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

<span class="kw">let </span><font color="Maroon" title="c1">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">X</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>   st  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> &amp; ( for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> &amp; <font color="Olive" title="b3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">y</font> holds <br/> not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="int_4.html#K1" title="INT_4:func.1">Cong</a> <font color="Maroon" title="c1">m</font> ) holds <br/><font color="Olive" title="b1">X</font> <a href="int_4.html#R1" title="INT_4:pred.1">is_CRS_of</a> <font color="Maroon" title="c1">m</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">X</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> &amp; ( for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> holds <br/> not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="int_4.html#K1" title="INT_4:func.1">Cong</a> <font color="Maroon" title="c1">m</font> ) implies <font color="Maroon" title="c2">X</font> <a href="int_4.html#R1" title="INT_4:pred.1">is_CRS_of</a> <font color="Maroon" title="c1">m</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:70"/><span class="lab"><font color="Green" title="E49">A1</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font>
 <span class="kw">and </span><br/><a NAME="E2:70"/><span class="lab"><font color="Green" title="E50">A2</font></span>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> holds <br/> not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="int_4.html#K1" title="INT_4:func.1">Cong</a> <font color="Maroon" title="c1">m</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="c2">X</font> <a href="int_4.html#R1" title="INT_4:pred.1">is_CRS_of</a> <font color="Maroon" title="c1">m</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_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="E1:70_1"/>
( <font color="Maroon" title="c2">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or  not <font color="Maroon" title="c2">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1"><span class="kw">suppose </span></a><a NAME="E1:70_1_1"/>
<font color="Maroon" title="c2">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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="c2">X</font> <a href="int_4.html#R1" title="INT_4:pred.1">is_CRS_of</a> <font color="Maroon" title="c1">m</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:70_1_1"/>
<font color="Maroon" title="c2">X</font> <a href="int_4.html#R1" title="INT_4:pred.1">is_CRS_of</a> <font color="Maroon" title="c1">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="ref" href="int_4.html#T50" target="_self" title="INT_4:th.50">Th50</a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</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="70_1_2"><span class="kw">suppose </span></a><a NAME="E1:70_1_2"/>
not <font color="Maroon" title="c2">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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="c2">X</font> <a href="int_4.html#R1" title="INT_4:pred.1">is_CRS_of</a> <font color="Maroon" title="c1">m</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">X</font> = <font color="Maroon" title="c2">X</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  ;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>$2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="int_4.html#K1" title="INT_4:func.1">Cong</a> <font color="Maroon" title="c1">m</font></span>)</span>,<span class="p1">(<span class="default">$1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>;<br/><a NAME="E3:70_1_2"/><span class="lab"><font color="Green" title="E51">A3</font></span>: 
<font color="Maroon" title="c3">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E4:70_1_2"/><span class="lab"><font color="Green" title="E52">A4</font></span>: 
for <font color="Olive" title="b1">a</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">X</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">y</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c4">Y</font> =  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c1">m</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">a</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> implies  ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">X</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">a</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:70_1_2_1"/><span class="lab"><font color="Green" title="E53">A5</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</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">y</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">X</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">a</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<a NAME="E2:70_1_2_1"/>
<font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1_2_1"><span class="lab"><font color="Green" title="E53">A5</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E3:70_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A6</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T149" title="XREAL_1:th.149">XREAL_1:149</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">fp</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E5:70_1_2_1"/><span class="lab"><font color="Green" title="E55">A7</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">fp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font>
 <span class="kw">and </span><br/><a NAME="E6:70_1_2_1"/><span class="lab"><font color="Green" title="E56">A8</font></span>: 
for <font color="Olive" title="b1">b</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">fp</font> holds <br/><font color="Maroon" title="c6">fp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">and </span><br/><a NAME="E7:70_1_2_1"/><span class="lab"><font color="Green" title="E57">A9</font></span>: 
<font color="Maroon" title="c6">fp</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="ref" href="int_4.html#T51" target="_self" title="INT_4:th.51">Th51</a></span>;<br/>
<a NAME="E8:70_1_2_1"/>
for <font color="Olive" title="b1">b</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">fp</font> holds <br/><font color="Maroon" title="c6">fp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:70_1_2_1"><span class="lab"><font color="Green" title="E56">A8</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">fp</font> = <font color="Maroon" title="c6">fp</font> as    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of <font color="Maroon" title="c3">X</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T14" title="FINSEQ_2:th.14">FINSEQ_2:14</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span><font color="Maroon" title="c7">fp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="int_4.html#K1" title="INT_4:func.1">Cong</a> <font color="Maroon" title="c1">m</font></span>)</span>,$2;<br/>
<a NAME="E10:70_1_2_1"/><span class="lab"><font color="Green" title="E58">A10</font></span>: 
for <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> holds <br/> ex <font color="Olive" title="b2">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c1">m</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">c</font>,<font color="Olive" title="b2">r</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">c</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> implies  ex <font color="Olive" title="b1">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c1">m</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">c</font>,<font color="Olive" title="b1">r</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:70_1_2_1_1"/>
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</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">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c1">m</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">c</font>,<font color="Olive" title="b1">r</font>]</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c9">q</font>, <font color="Maroon" title="c10">r</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E3:70_1_2_1_1"/><span class="lab"><font color="Green" title="E59">A11</font></span>: 
<font color="Maroon" title="c7">fp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c9">q</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c10">r</font>
 <span class="kw">and </span><br/><a NAME="E4:70_1_2_1_1"/><span class="lab"><font color="Green" title="E60">A12</font></span>: 
<font color="Maroon" title="c10">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E5:70_1_2_1_1"/><span class="lab"><font color="Green" title="E61">A13</font></span>: 
<font color="Maroon" title="c10">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="ref" href="int_4.html#T13" target="_self" title="INT_4:th.13">Th13</a></span>;<br/>
<a NAME="E6:70_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">fp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">c</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">r</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70_1_2_1_1"><span class="lab"><font color="Green" title="E59">A11</font></span></a>, <a class="ref" href="int_3.html#T8" title="INT_3:th.8">INT_3:8</a></span>;<br/>
<a NAME="E7:70_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">r</font>,<font color="Maroon" title="c7">fp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">c</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="ref" href="int_3.html#T12" title="INT_3:th.12">INT_3:12</a></span>;<br/>
<a NAME="E8:70_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A14</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">r</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">fp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">c</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="int_4.html#K1" title="INT_4:func.1">Cong</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_4.html#D1" target="_self" title="INT_4:def.1">Def1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">r</font> = <font color="Maroon" title="c10">r</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="#E4:70_1_2_1_1"><span class="lab"><font color="Green" title="E60">A12</font></span></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">r</font> = <font color="Maroon" title="c11">r</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c1">m</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:70_1_2_1_1"><span class="lab"><font color="Green" title="E61">A13</font></span></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c12">r</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">S<sub>2</sub></font>[<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c12">r</font>]</span><br/>

<span class="kw">thus </span><a NAME="E11:70_1_2_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c12">r</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:70_1_2_1_1"><span class="lab"><font color="Green" title="E62">A14</font></span></a>, <a class="ref" href="eqrel_1.html#T26" title="EQREL_1:th.26">EQREL_1:26</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">reconsider </span><font color="Maroon" title="c8">Y</font> =  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> <font color="Maroon" title="c1">m</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="txt" href="#E3:70_1_2"><span class="lab"><font color="Green" title="E51">A3</font></span></a></span>;<br/>
<a NAME="E12:70_1_2_1"/><span class="lab"><font color="Green" title="E59">A15</font></span>: 
for <font color="Olive" title="b1">c</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> holds <br/> ex <font color="Olive" title="b2">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">Y</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">c</font>,<font color="Olive" title="b2">r</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:70_1_2_1"><span class="lab"><font color="Green" title="E58">A10</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">fr</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c8">Y</font><span class="kw"> such that </span><br/><a NAME="E14:70_1_2_1"/><span class="lab"><font color="Green" title="E60">A16</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">fr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> &amp; ( for <font color="Olive" title="b1">c</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">c</font>,<font color="Maroon" title="c9">fr</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">c</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_1.html#S5" title="FINSEQ_1:sch.5">FINSEQ_1:sch 5</a>(<a class="txt" href="#E12:70_1_2_1"><span class="lab"><font color="Green" title="E59">A15</font></span></a>);<br/></span>
<a NAME="E15:70_1_2_1"/>
for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">fr</font> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">fr</font> &amp; <font color="Maroon" title="c9">fr</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">fr</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x2</font> holds <br/><font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_2_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">x1</font>, <font color="Maroon" title="c11">x2</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">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">fr</font> &amp; <font color="Maroon" title="c11">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">fr</font> &amp; <font color="Maroon" title="c9">fr</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">fr</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x2</font> implies <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:70_1_2_1_2"/><span class="lab"><font color="Green" title="E61">A17</font></span>: 
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">fr</font>
 <span class="kw">and </span><br/><a NAME="E2:70_1_2_1_2"/><span class="lab"><font color="Green" title="E62">A18</font></span>: 
<font color="Maroon" title="c11">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">fr</font>
 <span class="kw">and </span><br/><a NAME="E3:70_1_2_1_2"/><span class="lab"><font color="Green" title="E63">A19</font></span>: 
<font color="Maroon" title="c9">fr</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">fr</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x2</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="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x2</font></span><br/>

<a NAME="E4:70_1_2_1_2"/>
( <font color="Maroon" title="c7">fp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="int_4.html#K1" title="INT_4:func.1">Cong</a> <font color="Maroon" title="c1">m</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">fr</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x1</font></span>)</span> &amp; <font color="Maroon" title="c7">fp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="int_4.html#K1" title="INT_4:func.1">Cong</a> <font color="Maroon" title="c1">m</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">fr</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x1</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:70_1_2_1"><span class="lab"><font color="Green" title="E60">A16</font></span></a>, <a class="txt" href="#E1:70_1_2_1_2"><span class="lab"><font color="Green" title="E61">A17</font></span></a>, <a class="txt" href="#E2:70_1_2_1_2"><span class="lab"><font color="Green" title="E62">A18</font></span></a>, <a class="txt" href="#E3:70_1_2_1_2"><span class="lab"><font color="Green" title="E63">A19</font></span></a></span>;<br/>
<a NAME="E5:70_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A20</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">fp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">fp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="int_4.html#K1" title="INT_4:func.1">Cong</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T30" title="EQREL_1:th.30">EQREL_1:30</a></span>;<br/>
<span class="kw">assume </span><a NAME="E6:70_1_2_1_2"/><span class="lab"><font color="Green" title="E65">A21</font></span>: 
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">x2</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">reconsider </span><font color="Maroon" title="c12">x1</font> = <font color="Maroon" title="c10">x1</font>, <font color="Maroon" title="c13">x2</font> = <font color="Maroon" title="c11">x2</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1_2_1_2"><span class="lab"><font color="Green" title="E61">A17</font></span></a>, <a class="txt" href="#E2:70_1_2_1_2"><span class="lab"><font color="Green" title="E62">A18</font></span></a></span>;<br/>
<a NAME="E8:70_1_2_1_2"/><span class="lab"><font color="Green" title="E66">A22</font></span>: 
<font color="Maroon" title="c12">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">fp</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:70_1_2_1"><span class="lab"><font color="Green" title="E55">A7</font></span></a>, <a class="txt" href="#E14:70_1_2_1"><span class="lab"><font color="Green" title="E60">A16</font></span></a>, <a class="txt" href="#E1:70_1_2_1_2"><span class="lab"><font color="Green" title="E61">A17</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E9:70_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A23</font></span>: 
<font color="Maroon" title="c7">fp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T13" title="FINSEQ_2:th.13">FINSEQ_2:13</a></span>;<br/>
<a NAME="E10:70_1_2_1_2"/><span class="lab"><font color="Green" title="E68">A24</font></span>: 
<font color="Maroon" title="c13">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">fp</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:70_1_2_1"><span class="lab"><font color="Green" title="E55">A7</font></span></a>, <a class="txt" href="#E14:70_1_2_1"><span class="lab"><font color="Green" title="E60">A16</font></span></a>, <a class="txt" href="#E2:70_1_2_1_2"><span class="lab"><font color="Green" title="E62">A18</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E11:70_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A25</font></span>: 
<font color="Maroon" title="c7">fp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T13" title="FINSEQ_2:th.13">FINSEQ_2:13</a></span>;<br/>
<a NAME="E12:70_1_2_1_2"/>
<font color="Maroon" title="c7">fp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">fp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:70_1_2_1"><span class="lab"><font color="Green" title="E57">A9</font></span></a>, <a class="txt" href="#E6:70_1_2_1_2"><span class="lab"><font color="Green" title="E65">A21</font></span></a>, <a class="txt" href="#E8:70_1_2_1_2"><span class="lab"><font color="Green" title="E66">A22</font></span></a>, <a class="txt" href="#E10:70_1_2_1_2"><span class="lab"><font color="Green" title="E68">A24</font></span></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:70_1_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:70"><span class="lab"><font color="Green" title="E50">A2</font></span></a>, <a class="txt" href="#E5:70_1_2_1_2"><span class="lab"><font color="Green" title="E64">A20</font></span></a>, <a class="txt" href="#E9:70_1_2_1_2"><span class="lab"><font color="Green" title="E67">A23</font></span></a>, <a class="txt" href="#E11:70_1_2_1_2"><span class="lab"><font color="Green" title="E69">A25</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="E16:70_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">fr</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E61">A26</font></span>:  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">fr</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">fr</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T77" title="FINSEQ_4:th.77">FINSEQ_4:77</a></span>
<br/>.= 
<font color="Maroon" title="c1">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:70_1_2_1"><span class="lab"><font color="Green" title="E60">A16</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">Y</font> = <font color="Maroon" title="c8">Y</font> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<a NAME="E19:70_1_2_1"/>
<font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1_2_1"><span class="lab"><font color="Green" title="E53">A5</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E20:70_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">a</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70_1_2_1"><span class="lab"><font color="Green" title="E54">A6</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/>
<a NAME="E21:70_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A27</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Y</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="E22:70_1_2_1"/>
<font color="Maroon" title="c1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">l</font> where <font color="Olive" title="b1">l</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">m</font> </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="axioms.html#T30" title="AXIOMS:th.30">AXIOMS:30</a></span>;<br/>
<a NAME="E23:70_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">Y</font> <a href="int_4.html#R1" title="INT_4:pred.1">is_CRS_of</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_4.html#T48" target="_self" title="INT_4:th.48">Th48</a></span>;<br/>
<a NAME="E24:70_1_2_1"/><span class="kw">then </span>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_4.html#T49" target="_self" title="INT_4:th.49">Th49</a></span>;<br/>
<a NAME="E25:70_1_2_1"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">fr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:70_1_2_1"><span class="lab"><font color="Green" title="E61">A26</font></span></a>, <a class="ref" href="card_fin.html#T1" title="CARD_FIN:th.1">CARD_FIN:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">w</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E27:70_1_2_1"/><span class="lab"><font color="Green" title="E63">A28</font></span>: 
<font color="Maroon" title="c11">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">fr</font>
 <span class="kw">and </span><br/><a NAME="E28:70_1_2_1"/><span class="lab"><font color="Green" title="E64">A29</font></span>: 
<font color="Maroon" title="c9">fr</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:70_1_2_1"><span class="lab"><font color="Green" title="E62">A27</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">w</font> = <font color="Maroon" title="c11">w</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="#E27:70_1_2_1"><span class="lab"><font color="Green" title="E63">A28</font></span></a></span>;<br/>
<a NAME="E30:70_1_2_1"/>
<font color="Maroon" title="c12">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">fp</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:70_1_2_1"><span class="lab"><font color="Green" title="E55">A7</font></span></a>, <a class="txt" href="#E14:70_1_2_1"><span class="lab"><font color="Green" title="E60">A16</font></span></a>, <a class="txt" href="#E27:70_1_2_1"><span class="lab"><font color="Green" title="E63">A28</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">y</font> = <font color="Maroon" title="c7">fp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">w</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">X</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T13" title="FINSEQ_2:th.13">FINSEQ_2:13</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c13">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">S<sub>1</sub></font>[<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c13">y</font>]</span><br/>

<span class="kw">thus </span><a NAME="E32:70_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c13">y</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:70_1_2_1"><span class="lab"><font color="Green" title="E60">A16</font></span></a>, <a class="txt" href="#E27:70_1_2_1"><span class="lab"><font color="Green" title="E63">A28</font></span></a>, <a class="txt" href="#E28:70_1_2_1"><span class="lab"><font color="Green" title="E64">A29</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="c4">fp</font> being    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of <font color="Maroon" title="c3">X</font><span class="kw"> such that </span><br/><a NAME="E6:70_1_2"/><span class="lab"><font color="Green" title="E53">A30</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">fp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> &amp; ( for <font color="Olive" title="b1">a</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c4">fp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_1.html#S5" title="FINSEQ_1:sch.5">FINSEQ_1:sch 5</a>(<a class="txt" href="#E4:70_1_2"><span class="lab"><font color="Green" title="E52">A4</font></span></a>);<br/></span><a NAME="E7:70_1_2"/><span class="lab"><font color="Green" title="E54">A31</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">fp</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/><a NAME="E8:70_1_2"/><span class="lab"><font color="Green" title="E55">A32</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">fp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:70_1_2"><span class="lab"><font color="Green" title="E53">A30</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E9:70_1_2"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">fp</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">fp</font> &amp; <font color="Maroon" title="c4">fp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">fp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</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="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">fp</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">fp</font> &amp; <font color="Maroon" title="c4">fp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">fp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font> implies <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:70_1_2_2"/><span class="lab"><font color="Green" title="E56">A33</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">fp</font>
 <span class="kw">and </span><br/><a NAME="E2:70_1_2_2"/><span class="lab"><font color="Green" title="E57">A34</font></span>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">fp</font>
 <span class="kw">and </span><br/><a NAME="E3:70_1_2_2"/><span class="lab"><font color="Green" title="E58">A35</font></span>: 
<font color="Maroon" title="c4">fp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">fp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</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="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">a</font> = <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c8">b</font> = <font color="Maroon" title="c6">b</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1_2_2"><span class="lab"><font color="Green" title="E56">A33</font></span></a>, <a class="txt" href="#E2:70_1_2_2"><span class="lab"><font color="Green" title="E57">A34</font></span></a></span>;<br/>
<a NAME="E5:70_1_2_2"/><span class="lab"><font color="Green" title="E59">A36</font></span>: 
<font color="Maroon" title="c8">b</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:70_1_2"><span class="lab"><font color="Green" title="E53">A30</font></span></a>, <a class="txt" href="#E2:70_1_2_2"><span class="lab"><font color="Green" title="E57">A34</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E6:70_1_2_2"/><span class="lab"><font color="Green" title="E60">A37</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:70_1_2"><span class="lab"><font color="Green" title="E53">A30</font></span></a>, <a class="txt" href="#E1:70_1_2_2"><span class="lab"><font color="Green" title="E56">A33</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E61">A38</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:70_1_2_2"><span class="lab"><font color="Green" title="E59">A36</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>
<br/>.= 
<font color="Maroon" title="c7">a</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">b</font>

;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">l</font> = <font color="Maroon" title="c4">fp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c10">ll</font> = <font color="Maroon" title="c4">fp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">b</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1_2_2"><span class="lab"><font color="Green" title="E56">A33</font></span></a>, <a class="txt" href="#E2:70_1_2_2"><span class="lab"><font color="Green" title="E57">A34</font></span></a>, <a class="ref" href="finseq_2.html#T13" title="FINSEQ_2:th.13">FINSEQ_2:13</a></span>;<br/>
<a NAME="E9:70_1_2_2"/>
<font color="Maroon" title="c4">fp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="int_4.html#K1" title="INT_4:func.1">Cong</a> <font color="Maroon" title="c1">m</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:70_1_2"><span class="lab"><font color="Green" title="E53">A30</font></span></a>, <a class="txt" href="#E2:70_1_2_2"><span class="lab"><font color="Green" title="E57">A34</font></span></a></span>;<br/>
<a NAME="E10:70_1_2_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">fp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">b</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="int_4.html#K1" title="INT_4:func.1">Cong</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T26" title="EQREL_1:th.26">EQREL_1:26</a></span>;<br/>
<a NAME="E11:70_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">b</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1,<font color="Maroon" title="c10">ll</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_4.html#D1" target="_self" title="INT_4:def.1">Def1</a></span>;<br/>
<a NAME="E12:70_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A39</font></span>: 
<font color="Maroon" title="c9">l</font>,<font color="Maroon" title="c8">b</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70_1_2_2"><span class="lab"><font color="Green" title="E58">A35</font></span></a>, <a class="ref" href="int_1.html#T35" title="INT_1:th.35">INT_1:35</a></span>;<br/>
<a NAME="E13:70_1_2_2"/>
<font color="Maroon" title="c8">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:70_1_2"><span class="lab"><font color="Green" title="E53">A30</font></span></a>, <a class="txt" href="#E2:70_1_2_2"><span class="lab"><font color="Green" title="E57">A34</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E14:70_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">a</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">b</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:70_1_2_2"><span class="lab"><font color="Green" title="E60">A37</font></span></a>, <a class="ref" href="xreal_1.html#T15" title="XREAL_1:th.15">XREAL_1:15</a></span>;<br/>
<a NAME="E15:70_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A40</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T147" title="XREAL_1:th.147">XREAL_1:147</a></span>;<br/>
<a NAME="E16:70_1_2_2"/>
<font color="Maroon" title="c7">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:70_1_2"><span class="lab"><font color="Green" title="E53">A30</font></span></a>, <a class="txt" href="#E1:70_1_2_2"><span class="lab"><font color="Green" title="E56">A33</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E17:70_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">a</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:70_1_2_2"><span class="lab"><font color="Green" title="E59">A36</font></span></a>, <a class="ref" href="xreal_1.html#T15" title="XREAL_1:th.15">XREAL_1:15</a></span>;<br/>
<a NAME="E18:70_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">a</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T149" title="XREAL_1:th.149">XREAL_1:149</a></span>;<br/>
<a NAME="E19:70_1_2_2"/><span class="kw">then </span>
 <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:70_1_2_2"><span class="lab"><font color="Green" title="E63">A40</font></span></a>, <a class="ref" href="seq_2.html#T9" title="SEQ_2:th.9">SEQ_2:9</a></span>;<br/>
<a NAME="E20:70_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A41</font></span>: 
 <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/>
<a NAME="E21:70_1_2_2"/>
<font color="Maroon" title="c4">fp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="int_4.html#K1" title="INT_4:func.1">Cong</a> <font color="Maroon" title="c1">m</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:70_1_2"><span class="lab"><font color="Green" title="E53">A30</font></span></a>, <a class="txt" href="#E1:70_1_2_2"><span class="lab"><font color="Green" title="E56">A33</font></span></a></span>;<br/>
<a NAME="E22:70_1_2_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">fp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="int_4.html#K1" title="INT_4:func.1">Cong</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T26" title="EQREL_1:th.26">EQREL_1:26</a></span>;<br/>
<a NAME="E23:70_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">a</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1,<font color="Maroon" title="c9">l</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_4.html#D1" target="_self" title="INT_4:def.1">Def1</a></span>;<br/>
<a NAME="E24:70_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">a</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1,<font color="Maroon" title="c8">b</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:70_1_2_2"><span class="lab"><font color="Green" title="E62">A39</font></span></a>, <a class="ref" href="int_1.html#T36" title="INT_1:th.36">INT_1:36</a></span>;<br/>
<a NAME="E25:70_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A42</font></span>: 
<font color="Maroon" title="c1">m</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c7">a</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:70_1_2_2"><span class="lab"><font color="Green" title="E61">A38</font></span></a>, <a class="ref" href="int_2.html#T19" title="INT_2:th.19">INT_2:19</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_2_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:70_1_2_2_2"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b</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/><a NAME="E2:70_1_2_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">a</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><span class="kw">hence </span><a NAME="E3:70_1_2_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:70_1_2_2"><span class="lab"><font color="Green" title="E65">A42</font></span></a>, <a class="txt" href="#E20:70_1_2_2"><span class="lab"><font color="Green" title="E64">A41</font></span></a>, <a class="ref" href="int_4.html#T6" target="_self" title="INT_4:th.6">Th6</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">hence </span><a NAME="E27:70_1_2_2"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font>
 ; <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="E10:70_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">fp</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></span>;<br/><a NAME="E11:70_1_2"/><span class="kw">then </span>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">fp</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="txt" href="#E8:70_1_2"><span class="lab"><font color="Green" title="E55">A32</font></span></a>, <a class="ref" href="finseq_4.html#T77" title="FINSEQ_4:th.77">FINSEQ_4:77</a></span>;<br/><a NAME="E12:70_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">fp</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:70_1_2"><span class="lab"><font color="Green" title="E54">A31</font></span></a>, <a class="ref" href="card_fin.html#T1" title="CARD_FIN:th.1">CARD_FIN:1</a></span>;<br/><span class="kw">hence </span><a NAME="E13:70_1_2"/>
<font color="Maroon" title="c2">X</font> <a href="int_4.html#R1" title="INT_4:pred.1">is_CRS_of</a> <font color="Maroon" title="c1">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:70_1_2"><span class="lab"><font color="Green" title="E53">A30</font></span></a>, <a class="txt" href="#E8:70_1_2"><span class="lab"><font color="Green" title="E55">A32</font></span></a>, <a class="ref" href="int_4.html#D2" target="_self" title="INT_4:def.2">Def2</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>
