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

<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">H1</font>, <font color="Olive" title="b2">H2</font> being    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> holds  <font color="Olive" title="b1">H1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Olive" title="b2">H2</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a>  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">H1</font> <a href="group_4.html#K7" title="GROUP_4:func.7">*</a> <font color="Olive" title="b2">H2</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">H1</font>, <font color="Maroon" title="c3">H2</font> be    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">H1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">H2</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a>  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="group_4.html#K7" title="GROUP_4:func.7">*</a> <font color="Maroon" title="c3">H2</font></span>)</span></span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">H</font> =  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H1</font></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p2">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H2</font></span>)</span></span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H1</font></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p2">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H2</font></span>)</span></span>)</span> implies <font color="Maroon" title="c5">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">H1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">H2</font> )</span><br/><span class="kw">assume </span><a NAME="E1:58_1"/>
<font color="Maroon" title="c5">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H1</font></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p2">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H2</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">H1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">H2</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">F</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c7">I</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <span class="kw"> such that </span><br/><a NAME="E3:58_1"/><span class="lab"><font color="Green" title="E30">A1</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">I</font>
 <span class="kw">and </span><br/><a NAME="E4:58_1"/><span class="lab"><font color="Green" title="E31">A2</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H1</font></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H2</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:58_1"/><span class="lab"><font color="Green" title="E32">A3</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c7">I</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_4.html#T28" target="_self" title="GROUP_4:th.28">Th37</a></span>;<br/><a NAME="E6:58_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">H2</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c8">f</font> = <font color="Maroon" title="c6">F</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c7">I</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c7">I</font></span>)</span> or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">H2</font></span>)</span> )</span><br/>

<a NAME="E1:58_1_1"/><span class="lab"><font color="Green" title="E33">A4</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:58_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c7">I</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">H2</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:58_1_1"/><span class="lab"><font color="Green" title="E34">A5</font></span>: 
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c7">I</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:58_1_1"/><span class="lab"><font color="Green" title="E35">A6</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">y</font> = <font color="Maroon" title="c10">y</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_1"><span class="lab"><font color="Green" title="E34">A5</font></span></a></span>;<br/>
<a NAME="E7:58_1_1"/><span class="lab"><font color="Green" title="E36">A7</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_4.html#D3" target="_self" title="GROUP_4:def.3">Def4</a></span>;<br/>
<a NAME="E8:58_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A8</font></span>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:58_1"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="#E4:58_1_1"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/>
<a NAME="E9:58_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A9</font></span>: 
<font color="Maroon" title="c7">I</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<a NAME="E10:58_1_1"/>
<font color="Maroon" title="c7">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:58_1_1"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">i</font> = <font color="Maroon" title="c7">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_1"><span class="lab"><font color="Green" title="E33">A4</font></span></a></span>;<br/>
<a NAME="E12:58_1_1"/><span class="lab"><font color="Green" title="E39">A10</font></span>: 
 <a href="group_4.html#K2" title="GROUP_4:func.2">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">I</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y</font>
 
;<br/>
<a NAME="E13:58_1_1"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_1"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="txt" href="#E7:58_1_1"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/>
<a NAME="E14:58_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c6">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font> &amp; <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">b</font>, <font color="Maroon" title="c14">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E16:58_1_1"/><span class="lab"><font color="Green" title="E40">A11</font></span>: 
<font color="Maroon" title="c6">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c14">c</font>
 <span class="kw">and </span><br/><a NAME="E17:58_1_1"/><span class="lab"><font color="Green" title="E41">A12</font></span>: 
<font color="Maroon" title="c13">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H1</font>
 <span class="kw">and </span><br/><a NAME="E18:58_1_1"/><span class="lab"><font color="Green" title="E42">A13</font></span>: 
<font color="Maroon" title="c14">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="ref" href="group_2.html#T8" title="GROUP_2:th.8">GROUP_2:8</a></span>;<br/>
<a NAME="E19:58_1_1"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_1"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="txt" href="#E7:58_1_1"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/>
<a NAME="E20:58_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A14</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c12">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1"><span class="lab"><font color="Green" title="E35">A6</font></span></a>, <a class="txt" href="#E9:58_1_1"><span class="lab"><font color="Green" title="E38">A9</font></span></a>, <a class="txt" href="#E12:58_1_1"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="ref" href="group_4.html#D3" target="_self" title="GROUP_4:def.3">Def4</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_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="E1:58_1_1_1_1"/>
( <font color="Maroon" title="c12">i</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>  or <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:58_1_1_1_1_1"/>
<font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">H2</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">n</font> = <font color="Maroon" title="c12">i</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#T3" title="INT_1:th.3">INT_1:3</a></span>;<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>( ( $1 <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> ) &amp; ( $1 <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">c</font> ) );<br/><a NAME="E3:58_1_1_1_1_1"/><span class="lab"><font color="Green" title="E44">A15</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c15">n</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">x</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c15">n</font></span>)</span> implies  ex <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c16">k</font>,<font color="Olive" title="b1">x</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:58_1_1_1_1_1_1"/>
<font color="Maroon" title="c16">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c15">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c16">k</font>,<font color="Olive" title="b1">x</font>]</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_1_1_1_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="E1:58_1_1_1_1_1_1_1_1"/>
( <font color="Maroon" title="c16">k</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c16">k</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T12" title="NAT_D:th.12">NAT_D:12</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:58_1_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E45">A16</font></span>: 
<font color="Maroon" title="c16">k</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( ( <font color="Maroon" title="c16">k</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> ) &amp; ( <font color="Maroon" title="c16">k</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">c</font> ) )</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c17">x</font> = <font color="Maroon" title="c13">b</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><span class="kw">take </span><font color="Maroon" title="c18">x</font> = <font color="Maroon" title="c17">x</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c16">k</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> ) &amp; ( <font color="Maroon" title="c16">k</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">c</font> ) )</span><br/><span class="kw">thus </span><a NAME="E3:58_1_1_1_1_1_1_1_1_1"/>
( ( <font color="Maroon" title="c16">k</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> ) &amp; ( <font color="Maroon" title="c16">k</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">c</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A16</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:58_1_1_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E45">A17</font></span>: 
<font color="Maroon" title="c16">k</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( ( <font color="Maroon" title="c16">k</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> ) &amp; ( <font color="Maroon" title="c16">k</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">c</font> ) )</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c17">x</font> = <font color="Maroon" title="c14">c</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><span class="kw">take </span><font color="Maroon" title="c18">x</font> = <font color="Maroon" title="c17">x</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c16">k</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> ) &amp; ( <font color="Maroon" title="c16">k</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">c</font> ) )</span><br/><span class="kw">thus </span><a NAME="E3:58_1_1_1_1_1_1_1_1_2"/>
( ( <font color="Maroon" title="c16">k</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> ) &amp; ( <font color="Maroon" title="c16">k</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">c</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E45">A17</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:58_1_1_1_1_1_1"/>
 ex <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c16">k</font>,<font color="Olive" title="b1">x</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">consider </span><font color="Maroon" title="c16">p</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E5:58_1_1_1_1_1"/><span class="lab"><font color="Green" title="E45">A18</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c15">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:58_1_1_1_1_1"/><span class="lab"><font color="Green" title="E46">A19</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c15">n</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c16">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_1.html#S1" title="FINSEQ_1:sch.1">FINSEQ_1:sch 1</a>(<a class="txt" href="#E3:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E44">A15</font></span></a>);<br/></span><a NAME="E7:58_1_1_1_1_1"/><span class="lab"><font color="Green" title="E47">A20</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c15">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">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="E8:58_1_1_1_1_1"/><span class="lab"><font color="Green" title="E48">A21</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c14">c</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">p</font> or <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c14">c</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:58_1_1_1_1_1_2"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c14">c</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:58_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E49">A22</font></span>: 
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">p</font>
 <span class="kw">and </span><br/><a NAME="E4:58_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E50">A23</font></span>: 
<font color="Maroon" title="c16">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">y</font> = <font color="Maroon" title="c18">y</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:58_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E49">A22</font></span></a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_1_2_1_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="E1:58_1_1_1_1_1_2_1_1"/>
( <font color="Maroon" title="c19">y</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c19">y</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T12" title="NAT_D:th.12">NAT_D:12</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:58_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c19">y</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c14">c</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><br/><div class="add"><a NAME="E2:58_1_1_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E6:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E46">A19</font></span></a>, <a class="txt" href="#E3:58_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E49">A22</font></span></a>, <a class="txt" href="#E4:58_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E50">A23</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:58_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c14">c</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:58_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c19">y</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c14">c</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><br/><div class="add"><a NAME="E2:58_1_1_1_1_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E6:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E46">A19</font></span></a>, <a class="txt" href="#E3:58_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E49">A22</font></span></a>, <a class="txt" href="#E4:58_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E50">A23</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:58_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c14">c</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:58_1_1_1_1_1_2"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c14">c</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E9:58_1_1_1_1_1"/><span class="kw">then </span>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">p</font> = <font color="Maroon" title="c16">p</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</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="E11:58_1_1_1_1_1"/><span class="lab"><font color="Green" title="E49">A24</font></span>: 
( <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H2</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <span class="p1">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H2</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H2</font></span>)</span></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_4.html#D4" target="_self" title="GROUP_4:def.4">Def5</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>] <span class="kw">means </span>( $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c15">n</font> &amp; $1 <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  for <font color="Olive" title="b1">F1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> $1</span>)</span> holds <br/> <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Olive" title="b1">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default">$1 <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2</span>)</span> );<br/><a NAME="E12:58_1_1_1_1_1"/><span class="lab"><font color="Green" title="E50">A25</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st (  for <font color="Olive" title="b2">l</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">l</font>] ) holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">l</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">k</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">l</font>] ) implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c18">k</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:58_1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E51">A26</font></span>: 
 for <font color="Olive" title="b1">l</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">k</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">l</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c18">k</font>]</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:58_1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E52">A27</font></span>: 
<font color="Maroon" title="c18">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c15">n</font>
 <span class="kw">and </span><br/><a NAME="E3:58_1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E53">A28</font></span>: 
<font color="Maroon" title="c18">k</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">F1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c18">k</font></span>)</span> holds <br/> <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Olive" title="b1">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">k</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2</span>)</span></span><br/>

<span class="kw">let </span><font color="Maroon" title="c19">F1</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c19">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c18">k</font></span>)</span> implies  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c19">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">k</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E4:58_1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E54">A29</font></span>: 
<font color="Maroon" title="c19">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c18">k</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c19">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">k</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2</span>)</span></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1_1_1_3_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_1_3_1_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="E1:58_1_1_1_1_1_3_1_1"/>
( <font color="Maroon" title="c18">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c18">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_1_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:58_1_1_1_1_1_3_1_1_1"/><span class="lab"><font color="Green" title="E55">A30</font></span>: 
<font color="Maroon" title="c18">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c19">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">k</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2</span>)</span></span><br/><div class="add"><a NAME="E2:58_1_1_1_1_1_3_1_1_1"/>
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E3:58_1_1_1_1_1_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A31</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T18" title="NAT_D:th.18">NAT_D:18</a></span>;<br/><a NAME="E4:58_1_1_1_1_1_3_1_1_1"/>
<font color="Maroon" title="c19">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E54">A29</font></span></a>, <a class="txt" href="#E1:58_1_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A30</font></span></a></span>;<br/><a NAME="E5:58_1_1_1_1_1_3_1_1_1"/><span class="kw">then </span>
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c19">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_4.html#T8" target="_self" title="GROUP_4:th.8">Th11</a></span>;<br/><span class="kw">hence </span><a NAME="E6:58_1_1_1_1_1_3_1_1_1"/>
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c19">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">k</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A30</font></span></a>, <a class="txt" href="#E3:58_1_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E56">A31</font></span></a>, <a class="ref" href="group_1.html#T25" title="GROUP_1:th.25">GROUP_1:25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_1_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:58_1_1_1_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E55">A32</font></span>: 
<font color="Maroon" title="c18">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c19">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">k</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2</span>)</span></span><br/><div class="add"><a NAME="E2:58_1_1_1_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E56">A33</font></span>: 
<font color="Maroon" title="c18">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:58_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E53">A28</font></span></a>, <a class="ref" href="nat_d.html#T14" title="NAT_D:th.14">NAT_D:14</a></span>;<br/><a NAME="E3:58_1_1_1_1_1_3_1_1_2"/>
<font color="Maroon" title="c18">k</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="#E1:58_1_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E55">A32</font></span></a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/><a NAME="E4:58_1_1_1_1_1_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">k</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="txt" href="#E2:58_1_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E56">A33</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E5:58_1_1_1_1_1_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <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="E6:58_1_1_1_1_1_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">m</font> = <font color="Maroon" title="c18">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 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#T3" title="INT_1:th.3">INT_1:3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c21">q</font> = <font color="Maroon" title="c17">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c20">m</font></span>)</span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></span>;<br/><a NAME="E9:58_1_1_1_1_1_3_1_1_2"/>
1 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ;<br/><a NAME="E10:58_1_1_1_1_1_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A34</font></span>: 
<font color="Maroon" title="c20">m</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:58_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E53">A28</font></span></a>, <a class="ref" href="nat_d.html#T15" title="NAT_D:th.15">NAT_D:15</a></span>;<br/><a NAME="E11:58_1_1_1_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E58">A35</font></span>: 
<font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">k</font>
 ;<br/><a NAME="E12:58_1_1_1_1_1_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A36</font></span>: 
<font color="Maroon" title="c20">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c15">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E52">A27</font></span></a>, <a class="ref" href="nat_1.html#T16" title="NAT_1:th.16">NAT_1:16</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E13:58_1_1_1_1_1_3_1_1_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">o</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c15">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">o</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></span>;<br/><a NAME="E14:58_1_1_1_1_1_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A37</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E47">A20</font></span></a>, <a class="ref" href="finseq_3.html#T53" title="FINSEQ_3:th.53">FINSEQ_3:53</a></span>;<br/><a NAME="E15:58_1_1_1_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E61">A38</font></span>: 
 ex <font color="Olive" title="b1">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c15">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b1">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E52">A27</font></span></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></span>;<br/><a NAME="E16:58_1_1_1_1_1_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A39</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E47">A20</font></span></a>, <a class="txt" href="#E4:58_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E54">A29</font></span></a>, <a class="ref" href="finseq_3.html#T53" title="FINSEQ_3:th.53">FINSEQ_3:53</a></span>;<br/><div><span class="lab"><font color="Green" title="E63">A40</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1_1_1_3_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c22">l</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">q</font> implies <font color="Maroon" title="c19">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">l</font> )</span><br/><span class="kw">assume </span><a NAME="E1:58_1_1_1_1_1_3_1_1_2_1"/><span class="lab"><font color="Green" title="E64">A41</font></span>: 
<font color="Maroon" title="c22">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">l</font></span><br/><a NAME="E2:58_1_1_1_1_1_3_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c22">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E3:58_1_1_1_1_1_3_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A42</font></span>: 
<font color="Maroon" title="c22">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">F1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:58_1_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E58">A35</font></span></a>, <a class="txt" href="#E16:58_1_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E62">A39</font></span></a>, <a class="txt" href="#E14:58_1_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E60">A37</font></span></a>, <a class="ref" href="nat_1.html#T16" title="NAT_1:th.16">NAT_1:16</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E4:58_1_1_1_1_1_3_1_1_2_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_1_1_1_1_3_1_1_2_1"><span class="lab"><font color="Green" title="E64">A41</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E5:58_1_1_1_1_1_3_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c22">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">F1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:58_1_1_1_1_1_3_1_1_2_1"><span class="lab"><font color="Green" title="E65">A42</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E6:58_1_1_1_1_1_3_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E54">A29</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>;<br/><span class="kw">hence </span><a NAME="E7:58_1_1_1_1_1_3_1_1_2_1"/>
<font color="Maroon" title="c19">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_1_1_1_1_3_1_1_2_1"><span class="lab"><font color="Green" title="E64">A41</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E18:58_1_1_1_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E64">A43</font></span>: 
<font color="Maroon" title="c20">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:58_1_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E58">A35</font></span></a>, <a class="ref" href="nat_1.html#T16" title="NAT_1:th.16">NAT_1:16</a></span>;<br/><a NAME="E19:58_1_1_1_1_1_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A44</font></span>: 
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c21">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">m</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E51">A26</font></span></a>, <a class="txt" href="#E12:58_1_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E59">A36</font></span></a>, <a class="txt" href="#E10:58_1_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E57">A34</font></span></a></span>;<br/><div><span class="lab"><font color="Green" title="E66">A45</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1_1_1_3_1_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c22">l</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c14">c</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> implies <font color="Maroon" title="c19">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c14">c</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">l</font> )</span><br/><span class="kw">assume </span><a NAME="E1:58_1_1_1_1_1_3_1_1_2_2"/>
<font color="Maroon" title="c22">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c14">c</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c14">c</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">l</font></span><br/><a NAME="E2:58_1_1_1_1_1_3_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A46</font></span>: 
<font color="Maroon" title="c22">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default">1,2</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T2" title="FINSEQ_1:th.2">FINSEQ_1:2</a>, <a class="ref" href="finseq_1.html#T89" title="FINSEQ_1:th.89">FINSEQ_1:89</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1_1_1_3_1_1_2_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_1_3_1_1_2_2_1_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="E1:58_1_1_1_1_1_3_1_1_2_2_1_1"/>
( <font color="Maroon" title="c22">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c22">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58_1_1_1_1_1_3_1_1_2_2"><span class="lab"><font color="Green" title="E67">A46</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_1_3_1_1_2_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:58_1_1_1_1_1_3_1_1_2_2_1_1_1"/><span class="lab"><font color="Green" title="E68">A47</font></span>: 
<font color="Maroon" title="c22">l</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="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c14">c</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">l</font></span><br/><div class="add"><a NAME="E2:58_1_1_1_1_1_3_1_1_2_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A48</font></span>: 
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c14">c</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>;<br/><a NAME="E3:58_1_1_1_1_1_3_1_1_2_2_1_1_1"/><span class="lab"><font color="Green" title="E70">A49</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">F1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c17">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E54">A29</font></span></a>, <a class="txt" href="#E10:58_1_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E57">A34</font></span></a>, <a class="ref" href="nat_d.html#T16" title="NAT_D:th.16">NAT_D:16</a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a></span>;<br/><a NAME="E4:58_1_1_1_1_1_3_1_1_2_2_1_1_1"/>
( <font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">k</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:58_1_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E64">A43</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E5:58_1_1_1_1_1_3_1_1_2_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A50</font></span>: 
<font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">F1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:58_1_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E62">A39</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E6:58_1_1_1_1_1_3_1_1_2_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E54">A29</font></span></a>, <a class="txt" href="#E14:58_1_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E60">A37</font></span></a>, <a class="txt" href="#E1:58_1_1_1_1_1_3_1_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E68">A47</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>;<br/><span class="kw">hence </span><a NAME="E7:58_1_1_1_1_1_3_1_1_2_2_1_1_1"/>
<font color="Maroon" title="c19">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c14">c</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E6:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E46">A19</font></span></a>, <a class="txt" href="#E2:58_1_1_1_1_1_3_1_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E69">A48</font></span></a>, <a class="txt" href="#E5:58_1_1_1_1_1_3_1_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E71">A50</font></span></a>, <a class="txt" href="#E3:58_1_1_1_1_1_3_1_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E70">A49</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_1_3_1_1_2_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:58_1_1_1_1_1_3_1_1_2_2_1_1_2"/><span class="lab"><font color="Green" title="E68">A51</font></span>: 
<font color="Maroon" title="c22">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c14">c</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">l</font></span><br/><div class="add"><a NAME="E2:58_1_1_1_1_1_3_1_1_2_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A52</font></span>: 
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c14">c</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>;<br/><a NAME="E3:58_1_1_1_1_1_3_1_1_2_2_1_1_2"/><span class="lab"><font color="Green" title="E70">A53</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">F1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c17">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E54">A29</font></span></a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a></span>;<br/><a NAME="E4:58_1_1_1_1_1_3_1_1_2_2_1_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E5:58_1_1_1_1_1_3_1_1_2_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A54</font></span>: 
<font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">F1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:58_1_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E62">A39</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E6:58_1_1_1_1_1_3_1_1_2_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c19">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E54">A29</font></span></a>, <a class="txt" href="#E14:58_1_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E60">A37</font></span></a>, <a class="txt" href="#E1:58_1_1_1_1_1_3_1_1_2_2_1_1_2"><span class="lab"><font color="Green" title="E68">A51</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>;<br/><span class="kw">hence </span><a NAME="E7:58_1_1_1_1_1_3_1_1_2_2_1_1_2"/>
<font color="Maroon" title="c19">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c14">c</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E6:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E46">A19</font></span></a>, <a class="txt" href="#E3:58_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E53">A28</font></span></a>, <a class="txt" href="#E2:58_1_1_1_1_1_3_1_1_2_2_1_1_2"><span class="lab"><font color="Green" title="E69">A52</font></span></a>, <a class="txt" href="#E5:58_1_1_1_1_1_3_1_1_2_2_1_1_2"><span class="lab"><font color="Green" title="E71">A54</font></span></a>, <a class="txt" href="#E3:58_1_1_1_1_1_3_1_1_2_2_1_1_2"><span class="lab"><font color="Green" title="E70">A53</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:58_1_1_1_1_1_3_1_1_2_2"/>
<font color="Maroon" title="c19">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c14">c</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">l</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="lab"><font color="Green" title="E67">A55</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c20">m</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c20">m</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">2 <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T18" title="NAT_D:th.18">NAT_D:18</a></span>
<br/>.= 
<font color="Maroon" title="c18">k</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:58_1_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E58">A35</font></span></a>, <a class="txt" href="#E10:58_1_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E57">A34</font></span></a>, <a class="ref" href="nat_d.html#T19" title="NAT_D:th.19">NAT_D:19</a></span>
;<br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">F1</font> = 
<font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E47">A20</font></span></a>, <a class="txt" href="#E4:58_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E54">A29</font></span></a>, <a class="txt" href="#E15:58_1_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E61">A38</font></span></a>, <a class="ref" href="finseq_3.html#T53" title="FINSEQ_3:th.53">FINSEQ_3:53</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c14">c</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:58_1_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E60">A37</font></span></a>, <a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>
;<br/><a NAME="E23:58_1_1_1_1_1_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c19">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">q</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c14">c</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:58_1_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E63">A40</font></span></a>, <a class="txt" href="#E20:58_1_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E66">A45</font></span></a>, <a class="ref" href="finseq_3.html#T38" title="FINSEQ_3:th.38">FINSEQ_3:38</a></span>;<br/><span class="kw">then </span> <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c19">F1</font> = 
<span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c21">q</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p2"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c14">c</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finsop_1.html#T5" title="FINSOP_1:th.5">FINSOP_1:5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c21">q</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:58_1_1"><span class="lab"><font color="Green" title="E40">A11</font></span></a>, <a class="ref" href="finsop_1.html#T12" title="FINSOP_1:th.12">FINSOP_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">m</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:58_1_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E65">A44</font></span></a>, <a class="ref" href="group_1.html#T34" title="GROUP_1:th.34">GROUP_1:34</a></span>
;<br/><span class="kw">hence </span><a NAME="E25:58_1_1_1_1_1_3_1_1_2"/>
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c19">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">k</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:58_1_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E67">A55</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:58_1_1_1_1_1_3"/>
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c19">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">k</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E13:58_1_1_1_1_1"/><span class="lab"><font color="Green" title="E51">A56</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S4" title="NAT_1:sch.4">NAT_1:sch 4</a>(<a class="txt" href="#E12:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E50">A25</font></span></a>);<br/></span><a NAME="E14:58_1_1_1_1_1"/>
( <font color="Maroon" title="c13">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H2</font></span>)</span> &amp; <font color="Maroon" title="c14">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H2</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:58_1_1"><span class="lab"><font color="Green" title="E41">A12</font></span></a>, <a class="txt" href="#E18:58_1_1"><span class="lab"><font color="Green" title="E42">A13</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E15:58_1_1_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c14">c</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T32" title="ZFMISC_1:th.32">ZFMISC_1:32</a></span>;<br/><a NAME="E16:58_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A57</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E48">A21</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E17:58_1_1_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c15">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">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="E18:58_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A58</font></span>: 
<font color="Maroon" title="c17">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T49" title="FINSEQ_3:th.49">FINSEQ_3:49</a></span>;<br/><a NAME="E19:58_1_1_1_1_1"/>
( <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T13" title="NAT_D:th.13">NAT_D:13</a>, <a class="ref" href="nat_d.html#T18" title="NAT_D:th.18">NAT_D:18</a></span>;<br/><a NAME="E20:58_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c17">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:58_1_1"><span class="lab"><font color="Green" title="E43">A14</font></span></a>, <a class="txt" href="#E13:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E51">A56</font></span></a>, <a class="txt" href="#E18:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E53">A58</font></span></a></span>;<br/><a NAME="E21:58_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H2</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E52">A57</font></span></a>, <a class="txt" href="#E11:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E49">A24</font></span></a>, <a class="ref" href="group_4.html#T18" target="_self" title="GROUP_4:th.18">Th21</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><span class="kw">hence </span><a NAME="E22:58_1_1_1_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">H2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:58_1_1_1_1_2"/><span class="lab"><font color="Green" title="E44">A59</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">H2</font></span>)</span></span><br/><div class="add"><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>( ( $1 <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">c</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  ) &amp; ( $1 <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  ) );<br/><span class="kw">set </span><font color="Maroon" title="c15">n</font> =  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c12">i</font>;<br/><a NAME="E2:58_1_1_1_1_2"/><span class="lab"><font color="Green" title="E45">A60</font></span>: 
( <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c12">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T13" title="NAT_D:th.13">NAT_D:13</a>, <a class="ref" href="nat_d.html#T18" title="NAT_D:th.18">NAT_D:18</a></span>;<br/><a NAME="E3:58_1_1_1_1_2"/><span class="lab"><font color="Green" title="E46">A61</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span> holds <br/> ex <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">x</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span> implies  ex <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c16">k</font>,<font color="Olive" title="b1">x</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:58_1_1_1_1_2_1"/>
<font color="Maroon" title="c16">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c16">k</font>,<font color="Olive" title="b1">x</font>]</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1_1_2_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_2_1_1_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="E1:58_1_1_1_1_2_1_1_1"/>
( <font color="Maroon" title="c16">k</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c16">k</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T12" title="NAT_D:th.12">NAT_D:12</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_2_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:58_1_1_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E47">A62</font></span>: 
<font color="Maroon" title="c16">k</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( ( <font color="Maroon" title="c16">k</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">c</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  ) &amp; ( <font color="Maroon" title="c16">k</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  ) )</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c17">x</font> = <font color="Maroon" title="c14">c</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><span class="kw">take </span><font color="Maroon" title="c18">x</font> = <font color="Maroon" title="c17">x</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c16">k</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">c</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  ) &amp; ( <font color="Maroon" title="c16">k</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  ) )</span><br/><span class="kw">thus </span><a NAME="E3:58_1_1_1_1_2_1_1_1_1"/>
( ( <font color="Maroon" title="c16">k</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">c</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  ) &amp; ( <font color="Maroon" title="c16">k</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_1_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E47">A62</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_2_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:58_1_1_1_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E47">A63</font></span>: 
<font color="Maroon" title="c16">k</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( ( <font color="Maroon" title="c16">k</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">c</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  ) &amp; ( <font color="Maroon" title="c16">k</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  ) )</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c17">x</font> = <font color="Maroon" title="c13">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><span class="kw">take </span><font color="Maroon" title="c18">x</font> = <font color="Maroon" title="c17">x</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c16">k</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">c</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  ) &amp; ( <font color="Maroon" title="c16">k</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  ) )</span><br/><span class="kw">thus </span><a NAME="E3:58_1_1_1_1_2_1_1_1_2"/>
( ( <font color="Maroon" title="c16">k</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">c</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  ) &amp; ( <font color="Maroon" title="c16">k</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E47">A63</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:58_1_1_1_1_2_1"/>
 ex <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c16">k</font>,<font color="Olive" title="b1">x</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">consider </span><font color="Maroon" title="c16">p</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E5:58_1_1_1_1_2"/><span class="lab"><font color="Green" title="E47">A64</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:58_1_1_1_1_2"/><span class="lab"><font color="Green" title="E48">A65</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c16">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_1.html#S1" title="FINSEQ_1:sch.1">FINSEQ_1:sch 1</a>(<a class="txt" href="#E3:58_1_1_1_1_2"><span class="lab"><font color="Green" title="E46">A61</font></span></a>);<br/></span><a NAME="E7:58_1_1_1_1_2"/><span class="lab"><font color="Green" title="E49">A66</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c12">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1_1_1_2"><span class="lab"><font color="Green" title="E47">A64</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="E8:58_1_1_1_1_2"/><span class="lab"><font color="Green" title="E50">A67</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">c</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">p</font> or <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">c</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:58_1_1_1_1_2_2"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">c</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:58_1_1_1_1_2_2"/><span class="lab"><font color="Green" title="E51">A68</font></span>: 
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">p</font>
 <span class="kw">and </span><br/><a NAME="E4:58_1_1_1_1_2_2"/><span class="lab"><font color="Green" title="E52">A69</font></span>: 
<font color="Maroon" title="c16">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">y</font> = <font color="Maroon" title="c18">y</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:58_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E51">A68</font></span></a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1_1_2_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_2_2_1_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="E1:58_1_1_1_1_2_2_1_1"/>
( <font color="Maroon" title="c19">y</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c19">y</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T12" title="NAT_D:th.12">NAT_D:12</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_2_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:58_1_1_1_1_2_2_1_1_1"/>
<font color="Maroon" title="c19">y</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">c</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><br/><div class="add"><a NAME="E2:58_1_1_1_1_2_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1_1_1_2"><span class="lab"><font color="Green" title="E47">A64</font></span></a>, <a class="txt" href="#E6:58_1_1_1_1_2"><span class="lab"><font color="Green" title="E48">A65</font></span></a>, <a class="txt" href="#E3:58_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E51">A68</font></span></a>, <a class="txt" href="#E4:58_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E52">A69</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:58_1_1_1_1_2_2_1_1_1"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">c</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_2_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:58_1_1_1_1_2_2_1_1_2"/>
<font color="Maroon" title="c19">y</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">c</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><br/><div class="add"><a NAME="E2:58_1_1_1_1_2_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">c</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1_1_1_2"><span class="lab"><font color="Green" title="E47">A64</font></span></a>, <a class="txt" href="#E6:58_1_1_1_1_2"><span class="lab"><font color="Green" title="E48">A65</font></span></a>, <a class="txt" href="#E3:58_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E51">A68</font></span></a>, <a class="txt" href="#E4:58_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E52">A69</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:58_1_1_1_1_2_2_1_1_2"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">c</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:58_1_1_1_1_2_2"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">c</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E9:58_1_1_1_1_2"/><span class="kw">then </span>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">p</font> = <font color="Maroon" title="c16">p</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</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="E11:58_1_1_1_1_2"/><span class="lab"><font color="Green" title="E51">A70</font></span>: 
( <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H2</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <span class="p1">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H2</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H2</font></span>)</span></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_4.html#D4" target="_self" title="GROUP_4:def.4">Def5</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>] <span class="kw">means </span>( $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c12">i</font></span>)</span> &amp; $1 <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  for <font color="Olive" title="b1">F1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> $1</span>)</span> holds <br/> <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Olive" title="b1">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default">$1 <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2</span>)</span></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  );<br/><a NAME="E12:58_1_1_1_1_2"/><span class="lab"><font color="Green" title="E52">A71</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st (  for <font color="Olive" title="b2">l</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">l</font>] ) holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">l</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">k</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">l</font>] ) implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c18">k</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:58_1_1_1_1_2_3"/><span class="lab"><font color="Green" title="E53">A72</font></span>: 
 for <font color="Olive" title="b1">l</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">k</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">l</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c18">k</font>]</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:58_1_1_1_1_2_3"/><span class="lab"><font color="Green" title="E54">A73</font></span>: 
<font color="Maroon" title="c18">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c12">i</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:58_1_1_1_1_2_3"/><span class="lab"><font color="Green" title="E55">A74</font></span>: 
<font color="Maroon" title="c18">k</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">F1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c18">k</font></span>)</span> holds <br/> <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Olive" title="b1">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">k</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2</span>)</span></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c19">F1</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c19">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c18">k</font></span>)</span> implies  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c19">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">k</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2</span>)</span></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E4:58_1_1_1_1_2_3"/><span class="lab"><font color="Green" title="E56">A75</font></span>: 
<font color="Maroon" title="c19">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c18">k</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c19">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">k</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2</span>)</span></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1_1_2_3_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_2_3_1_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="E1:58_1_1_1_1_2_3_1_1"/>
( <font color="Maroon" title="c18">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c18">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_2_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:58_1_1_1_1_2_3_1_1_1"/><span class="lab"><font color="Green" title="E57">A76</font></span>: 
<font color="Maroon" title="c18">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c19">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">k</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2</span>)</span></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span><br/><div class="add"><a NAME="E2:58_1_1_1_1_2_3_1_1_1"/>
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E3:58_1_1_1_1_2_3_1_1_1"/><span class="kw">then </span>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T18" title="NAT_D:th.18">NAT_D:18</a></span>;<br/><a NAME="E4:58_1_1_1_1_2_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A77</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">k</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_1_1_1_2_3_1_1_1"><span class="lab"><font color="Green" title="E57">A76</font></span></a>, <a class="ref" href="group_1.html#T25" title="GROUP_1:th.25">GROUP_1:25</a></span>;<br/><a NAME="E5:58_1_1_1_1_2_3_1_1_1"/>
<font color="Maroon" title="c19">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_1_1_1_2_3"><span class="lab"><font color="Green" title="E56">A75</font></span></a>, <a class="txt" href="#E1:58_1_1_1_1_2_3_1_1_1"><span class="lab"><font color="Green" title="E57">A76</font></span></a></span>;<br/><a NAME="E6:58_1_1_1_1_2_3_1_1_1"/><span class="kw">then </span>
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c19">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_4.html#T8" target="_self" title="GROUP_4:th.8">Th11</a></span>;<br/><span class="kw">hence </span><a NAME="E7:58_1_1_1_1_2_3_1_1_1"/>
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c19">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">k</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2</span>)</span></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_1_1_1_2_3_1_1_1"><span class="lab"><font color="Green" title="E58">A77</font></span></a>, <a class="ref" href="group_1.html#T8" title="GROUP_1:th.8">GROUP_1:8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_2_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:58_1_1_1_1_2_3_1_1_2"/><span class="lab"><font color="Green" title="E57">A78</font></span>: 
<font color="Maroon" title="c18">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c19">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">k</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2</span>)</span></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span><br/><div class="add"><a NAME="E2:58_1_1_1_1_2_3_1_1_2"/><span class="lab"><font color="Green" title="E58">A79</font></span>: 
<font color="Maroon" title="c18">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:58_1_1_1_1_2_3"><span class="lab"><font color="Green" title="E55">A74</font></span></a>, <a class="ref" href="nat_d.html#T14" title="NAT_D:th.14">NAT_D:14</a></span>;<br/><a NAME="E3:58_1_1_1_1_2_3_1_1_2"/>
<font color="Maroon" title="c18">k</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="#E1:58_1_1_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E57">A78</font></span></a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/><a NAME="E4:58_1_1_1_1_2_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">k</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="txt" href="#E2:58_1_1_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E58">A79</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E5:58_1_1_1_1_2_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <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="E6:58_1_1_1_1_2_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">m</font> = <font color="Maroon" title="c18">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 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#T3" title="INT_1:th.3">INT_1:3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c21">q</font> = <font color="Maroon" title="c17">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c20">m</font></span>)</span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></span>;<br/><a NAME="E9:58_1_1_1_1_2_3_1_1_2"/>
1 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ;<br/><a NAME="E10:58_1_1_1_1_2_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A80</font></span>: 
<font color="Maroon" title="c20">m</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:58_1_1_1_1_2_3"><span class="lab"><font color="Green" title="E55">A74</font></span></a>, <a class="ref" href="nat_d.html#T15" title="NAT_D:th.15">NAT_D:15</a></span>;<br/><a NAME="E11:58_1_1_1_1_2_3_1_1_2"/><span class="lab"><font color="Green" title="E60">A81</font></span>: 
<font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">k</font>
 ;<br/><a NAME="E12:58_1_1_1_1_2_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A82</font></span>: 
<font color="Maroon" title="c20">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c12">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58_1_1_1_1_2_3"><span class="lab"><font color="Green" title="E54">A73</font></span></a>, <a class="ref" href="nat_1.html#T16" title="NAT_1:th.16">NAT_1:16</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E13:58_1_1_1_1_2_3_1_1_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">o</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">o</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></span>;<br/><a NAME="E14:58_1_1_1_1_2_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A83</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:58_1_1_1_1_2"><span class="lab"><font color="Green" title="E49">A66</font></span></a>, <a class="ref" href="finseq_3.html#T53" title="FINSEQ_3:th.53">FINSEQ_3:53</a></span>;<br/><a NAME="E15:58_1_1_1_1_2_3_1_1_2"/><span class="lab"><font color="Green" title="E63">A84</font></span>: 
 ex <font color="Olive" title="b1">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b1">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58_1_1_1_1_2_3"><span class="lab"><font color="Green" title="E54">A73</font></span></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></span>;<br/><a NAME="E16:58_1_1_1_1_2_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A85</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:58_1_1_1_1_2"><span class="lab"><font color="Green" title="E49">A66</font></span></a>, <a class="txt" href="#E4:58_1_1_1_1_2_3"><span class="lab"><font color="Green" title="E56">A75</font></span></a>, <a class="ref" href="finseq_3.html#T53" title="FINSEQ_3:th.53">FINSEQ_3:53</a></span>;<br/><div><span class="lab"><font color="Green" title="E65">A86</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1_1_2_3_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c22">l</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">q</font> implies <font color="Maroon" title="c19">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">l</font> )</span><br/><span class="kw">assume </span><a NAME="E1:58_1_1_1_1_2_3_1_1_2_1"/><span class="lab"><font color="Green" title="E66">A87</font></span>: 
<font color="Maroon" title="c22">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">l</font></span><br/><a NAME="E2:58_1_1_1_1_2_3_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c22">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E3:58_1_1_1_1_2_3_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A88</font></span>: 
<font color="Maroon" title="c22">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">F1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:58_1_1_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E60">A81</font></span></a>, <a class="txt" href="#E16:58_1_1_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E64">A85</font></span></a>, <a class="txt" href="#E14:58_1_1_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E62">A83</font></span></a>, <a class="ref" href="nat_1.html#T16" title="NAT_1:th.16">NAT_1:16</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E4:58_1_1_1_1_2_3_1_1_2_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_1_1_1_2_3_1_1_2_1"><span class="lab"><font color="Green" title="E66">A87</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E5:58_1_1_1_1_2_3_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c22">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">F1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:58_1_1_1_1_2_3_1_1_2_1"><span class="lab"><font color="Green" title="E67">A88</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E6:58_1_1_1_1_2_3_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_1_1_1_2_3"><span class="lab"><font color="Green" title="E56">A75</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>;<br/><span class="kw">hence </span><a NAME="E7:58_1_1_1_1_2_3_1_1_2_1"/>
<font color="Maroon" title="c19">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_1_1_1_2_3_1_1_2_1"><span class="lab"><font color="Green" title="E66">A87</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E18:58_1_1_1_1_2_3_1_1_2"/><span class="lab"><font color="Green" title="E66">A89</font></span>: 
<font color="Maroon" title="c20">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:58_1_1_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E60">A81</font></span></a>, <a class="ref" href="nat_1.html#T16" title="NAT_1:th.16">NAT_1:16</a></span>;<br/><a NAME="E19:58_1_1_1_1_2_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A90</font></span>: 
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c21">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">m</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2</span>)</span></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_1_1_1_2_3"><span class="lab"><font color="Green" title="E53">A72</font></span></a>, <a class="txt" href="#E12:58_1_1_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E61">A82</font></span></a>, <a class="txt" href="#E10:58_1_1_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E59">A80</font></span></a></span>;<br/><div><span class="lab"><font color="Green" title="E68">A91</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1_1_2_3_1_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c22">l</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">c</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> implies <font color="Maroon" title="c19">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">c</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">l</font> )</span><br/><span class="kw">assume </span><a NAME="E1:58_1_1_1_1_2_3_1_1_2_2"/>
<font color="Maroon" title="c22">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">c</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">c</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">l</font></span><br/><a NAME="E2:58_1_1_1_1_2_3_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A92</font></span>: 
<font color="Maroon" title="c22">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default">1,2</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T2" title="FINSEQ_1:th.2">FINSEQ_1:2</a>, <a class="ref" href="finseq_1.html#T89" title="FINSEQ_1:th.89">FINSEQ_1:89</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1_1_2_3_1_1_2_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_2_3_1_1_2_2_1_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="E1:58_1_1_1_1_2_3_1_1_2_2_1_1"/>
( <font color="Maroon" title="c22">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c22">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58_1_1_1_1_2_3_1_1_2_2"><span class="lab"><font color="Green" title="E69">A92</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_2_3_1_1_2_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:58_1_1_1_1_2_3_1_1_2_2_1_1_1"/><span class="lab"><font color="Green" title="E70">A93</font></span>: 
<font color="Maroon" title="c22">l</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="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">c</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">l</font></span><br/><div class="add"><a NAME="E2:58_1_1_1_1_2_3_1_1_2_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A94</font></span>: 
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">c</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">c</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>;<br/><a NAME="E3:58_1_1_1_1_2_3_1_1_2_2_1_1_1"/><span class="lab"><font color="Green" title="E72">A95</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">F1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c17">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_1_1_1_2_3"><span class="lab"><font color="Green" title="E56">A75</font></span></a>, <a class="txt" href="#E10:58_1_1_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E59">A80</font></span></a>, <a class="ref" href="nat_d.html#T16" title="NAT_D:th.16">NAT_D:16</a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a></span>;<br/><a NAME="E4:58_1_1_1_1_2_3_1_1_2_2_1_1_1"/>
( <font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">k</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:58_1_1_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E66">A89</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E5:58_1_1_1_1_2_3_1_1_2_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A96</font></span>: 
<font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">F1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:58_1_1_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E64">A85</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E6:58_1_1_1_1_2_3_1_1_2_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_1_1_1_2_3"><span class="lab"><font color="Green" title="E56">A75</font></span></a>, <a class="txt" href="#E14:58_1_1_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E62">A83</font></span></a>, <a class="txt" href="#E1:58_1_1_1_1_2_3_1_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E70">A93</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>;<br/><span class="kw">hence </span><a NAME="E7:58_1_1_1_1_2_3_1_1_2_2_1_1_1"/>
<font color="Maroon" title="c19">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">c</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1_1_1_2"><span class="lab"><font color="Green" title="E47">A64</font></span></a>, <a class="txt" href="#E6:58_1_1_1_1_2"><span class="lab"><font color="Green" title="E48">A65</font></span></a>, <a class="txt" href="#E2:58_1_1_1_1_2_3_1_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E71">A94</font></span></a>, <a class="txt" href="#E5:58_1_1_1_1_2_3_1_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E73">A96</font></span></a>, <a class="txt" href="#E3:58_1_1_1_1_2_3_1_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E72">A95</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_2_3_1_1_2_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:58_1_1_1_1_2_3_1_1_2_2_1_1_2"/><span class="lab"><font color="Green" title="E70">A97</font></span>: 
<font color="Maroon" title="c22">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">c</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">l</font></span><br/><div class="add"><a NAME="E2:58_1_1_1_1_2_3_1_1_2_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A98</font></span>: 
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">c</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>;<br/><a NAME="E3:58_1_1_1_1_2_3_1_1_2_2_1_1_2"/><span class="lab"><font color="Green" title="E72">A99</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">F1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c17">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_1_1_1_2_3"><span class="lab"><font color="Green" title="E56">A75</font></span></a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a></span>;<br/><a NAME="E4:58_1_1_1_1_2_3_1_1_2_2_1_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E5:58_1_1_1_1_2_3_1_1_2_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A100</font></span>: 
<font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">F1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:58_1_1_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E64">A85</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E6:58_1_1_1_1_2_3_1_1_2_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c19">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_1_1_1_2_3"><span class="lab"><font color="Green" title="E56">A75</font></span></a>, <a class="txt" href="#E14:58_1_1_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E62">A83</font></span></a>, <a class="txt" href="#E1:58_1_1_1_1_2_3_1_1_2_2_1_1_2"><span class="lab"><font color="Green" title="E70">A97</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>;<br/><span class="kw">hence </span><a NAME="E7:58_1_1_1_1_2_3_1_1_2_2_1_1_2"/>
<font color="Maroon" title="c19">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">c</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1_1_1_2"><span class="lab"><font color="Green" title="E47">A64</font></span></a>, <a class="txt" href="#E6:58_1_1_1_1_2"><span class="lab"><font color="Green" title="E48">A65</font></span></a>, <a class="txt" href="#E3:58_1_1_1_1_2_3"><span class="lab"><font color="Green" title="E55">A74</font></span></a>, <a class="txt" href="#E2:58_1_1_1_1_2_3_1_1_2_2_1_1_2"><span class="lab"><font color="Green" title="E71">A98</font></span></a>, <a class="txt" href="#E5:58_1_1_1_1_2_3_1_1_2_2_1_1_2"><span class="lab"><font color="Green" title="E73">A100</font></span></a>, <a class="txt" href="#E3:58_1_1_1_1_2_3_1_1_2_2_1_1_2"><span class="lab"><font color="Green" title="E72">A99</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:58_1_1_1_1_2_3_1_1_2_2"/>
<font color="Maroon" title="c19">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">c</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">l</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="lab"><font color="Green" title="E69">A101</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c20">m</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c20">m</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">2 <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T18" title="NAT_D:th.18">NAT_D:18</a></span>
<br/>.= 
<font color="Maroon" title="c18">k</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:58_1_1_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E60">A81</font></span></a>, <a class="txt" href="#E10:58_1_1_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E59">A80</font></span></a>, <a class="ref" href="nat_d.html#T19" title="NAT_D:th.19">NAT_D:19</a></span>
;<br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">F1</font> = 
<font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:58_1_1_1_1_2"><span class="lab"><font color="Green" title="E49">A66</font></span></a>, <a class="txt" href="#E4:58_1_1_1_1_2_3"><span class="lab"><font color="Green" title="E56">A75</font></span></a>, <a class="txt" href="#E15:58_1_1_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E63">A84</font></span></a>, <a class="ref" href="finseq_3.html#T53" title="FINSEQ_3:th.53">FINSEQ_3:53</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">c</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:58_1_1_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E62">A83</font></span></a>, <a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>
;<br/><a NAME="E23:58_1_1_1_1_2_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c19">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">q</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">c</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:58_1_1_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E65">A86</font></span></a>, <a class="txt" href="#E20:58_1_1_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E68">A91</font></span></a>, <a class="ref" href="finseq_3.html#T38" title="FINSEQ_3:th.38">FINSEQ_3:38</a></span>;<br/><span class="kw">then </span> <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c19">F1</font> = 
<span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c21">q</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p2"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">c</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finsop_1.html#T5" title="FINSOP_1:th.5">FINSOP_1:5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c21">q</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">c</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finsop_1.html#T12" title="FINSOP_1:th.12">FINSOP_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c21">q</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:58_1_1"><span class="lab"><font color="Green" title="E40">A11</font></span></a>, <a class="ref" href="group_1.html#T17" title="GROUP_1:th.17">GROUP_1:17</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">m</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:58_1_1_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E67">A90</font></span></a>, <a class="ref" href="group_1.html#T17" title="GROUP_1:th.17">GROUP_1:17</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">m</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#T34" title="GROUP_1:th.34">GROUP_1:34</a></span>
;<br/><span class="kw">hence </span><a NAME="E25:58_1_1_1_1_2_3_1_1_2"/>
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c19">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">k</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2</span>)</span></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:58_1_1_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E69">A101</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:58_1_1_1_1_2_3"/>
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c19">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">k</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> 2</span>)</span></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E13:58_1_1_1_1_2"/><span class="lab"><font color="Green" title="E53">A102</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S4" title="NAT_1:sch.4">NAT_1:sch 4</a>(<a class="txt" href="#E12:58_1_1_1_1_2"><span class="lab"><font color="Green" title="E52">A71</font></span></a>);<br/></span><a NAME="E14:58_1_1_1_1_2"/>
<font color="Maroon" title="c14">c</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:58_1_1"><span class="lab"><font color="Green" title="E42">A13</font></span></a>, <a class="ref" href="group_2.html#T75" title="GROUP_2:th.75">GROUP_2:75</a></span>;<br/><a NAME="E15:58_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A103</font></span>: 
<font color="Maroon" title="c14">c</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E16:58_1_1_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c12">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1_1_1_2"><span class="lab"><font color="Green" title="E47">A64</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="E17:58_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A104</font></span>: 
<font color="Maroon" title="c17">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T49" title="FINSEQ_3:th.49">FINSEQ_3:49</a></span>;<br/><a NAME="E18:58_1_1_1_1_2"/>
<font color="Maroon" title="c13">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:58_1_1"><span class="lab"><font color="Green" title="E41">A12</font></span></a>, <a class="ref" href="group_2.html#T75" title="GROUP_2:th.75">GROUP_2:75</a></span>;<br/><a NAME="E19:58_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E20:58_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">c</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:58_1_1_1_1_2"><span class="lab"><font color="Green" title="E54">A103</font></span></a>, <a class="ref" href="zfmisc_1.html#T32" title="ZFMISC_1:th.32">ZFMISC_1:32</a></span>;<br/><a NAME="E21:58_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A105</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:58_1_1_1_1_2"><span class="lab"><font color="Green" title="E50">A67</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E22:58_1_1_1_1_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:58_1_1"><span class="lab"><font color="Green" title="E43">A14</font></span></a>, <a class="txt" href="#E1:58_1_1_1_1_2"><span class="lab"><font color="Green" title="E44">A59</font></span></a>, <a class="ref" href="group_1.html#T30" title="GROUP_1:th.30">GROUP_1:30</a></span>;<br/><a NAME="E23:58_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c17">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:58_1_1_1_1_2"><span class="lab"><font color="Green" title="E53">A102</font></span></a>, <a class="txt" href="#E2:58_1_1_1_1_2"><span class="lab"><font color="Green" title="E45">A60</font></span></a>, <a class="txt" href="#E17:58_1_1_1_1_2"><span class="lab"><font color="Green" title="E55">A104</font></span></a></span>;<br/><a NAME="E24:58_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H2</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:58_1_1_1_1_2"><span class="lab"><font color="Green" title="E56">A105</font></span></a>, <a class="txt" href="#E11:58_1_1_1_1_2"><span class="lab"><font color="Green" title="E51">A70</font></span></a>, <a class="ref" href="group_4.html#T18" target="_self" title="GROUP_4:th.18">Th21</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><span class="kw">hence </span><a NAME="E25:58_1_1_1_1_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">H2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E22:58_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">H2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E7:58_1"/>
<font color="Maroon" title="c5">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">H1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">H2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="ref" href="group_4.html#T18" target="_self" title="GROUP_4:th.18">Th21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E2:58"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A106</font></span>: 
 <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H1</font></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p2">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H2</font></span>)</span></span>)</span> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">H1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">H2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T58" title="GROUP_2:th.58">GROUP_2:58</a></span>;<br/>
<a NAME="E3:58"/>
<span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H1</font></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H2</font></span>)</span> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H1</font></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H2</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:58_2"/><span class="lab"><font color="Green" title="E31">A107</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H1</font></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H2</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">a</font> = <font color="Maroon" title="c5">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_2_1_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="E1:58_2_1_1"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H1</font> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H2</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_2"><span class="lab"><font color="Green" title="E31">A107</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:58_2_1_1_1"/><span class="lab"><font color="Green" title="E32">A108</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H1</font></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H2</font></span>)</span></span><br/><div class="add"><a NAME="E2:58_2_1_1_1"/>
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">H2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T46" title="GROUP_2:th.46">GROUP_2:46</a></span>;<br/><a NAME="E3:58_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A109</font></span>: 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/><a NAME="E4:58_2_1_1_1"/>
<font color="Maroon" title="c6">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E5:58_2_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H1</font></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_2_1_1_1"><span class="lab"><font color="Green" title="E32">A108</font></span></a>, <a class="txt" href="#E3:58_2_1_1_1"><span class="lab"><font color="Green" title="E33">A109</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:58_2_1_1_2"/><span class="lab"><font color="Green" title="E32">A110</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H1</font></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H2</font></span>)</span></span><br/><div class="add"><a NAME="E2:58_2_1_1_2"/>
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">H1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T46" title="GROUP_2:th.46">GROUP_2:46</a></span>;<br/><a NAME="E3:58_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A111</font></span>: 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/><a NAME="E4:58_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E5:58_2_1_1_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H1</font></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_2_1_1_2"><span class="lab"><font color="Green" title="E32">A110</font></span></a>, <a class="txt" href="#E3:58_2_1_1_2"><span class="lab"><font color="Green" title="E33">A111</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:58_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H1</font></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:58"/><span class="kw">then </span>
<font color="Maroon" title="c2">H1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">H2</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c2">H1</font></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p2">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">H2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_4.html#T32" target="_self" title="GROUP_4:th.32">Th41</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:58"/>
 <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="group_4.html#K7" title="GROUP_4:func.7">*</a> <font color="Maroon" title="c3">H2</font></span>)</span> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c2">H1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">H2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58"><span class="lab"><font color="Green" title="E30">A106</font></span></a>, <a class="ref" href="group_2.html#T55" title="GROUP_2:th.55">GROUP_2:55</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
