32">d</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"><font color="Maroon" title="c13">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="c32">d</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_28"><span class="lab"><font color="Green" title="E132">A296</font></span></a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>
<br/>.= 
<font color="Maroon" title="c13">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c32">d</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E119:56"><span class="lab"><font color="Green" title="E109">A100</font></span></a>, <a class="txt" href="#E1:56_28"><span class="lab"><font color="Green" title="E131">A295</font></span></a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E156:56"/><span class="kw">then </span>
<font color="Maroon" title="c13">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="c23">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="c13">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E154:56"><span class="lab"><font color="Green" title="E130">A294</font></span></a>, <a class="ref" href="finseq_1.html#T13" title="FINSEQ_1:th.13">FINSEQ_1:13</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E131">A297</font></span>:  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c13">f1</font> = 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">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="c23">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#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">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="kw">by</span> <span class="lab"><a class="txt" href="#E143:56"><span class="lab"><font color="Green" title="E123">A287</font></span></a>, <a class="txt" href="#E150:56"><span class="lab"><font color="Green" title="E126">A290</font></span></a>, <a class="ref" href="rvsum_1.html#T89" title="RVSUM_1:th.89">RVSUM_1:89</a></span>
<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#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c23">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#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">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="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T87" title="RVSUM_1:th.87">RVSUM_1:87</a></span>
;<br/>
<a NAME="E158:56"/><span class="lab"><font color="Green" title="E132">A298</font></span>: 
<span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">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="c29">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">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 class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/>
<a NAME="E159:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E133">A299</font></span>: 
<font color="Maroon" title="c31">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="c26">n1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E151:56"><span class="lab"><font color="Green" title="E127">A291</font></span></a>, <a class="txt" href="#E152:56"><span class="lab"><font color="Green" title="E128">A292</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E160:56"/><span class="lab"><font color="Green" title="E134">A300</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E125:56"><span class="lab"><font color="Green" title="E113">A126</font></span></a>, <a class="txt" href="#E152:56"><span class="lab"><font color="Green" title="E128">A292</font></span></a>, <a class="ref" href="finseq_4.html#T62" title="FINSEQ_4:th.62">FINSEQ_4:62</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c32">n</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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="c29">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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:56"><span class="lab"><font color="Green" title="E58">A18</font></span></a>, <a class="txt" href="#E110:56"><span class="lab"><font color="Green" title="E104">A95</font></span></a>, <a class="txt" href="#E136:56"><span class="lab"><font color="Green" title="E119">A182</font></span></a>, <a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a>, <a class="ref" href="nat_1.html#T43" title="NAT_1:th.43">NAT_1:43</a></span>;<br/>
<a NAME="E162:56"/><span class="lab"><font color="Green" title="E135">A301</font></span>: 
 <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfinseq.html#T8" title="RFINSEQ:th.8">RFINSEQ:8</a></span>;<br/>
<a NAME="E163:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E136">A302</font></span>: 
<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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E122:56"><span class="lab"><font color="Green" title="E111">A108</font></span></a>, <a class="ref" href="finseq_3.html#T91" title="FINSEQ_3:th.91">FINSEQ_3:91</a></span>;<br/>
<a NAME="E164:56"/>
 <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T24" title="FINSEQ_2:th.24">FINSEQ_2:24</a></span>;<br/>
<a NAME="E165:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E137">A303</font></span>: 
 <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">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#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E125:56"><span class="lab"><font color="Green" title="E113">A126</font></span></a>, <a class="txt" href="#E152:56"><span class="lab"><font color="Green" title="E128">A292</font></span></a>, <a class="txt" href="#E122:56"><span class="lab"><font color="Green" title="E111">A108</font></span></a>, <a class="ref" href="rfinseq.html#T9" title="RFINSEQ:th.9">RFINSEQ:9</a>, <a class="ref" href="rfinseq.html#T26" title="RFINSEQ:th.26">RFINSEQ:26</a></span>;<br/>
<a NAME="E166:56"/>
 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="c31">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="c29">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="56_30"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c33">k</font>, <font color="Maroon" title="c34">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c33">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c31">Y</font> &amp; <font color="Maroon" title="c34">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c29">X</font> implies <font color="Maroon" title="c33">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c34">l</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:56_30"/><span class="lab"><font color="Green" title="E138">A304</font></span>: 
<font color="Maroon" title="c33">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c31">Y</font>
 <span class="kw">and </span><br/><a NAME="E2:56_30"/><span class="lab"><font color="Green" title="E139">A305</font></span>: 
<font color="Maroon" title="c34">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c29">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c33">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c34">l</font></span><br/>

<a NAME="E3:56_30"/><span class="lab"><font color="Green" title="E140">A306</font></span>: 
 not <font color="Maroon" title="c33">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c29">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_30"><span class="lab"><font color="Green" title="E138">A304</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E4:56_30"/><span class="lab"><font color="Green" title="E141">A307</font></span>: 
 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="c34">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#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">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 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_30"><span class="lab"><font color="Green" title="E139">A305</font></span></a></span>;<br/>
<a NAME="E5:56_30"/>
<font color="Maroon" title="c33">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_30"><span class="lab"><font color="Green" title="E138">A304</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E6:56_30"/><span class="kw">then </span>
<font color="Maroon" title="c33">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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_30"><span class="lab"><font color="Green" title="E140">A306</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:56_30"/>
<font color="Maroon" title="c33">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c34">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_30"><span class="lab"><font color="Green" title="E141">A307</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E167:56"/><span class="kw">then </span>
 <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">Y</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c29">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="c31">Y</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.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">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E153:56"><span class="lab"><font color="Green" title="E129">A293</font></span></a>, <a class="txt" href="#E159:56"><span class="lab"><font color="Green" title="E133">A299</font></span></a>, <a class="ref" href="finseq_3.html#T42" title="FINSEQ_3:th.42">FINSEQ_3:42</a></span>;<br/>
<a NAME="E168:56"/><span class="kw">then </span>
 <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#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c29">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="c31">Y</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.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">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T39" title="XBOOLE_1:th.39">XBOOLE_1:39</a></span>;<br/>
<a NAME="E169:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E138">A308</font></span>: 
 <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">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="c31">Y</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.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">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E152:56"><span class="lab"><font color="Green" title="E128">A292</font></span></a>, <a class="txt" href="#E136:56"><span class="lab"><font color="Green" title="E119">A182</font></span></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></span>;<br/>
<a NAME="E170:56"/><span class="kw">then </span>
 <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">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#K3" title="WSIERP_1:func.3">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="c31">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#K3" title="WSIERP_1:func.3">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">X</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T75" title="RVSUM_1:th.75">RVSUM_1:75</a></span>;<br/>
<a NAME="E171:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E139">A309</font></span>: 
<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="nat_d.html#K7" title="NAT_D:func.7">-'</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#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c23">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#K3" title="WSIERP_1:func.3">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="c31">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#K3" title="WSIERP_1:func.3">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">X</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E157:56"><span class="lab"><font color="Green" title="E131">A297</font></span></a>, <a class="txt" href="#E165:56"><span class="lab"><font color="Green" title="E137">A303</font></span></a>, <a class="ref" href="rvsum_1.html#T87" title="RVSUM_1:th.87">RVSUM_1:87</a></span>;<br/>
<span class="lab"><font color="Green" title="E140">A310</font></span>:  <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="c31">Y</font></span>)</span> = 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c31">Y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E151:56"><span class="lab"><font color="Green" title="E127">A291</font></span></a>, <a class="txt" href="#E152:56"><span class="lab"><font color="Green" title="E128">A292</font></span></a>, <a class="txt" href="#E158:56"><span class="lab"><font color="Green" title="E132">A298</font></span></a>, <a class="ref" href="finseq_3.html#T39" title="FINSEQ_3:th.39">FINSEQ_3:39</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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="c29">X</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:56"><span class="lab"><font color="Green" title="E58">A18</font></span></a>, <a class="txt" href="#E110:56"><span class="lab"><font color="Green" title="E104">A95</font></span></a>, <a class="txt" href="#E136:56"><span class="lab"><font color="Green" title="E119">A182</font></span></a>, <a class="txt" href="#E160:56"><span class="lab"><font color="Green" title="E134">A300</font></span></a>, <a class="ref" href="card_2.html#T44" title="CARD_2:th.44">CARD_2:44</a></span>
;<br/>
<a NAME="E173:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E141">A311</font></span>: 
<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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c29">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E169:56"><span class="lab"><font color="Green" title="E138">A308</font></span></a>, <a class="ref" href="finseq_5.html#T37" title="FINSEQ_5:th.37">FINSEQ_5:37</a></span>;<br/>
<a NAME="E174:56"/><span class="lab"><font color="Green" title="E142">A312</font></span>: 
<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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c31">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E169:56"><span class="lab"><font color="Green" title="E138">A308</font></span></a>, <a class="txt" href="#E172:56"><span class="lab"><font color="Green" title="E140">A310</font></span></a>, <a class="ref" href="finseq_3.html#T113" title="FINSEQ_3:th.113">FINSEQ_3:113</a>, <a class="ref" href="finseq_6.html#T10" title="FINSEQ_6:th.10">FINSEQ_6:10</a></span>;<br/>
<a NAME="E175:56"/><span class="lab"><font color="Green" title="E143">A313</font></span>: 
<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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E122:56"><span class="lab"><font color="Green" title="E111">A108</font></span></a>, <a class="txt" href="#E162:56"><span class="lab"><font color="Green" title="E135">A301</font></span></a>, <a class="ref" href="finseq_3.html#T91" title="FINSEQ_3:th.91">FINSEQ_3:91</a></span>;<br/>
<a NAME="E176:56"/>
 <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#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c23">f2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_32"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c33">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="c29">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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c34">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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">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="c32">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="c29">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font></span>)</span></span>)</span>;<br/>
<a NAME="E1:56_32"/><span class="lab"><font color="Green" title="E144">A314</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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="nat_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T45" title="RELAT_1:th.45">RELAT_1:45</a></span>;<br/>
<a NAME="E2:56_32"/><span class="lab"><font color="Green" title="E145">A315</font></span>: 
<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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">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="c32">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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T24" title="FINSEQ_2:th.24">FINSEQ_2:24</a></span>;<br/>
<a NAME="E3:56_32"/><span class="lab"><font color="Green" title="E146">A316</font></span>: 
 <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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E127:56"><span class="lab"><font color="Green" title="E114">A127</font></span></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a>, <a class="ref" href="xreal_1.html#T43" title="XREAL_1:th.43">XREAL_1:43</a></span>;<br/>
<span class="lab"><font color="Green" title="E147">A317</font></span>:  <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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">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="c32">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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c32">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfinseq.html#T29" title="RFINSEQ:th.29">RFINSEQ:29</a></span>
<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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E127:56"><span class="lab"><font color="Green" title="E114">A127</font></span></a>, <a class="ref" href="xreal_1.html#T43" title="XREAL_1:th.43">XREAL_1:43</a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>
<br/>.= 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c29">X</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E127:56"><span class="lab"><font color="Green" title="E114">A127</font></span></a></span>
;<br/>
<span class="lab"><font color="Green" title="E148">A318</font></span>:  <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="c29">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">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="c29">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T12" title="VALUED_1:th.12">VALUED_1:12</a></span>
<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="c29">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<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#K2" title="RELSET_1:func.2">rng</a> <span class="p0">(<span class="default"><font color="Maroon" title="c13">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="c32">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_32"><span class="lab"><font color="Green" title="E147">A317</font></span></a>, <a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>
<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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font></span>)</span>

;<br/>
<a NAME="E6:56_32"/><span class="kw">then </span><span class="lab"><font color="Green" title="E149">A319</font></span>: 
 <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="c29">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/>
<a NAME="E7:56_32"/><span class="lab"><font color="Green" title="E150">A320</font></span>: 
 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="c29">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">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="c29">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">d</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_32_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c35">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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="c29">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font></span>)</span></span>)</span> implies <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="c29">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">n</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="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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">d</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_32_3"/><span class="lab"><font color="Green" title="E151">A321</font></span>: 
<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="c29">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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="c29">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">n</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="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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">d</font></span>)</span></span><br/>

<a NAME="E2:56_32_3"/><span class="kw">then </span>
<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="c29">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_32"><span class="lab"><font color="Green" title="E147">A317</font></span></a>, <a class="txt" href="#E5:56_32"><span class="lab"><font color="Green" title="E148">A318</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E3:56_32_3"/><span class="kw">then </span>
<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="c29">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="recdef_1.html#K1" title="RECDEF_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="c3">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T57" title="FINSEQ_2:th.57">FINSEQ_2:57</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:56_32_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="c29">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">n</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="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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">d</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_32_3"><span class="lab"><font color="Green" title="E151">A321</font></span></a>, <a class="ref" href="valued_1.html#T13" title="VALUED_1:th.13">VALUED_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:56_32"/><span class="lab"><font color="Green" title="E151">A322</font></span>: 
 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="c29">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">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="c29">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">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="c29">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">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="nat_d.html#K7" title="NAT_D:func.7">-'</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="56_32_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c35">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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="c29">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font></span>)</span></span>)</span> implies ( <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="c29">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">n</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="c29">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">n</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="c3">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 ) )</span><br/>

<span class="kw">reconsider </span><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="c29">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">n</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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:56_32_4"/><span class="lab"><font color="Green" title="E152">A323</font></span>: 
<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="c29">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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="c29">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">n</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="c29">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">n</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="c3">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 )</span><br/>

<a NAME="E3:56_32_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c29">X</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_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#K2" title="RELSET_1:func.2">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">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E173:56"><span class="lab"><font color="Green" title="E141">A311</font></span></a>, <a class="txt" href="#E5:56_32"><span class="lab"><font color="Green" title="E148">A318</font></span></a>, <a class="ref" href="funct_1.html#T3" title="FUNCT_1:th.3">FUNCT_1:3</a></span>;<br/>
<a NAME="E4:56_32_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c29">X</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_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="c29">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E153:56"><span class="lab"><font color="Green" title="E129">A293</font></span></a>, <a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></span>;<br/>
<a NAME="E5:56_32_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E153">A324</font></span>: 
 ex <font color="Olive" title="b1">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>  st <br/>( <font color="Olive" title="b1">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="c29">X</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">d</font> &amp; <font color="Olive" title="b1">ll</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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="b1">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 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E152:56"><span class="lab"><font color="Green" title="E128">A292</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c37">e</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E7:56_32_4"/><span class="lab"><font color="Green" title="E154">A325</font></span>: 
<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> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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 class="kw">and </span><br/><a NAME="E8:56_32_4"/><span class="lab"><font color="Green" title="E155">A326</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">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="c37">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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E173:56"><span class="lab"><font color="Green" title="E141">A311</font></span></a>, <a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></span>;<br/>
<a NAME="E9:56_32_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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_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="c13">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c37">e</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E111:56"><span class="lab"><font color="Green" title="E105">A96</font></span></a>, <a class="txt" href="#E7:56_32_4"><span class="lab"><font color="Green" title="E154">A325</font></span></a>, <a class="txt" href="#E8:56_32_4"><span class="lab"><font color="Green" title="E155">A326</font></span></a>, <a class="ref" href="euler_2.html#D1" title="EULER_2:def.1">EULER_2:def 1</a></span>;<br/>
<a NAME="E10:56_32_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E156">A327</font></span>: 
<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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_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="c3">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T1" title="NAT_D:th.1">NAT_D:1</a></span>;<br/>
<a NAME="E11:56_32_4"/><span class="lab"><font color="Green" title="E157">A328</font></span>: 
<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="c29">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">n</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="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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">d</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:56_32"><span class="lab"><font color="Green" title="E150">A320</font></span></a>, <a class="txt" href="#E2:56_32_4"><span class="lab"><font color="Green" title="E152">A323</font></span></a></span>;<br/>
<a NAME="E12:56_32_4"/><span class="kw">then </span>
<font color="Maroon" title="c36">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E173:56"><span class="lab"><font color="Green" title="E141">A311</font></span></a>, <a class="txt" href="#E5:56_32_4"><span class="lab"><font color="Green" title="E153">A324</font></span></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:56_32_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="c29">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">n</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="c29">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">n</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="c3">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E144:56"><span class="lab"><font color="Green" title="E124">A288</font></span></a>, <a class="txt" href="#E11:56_32_4"><span class="lab"><font color="Green" title="E157">A328</font></span></a>, <a class="txt" href="#E10:56_32_4"><span class="lab"><font color="Green" title="E156">A327</font></span></a>, <a class="ref" href="int_1.html#T54" title="INT_1:th.54">INT_1:54</a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E9:56_32"/>
 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="c29">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">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="c29">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">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="56_32_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c35">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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="c29">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font></span>)</span></span>)</span> implies <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="c29">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">n</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>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_32_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="c29">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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="c29">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">n</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> </span><br/>

<a NAME="E2:56_32_5"/><span class="kw">then </span>
<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="c29">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">n</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:56_32"><span class="lab"><font color="Green" title="E151">A322</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:56_32_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="c29">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">n</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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T3" title="INT_1:th.3">INT_1:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c35">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="c29">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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">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="c32">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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T12" title="FINSEQ_2:th.12">FINSEQ_2:12</a></span>;<br/>
<a NAME="E11:56_32"/>
<font color="Maroon" title="c35">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/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c36">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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">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="c32">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="c29">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T75" title="FINSEQ_1:th.75">FINSEQ_1:75</a></span>;<br/>
<a NAME="E13:56_32"/><span class="lab"><font color="Green" title="E152">A329</font></span>: 
<font color="Maroon" title="c36">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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T24" title="FINSEQ_2:th.24">FINSEQ_2:24</a></span>;<br/>
<a NAME="E14:56_32"/><span class="lab"><font color="Green" title="E153">A330</font></span>: 
<font color="Maroon" title="c32">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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E127:56"><span class="lab"><font color="Green" title="E114">A127</font></span></a>, <a class="ref" href="xreal_1.html#T43" title="XREAL_1:th.43">XREAL_1:43</a></span>;<br/>
<a NAME="E15:56_32"/><span class="lab"><font color="Green" title="E154">A331</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c35">f5</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_32_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:56_32_6"/>
 not  <a href="relset_1.html#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c35">f5</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c37">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:56_32_6"/><span class="lab"><font color="Green" title="E155">A332</font></span>: 
<font color="Maroon" title="c37">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:56_32_6"/><span class="lab"><font color="Green" title="E156">A333</font></span>: 
<font color="Maroon" title="c37">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c35">f5</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c38">e</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E6:56_32_6"/><span class="lab"><font color="Green" title="E157">A334</font></span>: 
<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="c35">f5</font>
 <span class="kw">and </span><br/><a NAME="E7:56_32_6"/><span class="lab"><font color="Green" title="E158">A335</font></span>: 
<font color="Maroon" title="c35">f5</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c38">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_32_6"><span class="lab"><font color="Green" title="E156">A333</font></span></a>, <a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></span>;<br/>
<a NAME="E8:56_32_6"/>
<font color="Maroon" title="c37">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c38">e</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:56_32"><span class="lab"><font color="Green" title="E150">A320</font></span></a>, <a class="txt" href="#E6:56_32_6"><span class="lab"><font color="Green" title="E157">A334</font></span></a>, <a class="txt" href="#E7:56_32_6"><span class="lab"><font color="Green" title="E158">A335</font></span></a></span>;<br/>
<a NAME="E9:56_32_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E159">A336</font></span>: 
<font color="Maroon" title="c37">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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">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="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c38">e</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c32">n</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:56_32"><span class="lab"><font color="Green" title="E153">A330</font></span></a>, <a class="txt" href="#E5:56_32"><span class="lab"><font color="Green" title="E148">A318</font></span></a>, <a class="txt" href="#E6:56_32_6"><span class="lab"><font color="Green" title="E157">A334</font></span></a>, <a class="ref" href="rfinseq.html#D1" title="RFINSEQ:def.1">RFINSEQ:def 1</a></span>;<br/>
<a NAME="E10:56_32_6"/>
<font color="Maroon" title="c38">e</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c32">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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E5:56_32"><span class="lab"><font color="Green" title="E148">A318</font></span></a>, <a class="txt" href="#E6:56_32_6"><span class="lab"><font color="Green" title="E157">A334</font></span></a>, <a class="ref" href="finseq_5.html#T26" title="FINSEQ_5:th.26">FINSEQ_5:26</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c39">e1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E12:56_32_6"/><span class="lab"><font color="Green" title="E160">A337</font></span>: 
<font color="Maroon" title="c39">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="c13">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 class="kw">and </span><br/><a NAME="E13:56_32_6"/><span class="lab"><font color="Green" title="E161">A338</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">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="c39">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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">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="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">e</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c32">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E152:56"><span class="lab"><font color="Green" title="E128">A292</font></span></a>, <a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a>, <a class="ref" href="funct_1.html#T3" title="FUNCT_1:th.3">FUNCT_1:3</a></span>;<br/>
<a NAME="E14:56_32_6"/><span class="lab"><font color="Green" title="E162">A339</font></span>: 
<font color="Maroon" title="c39">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:56"><span class="lab"><font color="Green" title="E58">A18</font></span></a>, <a class="txt" href="#E110:56"><span class="lab"><font color="Green" title="E104">A95</font></span></a>, <a class="txt" href="#E12:56_32_6"><span class="lab"><font color="Green" title="E160">A337</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E15:56_32_6"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">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 class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T19" title="FINSEQ_5:th.19">FINSEQ_5:19</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c40">d1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E17:56_32_6"/><span class="lab"><font color="Green" title="E163">A340</font></span>: 
<font color="Maroon" title="c40">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="c13">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 class="kw">and </span><br/><a NAME="E18:56_32_6"/><span class="lab"><font color="Green" title="E164">A341</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">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="c40">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E152:56"><span class="lab"><font color="Green" title="E128">A292</font></span></a>, <a class="txt" href="#E3:56_32_6"><span class="lab"><font color="Green" title="E155">A332</font></span></a>, <a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></span>;<br/>
<a NAME="E19:56_32_6"/>
<font color="Maroon" title="c40">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:56"><span class="lab"><font color="Green" title="E58">A18</font></span></a>, <a class="txt" href="#E110:56"><span class="lab"><font color="Green" title="E104">A95</font></span></a>, <a class="txt" href="#E17:56_32_6"><span class="lab"><font color="Green" title="E163">A340</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E20:56_32_6"/><span class="kw">then </span>
<font color="Maroon" title="c40">d1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c39">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="nat_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:56_32_6"><span class="lab"><font color="Green" title="E162">A339</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/>
<a NAME="E21:56_32_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E165">A342</font></span>: 
<font color="Maroon" title="c40">d1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c39">e1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:56"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E13:56"><span class="lab"><font color="Green" title="E53">A11</font></span></a>, <a class="ref" href="xreal_1.html#T146" title="XREAL_1:th.146">XREAL_1:146</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E22:56_32_6"/>
<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="c13">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c40">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E111:56"><span class="lab"><font color="Green" title="E105">A96</font></span></a>, <a class="txt" href="#E17:56_32_6"><span class="lab"><font color="Green" title="E163">A340</font></span></a>, <a class="txt" href="#E18:56_32_6"><span class="lab"><font color="Green" title="E164">A341</font></span></a>, <a class="ref" href="euler_2.html#D1" title="EULER_2:def.1">EULER_2:def 1</a></span>;<br/>
<a NAME="E23:56_32_6"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c40">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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">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="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c38">e</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c32">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:56_32_6"><span class="lab"><font color="Green" title="E159">A336</font></span></a></span>;<br/>
<a NAME="E24:56_32_6"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c40">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="c13">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c39">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E111:56"><span class="lab"><font color="Green" title="E105">A96</font></span></a>, <a class="txt" href="#E12:56_32_6"><span class="lab"><font color="Green" title="E160">A337</font></span></a>, <a class="txt" href="#E13:56_32_6"><span class="lab"><font color="Green" title="E161">A338</font></span></a>, <a class="ref" href="euler_2.html#D1" title="EULER_2:def.1">EULER_2:def 1</a></span>;<br/>
<a NAME="E25:56_32_6"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c40">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="c13">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c39">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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T25" title="NAT_D:th.25">NAT_D:25</a></span>;<br/>
<a NAME="E26:56_32_6"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c40">d1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c39">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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euler_2.html#T6" title="EULER_2:th.6">EULER_2:6</a></span>;<br/>
<a NAME="E27:56_32_6"/><span class="kw">then </span>
<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="c13">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c40">d1</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="c13">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c39">e1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pepin.html#T6" title="PEPIN:th.6">PEPIN:6</a></span>;<br/>
<a NAME="E28:56_32_6"/><span class="kw">then </span>
<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="c40">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="c13">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c39">e1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:56"><span class="lab"><font color="Green" title="E55">A13</font></span></a>, <a class="txt" href="#E111:56"><span class="lab"><font color="Green" title="E105">A96</font></span></a>, <a class="txt" href="#E17:56_32_6"><span class="lab"><font color="Green" title="E163">A340</font></span></a></span>;<br/>
<a NAME="E29:56_32_6"/><span class="kw">then </span>
<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="c40">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="c39">e1</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c4">q</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:56"><span class="lab"><font color="Green" title="E55">A13</font></span></a>, <a class="txt" href="#E111:56"><span class="lab"><font color="Green" title="E105">A96</font></span></a>, <a class="txt" href="#E12:56_32_6"><span class="lab"><font color="Green" title="E160">A337</font></span></a></span>;<br/>
<a NAME="E30:56_32_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E166">A343</font></span>: 
<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="c40">d1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c39">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="E31:56_32_6"/>
<font color="Maroon" title="c40">d1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:56_32_6"><span class="lab"><font color="Green" title="E163">A340</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E32:56_32_6"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="txt" href="#E30:56_32_6"><span class="lab"><font color="Green" title="E166">A343</font></span></a>, <a class="txt" href="#E21:56_32_6"><span class="lab"><font color="Green" title="E165">A342</font></span></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a>, <a class="ref" href="pepin.html#T3" title="PEPIN:th.3">PEPIN:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E16:56_32"/>
 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">f5</font> holds <br/><font color="Maroon" title="c35">f5</font> <a href="recdef_1.html#K1" title="RECDEF_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">f5</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">e</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_32_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c37">d</font>, <font color="Maroon" title="c38">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c37">d</font> &amp; <font color="Maroon" title="c37">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c38">e</font> &amp; <font color="Maroon" title="c38">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">f5</font> implies <font color="Maroon" title="c35">f5</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c37">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c35">f5</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c38">e</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:56_32_7"/><span class="lab"><font color="Green" title="E155">A344</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c37">d</font>
 <span class="kw">and </span><br/><a NAME="E2:56_32_7"/><span class="lab"><font color="Green" title="E156">A345</font></span>: 
<font color="Maroon" title="c37">d</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c38">e</font>
 <span class="kw">and </span><br/><a NAME="E3:56_32_7"/><span class="lab"><font color="Green" title="E157">A346</font></span>: 
<font color="Maroon" title="c38">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">f5</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c35">f5</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c37">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c35">f5</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c38">e</font></span><br/>

<a NAME="E4:56_32_7"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c38">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_32_7"><span class="lab"><font color="Green" title="E155">A344</font></span></a>, <a class="txt" href="#E2:56_32_7"><span class="lab"><font color="Green" title="E156">A345</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E5:56_32_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E158">A347</font></span>: 
<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="c35">f5</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_32_7"><span class="lab"><font color="Green" title="E157">A346</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E6:56_32_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E159">A348</font></span>: 
<font color="Maroon" title="c35">f5</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c38">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c38">e</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:56_32"><span class="lab"><font color="Green" title="E150">A320</font></span></a></span>;<br/>
<a NAME="E7:56_32_7"/>
<font color="Maroon" title="c37">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">f5</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_32_7"><span class="lab"><font color="Green" title="E156">A345</font></span></a>, <a class="txt" href="#E3:56_32_7"><span class="lab"><font color="Green" title="E157">A346</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E8:56_32_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E160">A349</font></span>: 
<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="c35">f5</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_32_7"><span class="lab"><font color="Green" title="E155">A344</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E9:56_32_7"/><span class="kw">then </span>
<font color="Maroon" title="c35">f5</font> <a href="recdef_1.html#K1" title="RECDEF_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="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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c37">d</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:56_32"><span class="lab"><font color="Green" title="E150">A320</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:56_32_7"/>
<font color="Maroon" title="c35">f5</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c37">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c35">f5</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c38">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E163:56"><span class="lab"><font color="Green" title="E136">A302</font></span></a>, <a class="txt" href="#E5:56_32"><span class="lab"><font color="Green" title="E148">A318</font></span></a>, <a class="txt" href="#E2:56_32_7"><span class="lab"><font color="Green" title="E156">A345</font></span></a>, <a class="txt" href="#E8:56_32_7"><span class="lab"><font color="Green" title="E160">A349</font></span></a>, <a class="txt" href="#E5:56_32_7"><span class="lab"><font color="Green" title="E158">A347</font></span></a>, <a class="txt" href="#E6:56_32_7"><span class="lab"><font color="Green" title="E159">A348</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E17:56_32"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c35">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#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c35">f5</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="graph_5.html#T7" title="GRAPH_5:th.7">GRAPH_5:7</a></span>;<br/>
<a NAME="E18:56_32"/><span class="kw">then </span>
<font color="Maroon" title="c35">f5</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T62" title="FINSEQ_4:th.62">FINSEQ_4:62</a></span>;<br/>
<a NAME="E19:56_32"/><span class="kw">then </span><span class="lab"><font color="Green" title="E155">A350</font></span>: 
<font color="Maroon" title="c36">f6</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E175:56"><span class="lab"><font color="Green" title="E143">A313</font></span></a>, <a class="txt" href="#E15:56_32"><span class="lab"><font color="Green" title="E154">A331</font></span></a>, <a class="ref" href="finseq_3.html#T91" title="FINSEQ_3:th.91">FINSEQ_3:91</a></span>;<br/>
<a NAME="E20:56_32"/><span class="lab"><font color="Green" title="E156">A351</font></span>: 
 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">f6</font> holds <br/>( <font color="Maroon" title="c36">f6</font> <a href="recdef_1.html#K1" title="RECDEF_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">f6</font> <a href="recdef_1.html#K1" title="RECDEF_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="nat_d.html#K7" title="NAT_D:func.7">-'</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="56_32_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c37">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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">f6</font> implies ( <font color="Maroon" title="c36">f6</font> <a href="recdef_1.html#K1" title="RECDEF_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">f6</font> <a href="recdef_1.html#K1" title="RECDEF_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="c3">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_32_8"/><span class="lab"><font color="Green" title="E157">A352</font></span>: 
<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">f6</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c36">f6</font> <a href="recdef_1.html#K1" title="RECDEF_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">f6</font> <a href="recdef_1.html#K1" title="RECDEF_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="c3">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_32_8_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:56_32_8_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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">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="c32">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="c35">f5</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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_32_8"><span class="lab"><font color="Green" title="E157">A352</font></span></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_32_8_1_1"><span class="kw">suppose </span></a><a NAME="E1:56_32_8_1_1"/><span class="lab"><font color="Green" title="E158">A353</font></span>: 
<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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c36">f6</font> <a href="recdef_1.html#K1" title="RECDEF_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">f6</font> <a href="recdef_1.html#K1" title="RECDEF_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="c3">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 )</span><br/><div class="add"><a NAME="E2:56_32_8_1_1"/><span class="kw">then </span>
<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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_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#K2" title="RELSET_1:func.2">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="c31">Y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E174:56"><span class="lab"><font color="Green" title="E142">A312</font></span></a>, <a class="ref" href="funct_1.html#T3" title="FUNCT_1:th.3">FUNCT_1:3</a></span>;<br/><a NAME="E3:56_32_8_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E159">A354</font></span>: 
<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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_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="c31">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E159:56"><span class="lab"><font color="Green" title="E133">A299</font></span></a>, <a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></span>;<br/><a NAME="E4:56_32_8_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E160">A355</font></span>: 
<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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_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#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">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 class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E5:56_32_8_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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_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">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_32_8_1_1"><span class="lab"><font color="Green" title="E159">A354</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E6:56_32_8_1_1"/><span class="kw">then </span>
<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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_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="c3">p</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_32_8_1_1"><span class="lab"><font color="Green" title="E160">A355</font></span></a></span>;<br/><a NAME="E7:56_32_8_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E161">A356</font></span>: 
<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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_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="c3">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E144:56"><span class="lab"><font color="Green" title="E124">A288</font></span></a>, <a class="ref" href="int_1.html#T54" title="INT_1:th.54">INT_1:54</a></span>;<br/><a NAME="E8:56_32_8_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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E121:56"><span class="lab"><font color="Green" title="E110">A107</font></span></a>, <a class="txt" href="#E152:56"><span class="lab"><font color="Green" title="E128">A292</font></span></a>, <a class="txt" href="#E4:56_32_8_1_1"><span class="lab"><font color="Green" title="E160">A355</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><a NAME="E9:56_32_8_1_1"/><span class="kw">then </span>
<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#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E10:56_32_8_1_1"/>
( <font color="Maroon" title="c36">f6</font> <a href="recdef_1.html#K1" title="RECDEF_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">f6</font> <a href="recdef_1.html#K1" title="RECDEF_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="c3">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_32_8_1_1"><span class="lab"><font color="Green" title="E158">A353</font></span></a>, <a class="txt" href="#E7:56_32_8_1_1"><span class="lab"><font color="Green" title="E161">A356</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_32_8_1_2"><span class="kw">suppose </span></a><a NAME="E1:56_32_8_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">f5</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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c36">f6</font> <a href="recdef_1.html#K1" title="RECDEF_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">f6</font> <a href="recdef_1.html#K1" title="RECDEF_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="c3">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 )</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E3:56_32_8_1_2"/><span class="lab"><font color="Green" title="E158">A357</font></span>: 
<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">f5</font>
 <span class="kw">and </span><br/><a NAME="E4:56_32_8_1_2"/><span class="lab"><font color="Green" title="E159">A358</font></span>: 
<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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c38">l</font>
 ;<br/><a NAME="E5:56_32_8_1_2"/>
<font color="Maroon" title="c36">f6</font> <a href="recdef_1.html#K1" title="RECDEF_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">f5</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c38">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_32_8_1_2"><span class="lab"><font color="Green" title="E158">A357</font></span></a>, <a class="txt" href="#E4:56_32_8_1_2"><span class="lab"><font color="Green" title="E159">A358</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E6:56_32_8_1_2"/>
( <font color="Maroon" title="c36">f6</font> <a href="recdef_1.html#K1" title="RECDEF_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">f6</font> <a href="recdef_1.html#K1" title="RECDEF_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="c3">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:56_32"><span class="lab"><font color="Green" title="E151">A322</font></span></a>, <a class="txt" href="#E3:56_32_8_1_2"><span class="lab"><font color="Green" title="E158">A357</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c36">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">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="c35">f5</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" title="FINSEQ_1:th.22">FINSEQ_1:22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_32"><span class="lab"><font color="Green" title="E146">A316</font></span></a>, <a class="txt" href="#E4:56_32"><span class="lab"><font color="Green" title="E147">A317</font></span></a>, <a class="txt" href="#E6:56_32"><span class="lab"><font color="Green" title="E149">A319</font></span></a></span>
;<br/>
<a NAME="E22:56_32"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c36">f6</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_32"><span class="lab"><font color="Green" title="E144">A314</font></span></a>, <a class="txt" href="#E19:56_32"><span class="lab"><font color="Green" title="E155">A350</font></span></a>, <a class="txt" href="#E20:56_32"><span class="lab"><font color="Green" title="E156">A351</font></span></a>, <a class="ref" href="int_5.html#T40" target="_self" title="INT_5:th.40">Th40</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c27">M</font> = 
 <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c36">f6</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:56_32"><span class="lab"><font color="Green" title="E155">A350</font></span></a>, <a class="txt" href="#E13:56_32"><span class="lab"><font color="Green" title="E152">A329</font></span></a>, <a class="ref" href="rfinseq.html#T9" title="RFINSEQ:th.9">RFINSEQ:9</a>, <a class="ref" href="rfinseq.html#T26" title="RFINSEQ:th.26">RFINSEQ:26</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">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#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c35">f5</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T75" title="RVSUM_1:th.75">RVSUM_1:75</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">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="c29">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#K3" title="WSIERP_1:func.3">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#K2" title="RELSET_1:func.2">rng</a> <span class="p0">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_32"><span class="lab"><font color="Green" title="E147">A317</font></span></a>, <a class="txt" href="#E2:56_32"><span class="lab"><font color="Green" title="E145">A315</font></span></a>, <a class="ref" href="int_5.html#T47" target="_self" title="INT_5:th.47">Th47</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">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#K2" title="RELSET_1:func.2">rng</a> <span class="p0">(<span class="default"><font color="Maroon" title="c13">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="c32">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="c29">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#K3" title="WSIERP_1:func.3">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#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c13">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="c32">n</font></span>)</span></span>)</span>

;<br/>
<a NAME="E24:56_32"/><span class="kw">then </span>
<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="c27">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#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c23">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#K3" title="WSIERP_1:func.3">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="c29">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="c29">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E171:56"><span class="lab"><font color="Green" title="E139">A309</font></span></a>, <a class="txt" href="#E173:56"><span class="lab"><font color="Green" title="E141">A311</font></span></a>, <a class="txt" href="#E174:56"><span class="lab"><font color="Green" title="E142">A312</font></span></a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E157">A359</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c27">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#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c23">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="c29">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#K3" title="WSIERP_1:func.3">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="c29">X</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:56"><span class="lab"><font color="Green" title="E66">A28</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>
<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#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c23">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="c29">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
<span class="kw">by</span> <span class="lab"><a class="ref" href="euler_1.html#T12" title="EULER_1:th.12">EULER_1:12</a></span>
;<br/>
<a NAME="E26:56_32"/>
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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c27">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:56"><span class="lab"><font color="Green" title="E75">A39</font></span></a>, <a class="ref" href="nat_d.html#T9" title="NAT_D:th.9">NAT_D:9</a></span>;<br/>
<a NAME="E27:56_32"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c27">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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pepin.html#T6" title="PEPIN:th.6">PEPIN:6</a></span>;<br/>
<a NAME="E28:56_32"/><span class="kw">then </span>
2 <a href="int_1.html#R1" title="INT_1:pred.1">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#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c23">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="c29">X</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:56_32"><span class="lab"><font color="Green" title="E157">A359</font></span></a>, <a class="txt" href="int_5.html#E3"><span class="lab"><font color="Green" title="E3">Lm1</font></span></a></span>;<br/>
<a NAME="E29:56_32"/><span class="kw">then </span>
2 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c23">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="c29">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E134:56"><span class="lab"><font color="Green" title="E118">A181</font></span></a>, <a class="ref" href="int_2.html#T25" title="INT_2:th.25">INT_2:25</a></span>;<br/>
<a NAME="E30:56_32"/><span class="kw">then </span>
 <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c23">f2</font>, <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c29">X</font> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T15" title="INT_2:th.15">INT_2:15</a></span>;<br/>
<a NAME="E31:56_32"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c23">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="c29">X</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T64" title="NAT_D:th.64">NAT_D:64</a></span>;<br/>
<a NAME="E32:56_32"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c23">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="c29">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_5.html#T45" target="_self" title="INT_5:th.45">Th45</a></span>;<br/>
<span class="kw">hence </span><a NAME="E33:56_32"/>
 <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#K3" title="WSIERP_1:func.3">Sum</a> <font color="Maroon" title="c23">f2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="txt" href="#E5:56"><span class="lab"><font color="Green" title="E47">A5</font></span></a>, <a class="txt" href="#E152:56"><span class="lab"><font color="Green" title="E128">A292</font></span></a>, <a class="ref" href="int_5.html#T41" target="_self" title="INT_5:th.41">Th41</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E177:56"/>
<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="nat_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E130:56"><span class="lab"><font color="Green" title="E117">A130</font></span></a>, <a class="txt" href="#E138:56"><span class="lab"><font color="Green" title="E121">A184</font></span></a>, <a class="ref" href="newton.html#T8" title="NEWTON:th.8">NEWTON:8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
