<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">p</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">p</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  &amp; <font color="Maroon" title="c1">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies  ex <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> )</span><br/>

<a NAME="E1:32"/>
 <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c1">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">p9</font> = <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></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="int_1.html#T16" title="INT_1:th.16">INT_1:16</a>, <a class="ref" href="int_1.html#T81" title="INT_1:th.81">INT_1:81</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c3">p99</font> = <font color="Maroon" title="c2">p9</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">X</font> =  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font>;<br/>
<span class="lab"><font color="Green" title="E31">A1</font></span>:  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font></span>)</span> = 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">p99</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D12" title="CARD_1:def.12">CARD_1:def 12</a></span>
<br/>.= 
1 <a href="binop_2.html#K20" title="BINOP_2:func.20">+</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"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c1">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="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a></span>
;<br/>
<a NAME="E5:32"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A2</font></span>: 
<span class="p1">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T52" title="INT_1:th.52">INT_1:52</a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></span>;<br/>
<span class="kw">assume </span><a NAME="E6:32"/><span class="lab"><font color="Green" title="E33">A3</font></span>: 
<font color="Maroon" title="c1">p</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c1">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  ex <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> )</span><br/>

<a NAME="E7:32"/><span class="kw">then </span>
<font color="Maroon" title="c1">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#D5" title="INT_2:def.5">INT_2:def 5</a></span>;<br/>
<a NAME="E8:32"/><span class="kw">then </span>
<font color="Maroon" title="c1">p</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E9:32"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A4</font></span>: 
<font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E10:32"/><span class="lab"><font color="Green" title="E35">A5</font></span>: 
<font color="Maroon" title="c2">p9</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:32_2"/>
<font color="Maroon" title="c2">p9</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">p99</font> =  <a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c1">p</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="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></span>;<br/>
<a NAME="E3:32_2"/>
<font color="Maroon" title="c5">p99</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></span>;<br/>
<a NAME="E4:32_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A6</font></span>: 
<font color="Maroon" title="c5">p99</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c5">p99</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#D3" title="SQUARE_1:def.3">SQUARE_1:def 3</a></span>;<br/>
<a NAME="E5:32_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A7</font></span>: 
<font color="Maroon" title="c5">p99</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D9" title="INT_1:def.9">INT_1:def 9</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_2_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="E1:32_2_1"/>
( <font color="Maroon" title="c5">p99</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c5">p99</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="txt" href="#E5:32_2"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:32_2_1_1"/>
<font color="Maroon" title="c5">p99</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:32_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="txt" href="#E4:32_2"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="32_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:32_2_1_2"/>
<font color="Maroon" title="c5">p99</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">then </span>1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c1">p</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="txt" href="#E4:32_2"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></span>
<br/>.= 
<font color="Maroon" title="c1">p</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c1">p</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T75" title="XCMPLX_1:th.75">XCMPLX_1:75</a></span>
<br/>.= 
<font color="Maroon" title="c1">p</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:32_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="E11:32"/>
<font color="Maroon" title="c2">p9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c1">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D4" title="INT_1:def.4">INT_1:def 4</a></span>;<br/>
<a NAME="E12:32"/><span class="kw">then </span>
<font color="Maroon" title="c2">p9</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#T77" title="SQUARE_1:th.77">SQUARE_1:77</a></span>;<br/>
<a NAME="E13:32"/><span class="kw">then </span>
<font color="Maroon" title="c2">p9</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></span>;<br/>
<a NAME="E14:32"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A8</font></span>: 
<font color="Maroon" title="c2">p9</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:32"><span class="lab"><font color="Green" title="E35">A5</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">assume </span><a NAME="E15:32"/><span class="lab"><font color="Green" title="E37">A9</font></span>: 
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span></span><br/>

<a NAME="E16:32"/><span class="kw">then </span>
<font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T24" title="NAT_D:th.24">NAT_D:24</a></span>;<br/>
<a NAME="E17:32"/><span class="kw">then </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">by</span> <span class="lab"><a class="txt" href="#E9:32"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E18:32"/><span class="kw">then </span>
 <a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1 <a href="int_5.html#R1" title="INT_5:pred.1">is_quadratic_residue_mod</a> <font color="Maroon" title="c1">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="txt" href="#E15:32"><span class="lab"><font color="Green" title="E37">A9</font></span></a>, <a class="ref" href="int_5.html#T37" title="INT_5:th.37">INT_5:37</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">s</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E20:32"/><span class="lab"><font color="Green" title="E38">A10</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_5.html#D2" title="INT_5:def.2">INT_5:def 2</a></span>;<br/>
<a NAME="E21:32"/>
 <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c1">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="square_1.html#T93" title="SQUARE_1:th.93">SQUARE_1:93</a></span>;<br/>
<a NAME="E22:32"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c1">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="ref" href="int_1.html#T52" title="INT_1:th.52">INT_1:52</a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></span>;<br/>
<a NAME="E23:32"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E24:32"/><span class="kw">then </span>
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#T97" title="SQUARE_1:th.97">SQUARE_1:97</a></span>;<br/>
<a NAME="E25:32"/><span class="kw">then </span>
 <a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#D3" title="SQUARE_1:def.3">SQUARE_1:def 3</a></span>;<br/>
<a NAME="E26:32"/><span class="kw">then </span>
<font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#T89" title="SQUARE_1:th.89">SQUARE_1:89</a></span>;<br/>
<a NAME="E27:32"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A11</font></span>: 
<font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font></span>)</span>,<span class="p2">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font></span>)</span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T65" title="CARD_2:th.65">CARD_2:65</a></span>;<br/>
<a NAME="E28:32"/>
for <font color="Olive" title="b1">s</font> being  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   ex <font color="Olive" title="b2">x9</font>, <font color="Olive" title="b3">x99</font>, <font color="Olive" title="b4">y9</font>, <font color="Olive" title="b5">y99</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b2">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font> &amp; <font color="Olive" title="b3">x99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font> &amp; <font color="Olive" title="b4">y9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font> &amp; <font color="Olive" title="b5">y99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">x9</font>,<font color="Olive" title="b4">y9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">x99</font>,<font color="Olive" title="b5">y99</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b2">x9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">s</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Olive" title="b4">y9</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">x99</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">s</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Olive" title="b5">y99</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">p</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c6">A</font> = <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font></span>)</span>,<span class="p2">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font></span>)</span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">p9</font> = <font color="Maroon" title="c1">p</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#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c8">s</font> be   <a href="int_1.html#V1" title="INT_1:attr.1">integer</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x9</font>, <font color="Olive" title="b2">x99</font>, <font color="Olive" title="b3">y9</font>, <font color="Olive" title="b4">y99</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font> &amp; <font color="Olive" title="b2">x99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font> &amp; <font color="Olive" title="b3">y9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font> &amp; <font color="Olive" title="b4">y99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x9</font>,<font color="Olive" title="b3">y9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">x99</font>,<font color="Olive" title="b4">y99</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">x9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Olive" title="b3">y9</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">x99</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Olive" title="b4">y99</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">p</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c9">B</font> =  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c7">p9</font>;<br/>
<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">n1</font>, <font color="Olive" title="b2">n2</font>, <font color="Olive" title="b3">n3</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/>( $1 <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">n1</font>,<font color="Olive" title="b2">n2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">n3</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">n1</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Olive" title="b2">n2</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">n3</font> );<br/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c7">p9</font></span>)</span> = 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">p9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D12" title="CARD_1:def.12">CARD_1:def 12</a></span>
<br/>.= 
<font color="Maroon" title="c1">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a></span>
;<br/>
<a NAME="E3:32_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A12</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c7">p9</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font></span>)</span>,<span class="p2">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font></span>)</span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:32"><span class="lab"><font color="Green" title="E39">A11</font></span></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/>
<a NAME="E4:32_3"/><span class="lab"><font color="Green" title="E41">A13</font></span>: 
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"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font></span>)</span>,<span class="p2">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font></span>)</span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c7">p9</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font></span>)</span>,<span class="p2">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font></span>)</span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> implies  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c7">p9</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">x</font>,<font color="Olive" title="b1">y</font>] ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:32_3_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font></span>)</span>,<span class="p2">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font></span>)</span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c7">p9</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">n1</font>, <font color="Maroon" title="c12">n2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:32_3_2"/><span class="lab"><font color="Green" title="E42">A14</font></span>: 
( <font color="Maroon" title="c11">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font> &amp; <font color="Maroon" title="c12">n2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font> )
 <span class="kw">and </span><br/><a NAME="E4:32_3_2"/><span class="lab"><font color="Green" title="E43">A15</font></span>: 
<font color="Maroon" title="c10">x</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="c11">n1</font>,<font color="Maroon" title="c12">n2</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="c13">n1</font> = <font color="Maroon" title="c11">n1</font>, <font color="Maroon" title="c14">n2</font> = <font color="Maroon" title="c12">n2</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="#E3:32_3_2"><span class="lab"><font color="Green" title="E42">A14</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">y</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c13">n1</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c14">n2</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">p</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">y</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c13">n1</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c14">n2</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">p</font> 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="int_1.html#T16" title="INT_1:th.16">INT_1:16</a>, <a class="ref" href="int_1.html#T84" title="INT_1:th.84">INT_1:84</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c16">y</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c7">p9</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c16">y</font>] )</span><br/>

<a NAME="E7:32_3_2"/>
<font color="Maroon" title="c16">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="int_1.html#T85" title="INT_1:th.85">INT_1:85</a></span>;<br/>
<a NAME="E8:32_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:32_3_2"/>
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c7">p9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D12" title="CARD_1:def.12">CARD_1:def 12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c16">y</font>]</span><br/>

<span class="kw">thus </span><a NAME="E10:32_3_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c16">y</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_3_2"><span class="lab"><font color="Green" title="E43">A15</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="c10">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font></span>)</span>,<span class="p2">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font></span>)</span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>,<span class="p1">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c7">p9</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E6:32_3"/><span class="lab"><font color="Green" title="E42">A16</font></span>: 
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"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font></span>)</span>,<span class="p2">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font></span>)</span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c10">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="#E4:32_3"><span class="lab"><font color="Green" title="E41">A13</font></span></a>);<br/></span>
<a NAME="E7:32_3"/>
 <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c7">p9</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="#E6:32"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="algseq_1.html#T10" title="ALGSEQ_1:th.10">ALGSEQ_1:10</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">x1</font>, <font color="Maroon" title="c12">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E9:32_3"/><span class="lab"><font color="Green" title="E43">A17</font></span>: 
<font color="Maroon" title="c11">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font></span>)</span>,<span class="p2">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font></span>)</span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 <span class="kw">and </span><br/><a NAME="E10:32_3"/><span class="lab"><font color="Green" title="E44">A18</font></span>: 
<font color="Maroon" title="c12">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font></span>)</span>,<span class="p2">(<span class="default"><a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font></span>)</span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 <span class="kw">and </span><br/><a NAME="E11:32_3"/><span class="lab"><font color="Green" title="E45">A19</font></span>: 
<font color="Maroon" title="c11">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">x2</font>
 <span class="kw">and </span><br/><a NAME="E12:32_3"/><span class="lab"><font color="Green" title="E46">A20</font></span>: 
<font color="Maroon" title="c10">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_3"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="ref" href="finseq_4.html#T80" title="FINSEQ_4:th.80">FINSEQ_4:80</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">x9</font>, <font color="Maroon" title="c14">y9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E14:32_3"/><span class="lab"><font color="Green" title="E47">A21</font></span>: 
( <font color="Maroon" title="c13">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font> &amp; <font color="Maroon" title="c14">y9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font> )
 <span class="kw">and </span><br/><a NAME="E15:32_3"/><span class="lab"><font color="Green" title="E48">A22</font></span>: 
<font color="Maroon" title="c11">x1</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="c13">x9</font>,<font color="Maroon" title="c14">y9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32_3"><span class="lab"><font color="Green" title="E43">A17</font></span></a>, <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="c15">x9</font> = <font color="Maroon" title="c13">x9</font>, <font color="Maroon" title="c16">y9</font> = <font color="Maroon" title="c14">y9</font> as   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:32_3"><span class="lab"><font color="Green" title="E47">A21</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c17">x99</font>, <font color="Maroon" title="c18">y99</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E18:32_3"/><span class="lab"><font color="Green" title="E49">A23</font></span>: 
( <font color="Maroon" title="c17">x99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font> &amp; <font color="Maroon" title="c18">y99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font> )
 <span class="kw">and </span><br/><a NAME="E19:32_3"/><span class="lab"><font color="Green" title="E50">A24</font></span>: 
<font color="Maroon" title="c12">x2</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="c17">x99</font>,<font color="Maroon" title="c18">y99</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:32_3"><span class="lab"><font color="Green" title="E44">A18</font></span></a>, <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="c19">x99</font> = <font color="Maroon" title="c17">x99</font>, <font color="Maroon" title="c20">y99</font> = <font color="Maroon" title="c18">y99</font> as   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:32_3"><span class="lab"><font color="Green" title="E49">A23</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c15">x9</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x99</font>, <font color="Olive" title="b2">y9</font>, <font color="Olive" title="b3">y99</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Maroon" title="c15">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font> &amp; <font color="Olive" title="b1">x99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font> &amp; <font color="Olive" title="b2">y9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font> &amp; <font color="Olive" title="b3">y99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">x9</font>,<font color="Olive" title="b2">y9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x99</font>,<font color="Olive" title="b3">y99</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c15">x9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Olive" title="b2">y9</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x99</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Olive" title="b3">y99</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">p</font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c19">x99</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y9</font>, <font color="Olive" title="b2">y99</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Maroon" title="c15">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font> &amp; <font color="Maroon" title="c19">x99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font> &amp; <font color="Olive" title="b1">y9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font> &amp; <font color="Olive" title="b2">y99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">x9</font>,<font color="Olive" title="b1">y9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">x99</font>,<font color="Olive" title="b2">y99</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c15">x9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Olive" title="b1">y9</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">x99</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Olive" title="b2">y99</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">p</font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c16">y9</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y99</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Maroon" title="c15">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font> &amp; <font color="Maroon" title="c19">x99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font> &amp; <font color="Maroon" title="c16">y9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font> &amp; <font color="Olive" title="b1">y99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">x9</font>,<font color="Maroon" title="c16">y9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">x99</font>,<font color="Olive" title="b1">y99</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c15">x9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c16">y9</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">x99</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Olive" title="b1">y99</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">p</font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c20">y99</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font> &amp; <font color="Maroon" title="c19">x99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font> &amp; <font color="Maroon" title="c16">y9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font> &amp; <font color="Maroon" title="c20">y99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">x9</font>,<font color="Maroon" title="c16">y9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">x99</font>,<font color="Maroon" title="c20">y99</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c15">x9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c16">y9</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">x99</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c20">y99</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">p</font> )</span><br/>

<span class="kw">thus </span><a NAME="E21:32_3"/>
( <font color="Maroon" title="c15">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font> &amp; <font color="Maroon" title="c19">x99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font> &amp; <font color="Maroon" title="c16">y9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font> &amp; <font color="Maroon" title="c20">y99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:32_3"><span class="lab"><font color="Green" title="E47">A21</font></span></a>, <a class="txt" href="#E18:32_3"><span class="lab"><font color="Green" title="E49">A23</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">x9</font>,<font color="Maroon" title="c16">y9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">x99</font>,<font color="Maroon" title="c20">y99</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c15">x9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c16">y9</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">x99</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c20">y99</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">p</font> )</span><br/>

<span class="kw">thus </span><a NAME="E22:32_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">x9</font>,<font color="Maroon" title="c16">y9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">x99</font>,<font color="Maroon" title="c20">y99</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:32_3"><span class="lab"><font color="Green" title="E45">A19</font></span></a>, <a class="txt" href="#E15:32_3"><span class="lab"><font color="Green" title="E48">A22</font></span></a>, <a class="txt" href="#E19:32_3"><span class="lab"><font color="Green" title="E50">A24</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c16">y9</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">x99</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c20">y99</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">p</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c21">n11</font>, <font color="Maroon" title="c22">n12</font>, <font color="Maroon" title="c23">n13</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="E24:32_3"/><span class="lab"><font color="Green" title="E51">A25</font></span>: 
<font color="Maroon" title="c11">x1</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="c21">n11</font>,<font color="Maroon" title="c22">n12</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <span class="kw">and </span><br/><a NAME="E25:32_3"/><span class="lab"><font color="Green" title="E52">A26</font></span>: 
( <font color="Maroon" title="c10">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n13</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c21">n11</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c22">n12</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n13</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32_3"><span class="lab"><font color="Green" title="E42">A16</font></span></a>, <a class="txt" href="#E9:32_3"><span class="lab"><font color="Green" title="E43">A17</font></span></a></span>;<br/>
<a NAME="E26:32_3"/><span class="lab"><font color="Green" title="E53">A27</font></span>: 
( <font color="Maroon" title="c21">n11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x9</font> &amp; <font color="Maroon" title="c22">n12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">y9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:32_3"><span class="lab"><font color="Green" title="E48">A22</font></span></a>, <a class="txt" href="#E24:32_3"><span class="lab"><font color="Green" title="E51">A25</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c24">n21</font>, <font color="Maroon" title="c25">n22</font>, <font color="Maroon" title="c26">n23</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="E28:32_3"/><span class="lab"><font color="Green" title="E54">A28</font></span>: 
<font color="Maroon" title="c12">x2</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="c24">n21</font>,<font color="Maroon" title="c25">n22</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <span class="kw">and </span><br/><a NAME="E29:32_3"/><span class="lab"><font color="Green" title="E55">A29</font></span>: 
( <font color="Maroon" title="c10">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">n23</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c24">n21</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c25">n22</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">n23</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32_3"><span class="lab"><font color="Green" title="E42">A16</font></span></a>, <a class="txt" href="#E10:32_3"><span class="lab"><font color="Green" title="E44">A18</font></span></a></span>;<br/>
<a NAME="E30:32_3"/>
<font color="Maroon" title="c24">n21</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">x99</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:32_3"><span class="lab"><font color="Green" title="E50">A24</font></span></a>, <a class="txt" href="#E28:32_3"><span class="lab"><font color="Green" title="E54">A28</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/>
<span class="kw">hence </span><a NAME="E31:32_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c15">x9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c16">y9</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">x99</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c20">y99</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:32_3"><span class="lab"><font color="Green" title="E46">A20</font></span></a>, <a class="txt" href="#E19:32_3"><span class="lab"><font color="Green" title="E50">A24</font></span></a>, <a class="txt" href="#E25:32_3"><span class="lab"><font color="Green" title="E52">A26</font></span></a>, <a class="txt" href="#E28:32_3"><span class="lab"><font color="Green" title="E54">A28</font></span></a>, <a class="txt" href="#E29:32_3"><span class="lab"><font color="Green" title="E55">A29</font></span></a>, <a class="txt" href="#E26:32_3"><span class="lab"><font color="Green" title="E53">A27</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">consider </span><font color="Maroon" title="c6">x9</font>, <font color="Maroon" title="c7">x99</font>, <font color="Maroon" title="c8">y9</font>, <font color="Maroon" title="c9">y99</font> being   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E30:32"/><span class="lab"><font color="Green" title="E40">A30</font></span>: 
<font color="Maroon" title="c6">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font>
 <span class="kw">and </span><br/><a NAME="E31:32"/><span class="lab"><font color="Green" title="E41">A31</font></span>: 
<font color="Maroon" title="c7">x99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font>
 <span class="kw">and </span><br/><a NAME="E32:32"/><span class="lab"><font color="Green" title="E42">A32</font></span>: 
<font color="Maroon" title="c8">y9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font>
 <span class="kw">and </span><br/><a NAME="E33:32"/><span class="lab"><font color="Green" title="E43">A33</font></span>: 
<font color="Maroon" title="c9">y99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c3">p99</font>
 <span class="kw">and </span><br/><a NAME="E34:32"/><span class="lab"><font color="Green" title="E44">A34</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x9</font>,<font color="Maroon" title="c8">y9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x99</font>,<font color="Maroon" title="c9">y99</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E35:32"/><span class="lab"><font color="Green" title="E45">A35</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">x9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c8">y9</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x99</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c9">y99</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">p</font>
 ;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">m9</font> = <font color="Maroon" title="c8">y9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c9">y99</font>;<br/>
<span class="lab"><font color="Green" title="E46">A36</font></span>:  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">y9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c9">y99</font></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">p</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:32"><span class="lab"><font color="Green" title="E38">A10</font></span></a>, <a class="ref" href="int_4.html#T12" title="INT_4:th.12">INT_4:12</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="c5">s</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">y9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c9">y99</font></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_3.html#T15" title="INT_3:th.15">INT_3:15</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="c5">s</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">y9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c9">y99</font></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">y9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c9">y99</font></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">p</font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">y9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c9">y99</font></span>)</span></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">y9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c9">y99</font></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#T68" title="SQUARE_1:th.68">SQUARE_1:68</a></span>
;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">n9</font> = <font color="Maroon" title="c6">x9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c7">x99</font>;<br/>
<span class="lab"><font color="Green" title="E47">A37</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c7">x99</font></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">y9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c9">y99</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c7">x99</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">y9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c9">y99</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c7">x99</font></span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> 2</span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">y9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c9">y99</font></span>)</span></span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> 2</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fib_num3.html#T7" title="FIB_NUM3:th.7">FIB_NUM3:7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c7">x99</font></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">y9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c9">y99</font></span>)</span></span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> 2</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T53" title="POWER:th.53">POWER:53</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c7">x99</font></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">y9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c9">y99</font></span>)</span></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T53" title="POWER:th.53">POWER:53</a></span>
;<br/>
<a NAME="E38:32"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c8">y9</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">x99</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c9">y99</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:32"><span class="lab"><font color="Green" title="E45">A35</font></span></a></span>;<br/>
<a NAME="E39:32"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c8">y9</font></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">x99</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c9">y99</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_6.html#T7" title="INT_6:th.7">INT_6:7</a></span>;<br/>
<a NAME="E40:32"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c7">x99</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">y9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c9">y99</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_4.html#T12" title="INT_4:th.12">INT_4:12</a></span>;<br/>
<span class="kw">then </span> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">x9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c7">x99</font></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">y9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c9">y99</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">x9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c7">x99</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">y9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c9">y99</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_4.html#T12" title="INT_4:th.12">INT_4:12</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="c6">x9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c7">x99</font></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">y9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c9">y99</font></span>)</span></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">p</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:32"><span class="lab"><font color="Green" title="E47">A37</font></span></a>, <a class="ref" href="int_3.html#T15" title="INT_3:th.15">INT_3:15</a></span>
;<br/>
<span class="kw">then </span> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">x9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c7">x99</font></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c8">y9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c9">y99</font></span>)</span></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c8">y9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c9">y99</font></span>)</span></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">y9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c9">y99</font></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">p</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:32"><span class="lab"><font color="Green" title="E46">A36</font></span></a>, <a class="ref" href="int_4.html#T12" title="INT_4:th.12">INT_4:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">x9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c7">x99</font></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">y9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c9">y99</font></span>)</span></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">y9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c9">y99</font></span>)</span></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">y9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c9">y99</font></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_3.html#T14" title="INT_3:th.14">INT_3:14</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="c6">x9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c7">x99</font></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">y9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c9">y99</font></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">p</font>

;<br/>
<a NAME="E43:32"/><span class="kw">then </span>
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c7">x99</font></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">y9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c9">y99</font></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="int_1.html#T89" title="INT_1:th.89">INT_1:89</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E45:32"/><span class="lab"><font color="Green" title="E48">A38</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c7">x99</font></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">y9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c9">y99</font></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c12">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D9" title="INT_1:def.9">INT_1:def 9</a></span>;<br/>
<span class="lab"><font color="Green" title="E49">A39</font></span>: <font color="Maroon" title="c1">p</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c12">i</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">x9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c7">x99</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">y9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c9">y99</font></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:32"><span class="lab"><font color="Green" title="E48">A38</font></span></a>, <a class="ref" href="complex1.html#T161" title="COMPLEX1:th.161">COMPLEX1:161</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c7">x99</font></span>)</span></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">y9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c9">y99</font></span>)</span></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T161" title="COMPLEX1:th.161">COMPLEX1:161</a></span>
;<br/>
<a NAME="E47:32"/>
<font color="Maroon" title="c8">y9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">p99</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:32"><span class="lab"><font color="Green" title="E42">A32</font></span></a>, <a class="ref" href="card_1.html#D12" title="CARD_1:def.12">CARD_1:def 12</a></span>;<br/>
<a NAME="E48:32"/><span class="kw">then </span>
<font color="Maroon" title="c8">y9</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">p99</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/>
<a NAME="E49:32"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A40</font></span>: 
<font color="Maroon" title="c8">y9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">p9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E50:32"/>
<font color="Maroon" title="c7">x99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">p99</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:32"><span class="lab"><font color="Green" title="E41">A31</font></span></a>, <a class="ref" href="card_1.html#D12" title="CARD_1:def.12">CARD_1:def 12</a></span>;<br/>
<a NAME="E51:32"/><span class="kw">then </span>
<font color="Maroon" title="c7">x99</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">p99</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/>
<a NAME="E52:32"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A41</font></span>: 
<font color="Maroon" title="c7">x99</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">p9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E53:32"/>
<font color="Maroon" title="c9">y99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">p99</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:32"><span class="lab"><font color="Green" title="E43">A33</font></span></a>, <a class="ref" href="card_1.html#D12" title="CARD_1:def.12">CARD_1:def 12</a></span>;<br/>
<a NAME="E54:32"/><span class="kw">then </span>
<font color="Maroon" title="c9">y99</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">p99</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/>
<a NAME="E55:32"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A42</font></span>: 
<font color="Maroon" title="c9">y99</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">p9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E56:32"/>
<font color="Maroon" title="c6">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">p99</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:32"><span class="lab"><font color="Green" title="E40">A30</font></span></a>, <a class="ref" href="card_1.html#D12" title="CARD_1:def.12">CARD_1:def 12</a></span>;<br/>
<a NAME="E57:32"/><span class="kw">then </span>
<font color="Maroon" title="c6">x9</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">p99</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/>
<a NAME="E58:32"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A43</font></span>: 
<font color="Maroon" title="c6">x9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">p9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">m</font> =  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c9">y99</font></span>)</span>;<br/>
<a NAME="E59:32"/><span class="lab"><font color="Green" title="E54">A44</font></span>: 
( <font color="Maroon" title="c8">y9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c9">y99</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#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">n</font> =  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c7">x99</font></span>)</span>;<br/>
<a NAME="E60:32"/>
( <font color="Maroon" title="c6">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c7">x99</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#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">x9</font> = <font color="Maroon" title="c6">x9</font>, <font color="Maroon" title="c16">x99</font> = <font color="Maroon" title="c7">x99</font>, <font color="Maroon" title="c17">y9</font> = <font color="Maroon" title="c8">y9</font>, <font color="Maroon" title="c18">y99</font> = <font color="Maroon" title="c9">y99</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E59:32"><span class="lab"><font color="Green" title="E54">A44</font></span></a></span>;<br/>
<a NAME="E62:32"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">y9</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c18">y99</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">p9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:32"><span class="lab"><font color="Green" title="E50">A40</font></span></a>, <a class="txt" href="#E55:32"><span class="lab"><font color="Green" title="E52">A42</font></span></a>, <a class="ref" href="jgraph_1.html#T31" title="JGRAPH_1:th.31">JGRAPH_1:31</a></span>;<br/>
<a NAME="E63:32"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c9">y99</font></span>)</span></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">p9</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#T77" title="SQUARE_1:th.77">SQUARE_1:77</a></span>;<br/>
<a NAME="E64:32"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A45</font></span>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c9">y99</font></span>)</span></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:32"><span class="lab"><font color="Green" title="E36">A8</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E65:32"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x9</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c16">x99</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">p9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:32"><span class="lab"><font color="Green" title="E53">A43</font></span></a>, <a class="txt" href="#E52:32"><span class="lab"><font color="Green" title="E51">A41</font></span></a>, <a class="ref" href="jgraph_1.html#T31" title="JGRAPH_1:th.31">JGRAPH_1:31</a></span>;<br/>
<a NAME="E66:32"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c7">x99</font></span>)</span></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">p9</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#T77" title="SQUARE_1:th.77">SQUARE_1:77</a></span>;<br/>
<a NAME="E67:32"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c7">x99</font></span>)</span></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:32"><span class="lab"><font color="Green" title="E36">A8</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E68:32"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c7">x99</font></span>)</span></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">y9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c9">y99</font></span>)</span></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">p</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c1">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E64:32"><span class="lab"><font color="Green" title="E55">A45</font></span></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/>
<a NAME="E69:32"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c1">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="txt" href="#E46:32"><span class="lab"><font color="Green" title="E49">A39</font></span></a>, <a class="ref" href="xreal_1.html#T76" title="XREAL_1:th.76">XREAL_1:76</a></span>;<br/>
<a NAME="E70:32"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c1">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T75" title="XCMPLX_1:th.75">XCMPLX_1:75</a></span>;<br/>
<a NAME="E71:32"/><span class="kw">then </span>
<font color="Maroon" title="c12">i</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c1">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T75" title="XCMPLX_1:th.75">XCMPLX_1:75</a></span>;<br/>
<a NAME="E72:32"/><span class="kw">then </span>
<font color="Maroon" title="c12">i</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></span>;<br/>
<a NAME="E73:32"/><span class="kw">then </span>
<font color="Maroon" title="c12">i</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></span>;<br/>
<a NAME="E74:32"/><span class="kw">then </span>
<font color="Maroon" title="c12">i</font> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></span>;<br/>
<a NAME="E75:32"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A46</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> 1</span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/>
<span class="kw">take </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c7">x99</font></span>)</span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">m</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon" title="c1">p</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="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c7">x99</font></span>)</span></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span></span><br/>

<span class="kw">take </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c9">y99</font></span>)</span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">p</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="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c7">x99</font></span>)</span></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">y9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c9">y99</font></span>)</span></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span></span><br/>

<a NAME="E76:32"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c7">x99</font></span>)</span></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">y9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c9">y99</font></span>)</span></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:32_9"/><span class="lab"><font color="Green" title="E57">A47</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c7">x99</font></span>)</span></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">y9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c9">y99</font></span>)</span></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:32_9"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c9">y99</font></span>)</span></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
;<br/>
<a NAME="E3:32_9"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c9">y99</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c9">y99</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#D3" title="SQUARE_1:def.3">SQUARE_1:def 3</a></span>;<br/>
<a NAME="E4:32_9"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c9">y99</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
;<br/>
<a NAME="E5:32_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A48</font></span>: 
<font color="Maroon" title="c17">y9</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c18">y99</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></span>;<br/>
<a NAME="E6:32_9"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c7">x99</font></span>)</span></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_9"><span class="lab"><font color="Green" title="E57">A47</font></span></a></span>;<br/>
<a NAME="E7:32_9"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c7">x99</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c7">x99</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#D3" title="SQUARE_1:def.3">SQUARE_1:def 3</a></span>;<br/>
<a NAME="E8:32_9"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c7">x99</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
;<br/>
<a NAME="E9:32_9"/><span class="kw">then </span>
<font color="Maroon" title="c15">x9</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c16">x99</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:32_9"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:32"><span class="lab"><font color="Green" title="E44">A34</font></span></a>, <a class="txt" href="#E5:32_9"><span class="lab"><font color="Green" title="E58">A48</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E77:32"/><span class="kw">then </span>
<font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:32"><span class="lab"><font color="Green" title="E49">A39</font></span></a></span>;<br/>
<a NAME="E78:32"/><span class="kw">then </span>
<font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></span>;<br/>
<a NAME="E79:32"/><span class="kw">then </span>
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:32"><span class="lab"><font color="Green" title="E56">A46</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="E80:32"/>
<font color="Maroon" title="c1">p</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="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">x9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c7">x99</font></span>)</span></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">y9</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c9">y99</font></span>)</span></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:32"><span class="lab"><font color="Green" title="E49">A39</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
