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

<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a> <a href="yellow11.html#V1" title="YELLOW11:attr.1">modular</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>;<br/><b>let </b><font color="Maroon" title="c2">S</font> be    <a href="lattice8.html#M1" title="LATTICE8:mode.1">ExtensionSeq2</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>, <a href="lattice5.html#K17" title="LATTICE5:func.17">BasicDF</a> <font color="Maroon" title="c1">L</font>;<br/><b>let </b><font color="Maroon" title="c3">FS</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c4">FD</font> be   <a href="lattice5.html#NM4" title="LATTICE5:NM.4">distance_function</a> of <font color="Maroon" title="c3">FS</font>,<font color="Maroon" title="c1">L</font>;<br/><b>let </b><font color="Maroon" title="c5">f</font> be   <a href="lattice5.html#NM1" title="LATTICE5:NM.1">Homomorphism</a> of <font color="Maroon" title="c1">L</font>,<span class="p1">(<span class="default"><a href="lattice5.html#K1" title="LATTICE5:func.1">EqRelLATT</a> <font color="Maroon" title="c3">FS</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c6">e1</font>, <font color="Maroon" title="c7">e2</font> be   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c3">FS</font>;<br/><b>let </b><font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c9">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>













<b>assume </b><b>that </b><br/><a NAME="E1:55"/><i><font color="Green" title="E44">A1</font></i>: 
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="lattice5.html#K5" title="LATTICE5:func.5">alpha</a> <font color="Maroon" title="c4">FD</font>
 <b>and </b><br/><a NAME="E2:55"/><i><font color="Green" title="E45">A2</font></i>: 
( <font color="Maroon" title="c3">FS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span>  &amp; <font color="Maroon" title="c4">FD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : verum </span>}</span>  )
 <b>and </b><br/><a NAME="E3:55"/><i><font color="Green" title="E46">A3</font></i>: 
( <font color="Maroon" title="c6">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K2" title="LATTICE5:func.2">Image</a> <font color="Maroon" title="c5">f</font></span>)</span> &amp; <font color="Maroon" title="c7">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K2" title="LATTICE5:func.2">Image</a> <font color="Maroon" title="c5">f</font></span>)</span> )
 <b>and </b><br/><a NAME="E4:55"/><i><font color="Green" title="E47">A4</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">e1</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <font color="Maroon" title="c7">e2</font>
 ;<br/>

<a NAME="E5:55"/><i><font color="Green" title="E48">A5</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K2" title="LATTICE5:func.2">Image</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">f</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:55"/><i><font color="Green" title="E49">A6</font></i>: 
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">f</font>
 <b>and </b><br/><a NAME="E8:55"/><i><font color="Green" title="E50">A7</font></i>: 
<font color="Maroon" title="c6">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E3:55"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:55"/><i><font color="Green" title="E51">A8</font></i>: 
<font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">f</font>
 <b>and </b><br/><a NAME="E11:55"/><i><font color="Green" title="E52">A9</font></i>: 
<font color="Maroon" title="c7">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">b</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E3:55"><i><font color="Green" title="E46">A3</font></i></a>, <a class="txt" href="lattice8.html#E5:55"><i><font color="Green" title="E48">A5</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">a</font> = <font color="Maroon" title="c10">a</font>, <font color="Maroon" title="c13">b</font> = <font color="Maroon" title="c11">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="lattice8.html#E7:55"><i><font color="Green" title="E49">A6</font></i></a>, <a class="txt" href="lattice8.html#E10:55"><i><font color="Green" title="E51">A8</font></i></a></i>;<br/>
<a NAME="E13:55"/>
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">e1</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <font color="Maroon" title="c7">e2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">FS</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T97" title="ORDERS_1:th.97">ORDERS_1:97</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">u</font> = <font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c15">v</font> = <font color="Maroon" title="c9">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">FS</font> <b>by </b><i><a class="txt" href="lattice8.html#E4:55"><i><font color="Green" title="E47">A4</font></i></a>, <a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">a</font> = <font color="Maroon" title="c12">a</font>, <font color="Maroon" title="c17">b</font> = <font color="Maroon" title="c13">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<b>consider </b><font color="Maroon" title="c18">e1'</font> being   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c3">FS</font><b> such that </b><br/><a NAME="E17:55"/><i><font color="Green" title="E53">A10</font></i>: 
<font color="Maroon" title="c18">e1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">a</font>
 <b>and </b><br/><a NAME="E18:55"/><i><font color="Green" title="E54">A11</font></i>: 
for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">FS</font> holds <br/> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">v</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">e1'</font> iff <font color="Maroon" title="c4">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">v</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">a</font> )
 <b>by </b><i><a class="txt" href="lattice8.html#E1:55"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="lattice5.html#D9" title="LATTICE5:def.9">LATTICE5:def 9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c19">e2'</font> being   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c3">FS</font><b> such that </b><br/><a NAME="E20:55"/><i><font color="Green" title="E55">A12</font></i>: 
<font color="Maroon" title="c19">e2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">b</font>
 <b>and </b><br/><a NAME="E21:55"/><i><font color="Green" title="E56">A13</font></i>: 
for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">FS</font> holds <br/> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">v</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">e2'</font> iff <font color="Maroon" title="c4">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">v</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c17">b</font> )
 <b>by </b><i><a class="txt" href="lattice8.html#E1:55"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="lattice5.html#D9" title="LATTICE5:def.9">LATTICE5:def 9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c20">e</font> being   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c3">FS</font><b> such that </b><br/><a NAME="E23:55"/><i><font color="Green" title="E57">A14</font></i>: 
<font color="Maroon" title="c20">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c17">b</font></span>)</span>
 <b>and </b><br/><a NAME="E24:55"/><i><font color="Green" title="E58">A15</font></i>: 
for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">FS</font> holds <br/> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">v</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">e</font> iff <font color="Maroon" title="c4">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">v</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c17">b</font> )
 <b>by </b><i><a class="txt" href="lattice8.html#E1:55"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="lattice5.html#D9" title="LATTICE5:def.9">LATTICE5:def 9</a></i>;<br/>
<font color="Maroon" title="c20">e</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">b</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice8.html#E23:55"><i><font color="Green" title="E57">A14</font></i></a>, <a class="ref" href="waybel_6.html#T2" title="WAYBEL_6:th.2">WAYBEL_6:2</a></i>
<br/>.= 
<font color="Maroon" title="c6">e1</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <font color="Maroon" title="c7">e2</font>
<b>by </b><i><a class="txt" href="lattice8.html#E8:55"><i><font color="Green" title="E50">A7</font></i></a>, <a class="txt" href="lattice8.html#E11:55"><i><font color="Green" title="E52">A9</font></i></a>, <a class="ref" href="lattice5.html#T10" title="LATTICE5:th.10">LATTICE5:10</a></i>
;<br/>
<a NAME="E26:55"/><b>then </b><i><font color="Green" title="E59">A16</font></i>: 
<font color="Maroon" title="c4">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u</font>,<font color="Maroon" title="c15">v</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c17">b</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E4:55"><i><font color="Green" title="E47">A4</font></i></a>, <a class="txt" href="lattice8.html#E24:55"><i><font color="Green" title="E58">A15</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c21">z1</font>, <font color="Maroon" title="c22">z2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">FS</font><b> such that </b><br/><a NAME="E28:55"/><i><font color="Green" title="E60">A17</font></i>: 
<font color="Maroon" title="c4">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u</font>,<font color="Maroon" title="c21">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">a</font>
 <b>and </b><br/><a NAME="E29:55"/><i><font color="Green" title="E61">A18</font></i>: 
<font color="Maroon" title="c4">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c21">z1</font>,<font color="Maroon" title="c22">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u</font>,<font color="Maroon" title="c15">v</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c17">b</font>
 <b>and </b><br/><a NAME="E30:55"/><i><font color="Green" title="E62">A19</font></i>: 
<font color="Maroon" title="c4">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c22">z2</font>,<font color="Maroon" title="c15">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">a</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E2:55"><i><font color="Green" title="E45">A2</font></i></a>, <a class="ref" href="lattice8.html#T34" target="_self" title="LATTICE8:th.34">Th34</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( ( $1 <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">u</font> ) &amp; ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">z1</font> ) &amp; ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">z2</font> ) &amp; ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">v</font> ) );<br/>
<a NAME="E31:55"/><i><font color="Green" title="E63">A20</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">w1</font>, <font color="Olive" title="b3">w2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 4 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">w1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>,<font color="Olive" title="b3">w2</font>] holds <br/><font color="Olive" title="b2">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">w2</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E49"><i><font color="Green" title="E41">Lm3</font></i></a></i>;<br/>
<a NAME="E32:55"/><i><font color="Green" title="E64">A21</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 4 holds <br/> ex <font color="Olive" title="b2">w</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">m</font>,<font color="Olive" title="b2">w</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:55_2"/><i><font color="Green" title="E65">A22</font></i>: 
<font color="Maroon" title="c23">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 4
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_2_1"/>
( <font color="Maroon" title="c23">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c23">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c23">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c23">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 )
 </span><b>by </b><i><a class="txt" href="lattice8.html#E1:55_2"><i><font color="Green" title="E65">A22</font></i></a>, <a class="txt" href="lattice8.html#E49"><i><font color="Green" title="E41">Lm3</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_2_1_1"><b>suppose </b></a><a NAME="E1:55_2_1_1"/><i><font color="Green" title="E66">A23</font></i>: 
<font color="Maroon" title="c23">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c24">w</font> = <font color="Maroon" title="c8">x</font>;<br/><b>thus </b><a NAME="E2:55_2_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c23">m</font>,<font color="Maroon" title="c24">w</font>]
 <b>by </b><i><a class="txt" href="lattice8.html#E1:55_2_1_1"><i><font color="Green" title="E66">A23</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_2_1_2"><b>suppose </b></a><a NAME="E1:55_2_1_2"/><i><font color="Green" title="E66">A24</font></i>: 
<font color="Maroon" title="c23">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c24">w</font> = <font color="Maroon" title="c21">z1</font>;<br/><b>thus </b><a NAME="E2:55_2_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c23">m</font>,<font color="Maroon" title="c24">w</font>]
 <b>by </b><i><a class="txt" href="lattice8.html#E1:55_2_1_2"><i><font color="Green" title="E66">A24</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_2_1_3"><b>suppose </b></a><a NAME="E1:55_2_1_3"/><i><font color="Green" title="E66">A25</font></i>: 
<font color="Maroon" title="c23">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c24">w</font> = <font color="Maroon" title="c22">z2</font>;<br/><b>thus </b><a NAME="E2:55_2_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c23">m</font>,<font color="Maroon" title="c24">w</font>]
 <b>by </b><i><a class="txt" href="lattice8.html#E1:55_2_1_3"><i><font color="Green" title="E66">A25</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_2_1_4"><b>suppose </b></a><a NAME="E1:55_2_1_4"/><i><font color="Green" title="E66">A26</font></i>: 
<font color="Maroon" title="c23">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c24">w</font> = <font color="Maroon" title="c9">y</font>;<br/><b>thus </b><a NAME="E2:55_2_1_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c23">m</font>,<font color="Maroon" title="c24">w</font>]
 <b>by </b><i><a class="txt" href="lattice8.html#E1:55_2_1_4"><i><font color="Green" title="E66">A26</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E33:55"/>
 ex <font color="Olive" title="b1">p</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">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> 4 &amp; ( for <font color="Olive" title="b2">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">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> 4 holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">k</font>,<font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">k</font>] ) )
 
<b>from </b><i><a class="ref" href="finseq_1.html#S1" title="FINSEQ_1:sch.1">FINSEQ_1:sch 1</a>(<a class="txt" href="lattice8.html#E31:55"><i><font color="Green" title="E63">A20</font></i></a>, <a class="txt" href="lattice8.html#E32:55"><i><font color="Green" title="E64">A21</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon" title="c23">h</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E35:55"/><i><font color="Green" title="E65">A27</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 4
 <b>and </b><br/><a NAME="E36:55"/><i><font color="Green" title="E66">A28</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 4 holds <br/>( ( <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Maroon" title="c23">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">u</font> ) &amp; ( <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies <font color="Maroon" title="c23">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">z1</font> ) &amp; ( <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 implies <font color="Maroon" title="c23">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">z2</font> ) &amp; ( <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 implies <font color="Maroon" title="c23">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">v</font> ) )
 ;<br/>
<a NAME="E37:55"/>
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">n</font> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 4 ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D1" title="FINSEQ_1:def.1">FINSEQ_1:def 1</a></i>;<br/>
<a NAME="E38:55"/><b>then </b><i><font color="Green" title="E67">A29</font></i>: 
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 4 &amp; 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 4 &amp; 3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 4 &amp; 4 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 4 )
 
;<br/>
<a NAME="E39:55"/><i><font color="Green" title="E68">A30</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 
<b>by </b><i><a class="txt" href="lattice8.html#E35:55"><i><font color="Green" title="E65">A27</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E40:55"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c23">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">FS</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c24">w</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:55_3"/>
<font color="Maroon" title="c24">w</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="c23">h</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c25">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:55_3"/><i><font color="Green" title="E69">A31</font></i>: 
( <font color="Maroon" title="c25">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">h</font> &amp; <font color="Maroon" title="c24">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_3_1"/>
( <font color="Maroon" title="c25">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c25">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c25">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c25">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 )
 </span><b>by </b><i><a class="txt" href="lattice8.html#E35:55"><i><font color="Green" title="E65">A27</font></i></a>, <a class="txt" href="lattice8.html#E3:55_3"><i><font color="Green" title="E69">A31</font></i></a>, <a class="txt" href="lattice8.html#E49"><i><font color="Green" title="E41">Lm3</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_3_1_1"><b>suppose </b></a><a NAME="E1:55_3_1_1"/>
<font color="Maroon" title="c25">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:55_3_1_1"/><b>then </b>
<font color="Maroon" title="c23">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">u</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E36:55"><i><font color="Green" title="E66">A28</font></i></a>, <a class="txt" href="lattice8.html#E38:55"><i><font color="Green" title="E67">A29</font></i></a></i>;<br/><b>hence </b><a NAME="E3:55_3_1_1"/>
<font color="Maroon" title="c24">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">FS</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E3:55_3"><i><font color="Green" title="E69">A31</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_3_1_2"><b>suppose </b></a><a NAME="E1:55_3_1_2"/>
<font color="Maroon" title="c25">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:55_3_1_2"/><b>then </b>
<font color="Maroon" title="c23">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">z1</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E36:55"><i><font color="Green" title="E66">A28</font></i></a>, <a class="txt" href="lattice8.html#E38:55"><i><font color="Green" title="E67">A29</font></i></a></i>;<br/><b>hence </b><a NAME="E3:55_3_1_2"/>
<font color="Maroon" title="c24">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">FS</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E3:55_3"><i><font color="Green" title="E69">A31</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_3_1_3"><b>suppose </b></a><a NAME="E1:55_3_1_3"/>
<font color="Maroon" title="c25">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ;<br/><div class="add"><a NAME="E2:55_3_1_3"/><b>then </b>
<font color="Maroon" title="c23">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">z2</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E36:55"><i><font color="Green" title="E66">A28</font></i></a>, <a class="txt" href="lattice8.html#E38:55"><i><font color="Green" title="E67">A29</font></i></a></i>;<br/><b>hence </b><a NAME="E3:55_3_1_3"/>
<font color="Maroon" title="c24">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">FS</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E3:55_3"><i><font color="Green" title="E69">A31</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_3_1_4"><b>suppose </b></a><a NAME="E1:55_3_1_4"/>
<font color="Maroon" title="c25">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 ;<br/><div class="add"><a NAME="E2:55_3_1_4"/><b>then </b>
<font color="Maroon" title="c23">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">v</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E36:55"><i><font color="Green" title="E66">A28</font></i></a>, <a class="txt" href="lattice8.html#E38:55"><i><font color="Green" title="E67">A29</font></i></a></i>;<br/><b>hence </b><a NAME="E3:55_3_1_4"/>
<font color="Maroon" title="c24">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">FS</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E3:55_3"><i><font color="Green" title="E69">A31</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c24">h</font> = <font color="Maroon" title="c23">h</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c3">FS</font> <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E42:55"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">h</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="lattice8.html#E35:55"><i><font color="Green" title="E65">A27</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c25">h</font> = <font color="Maroon" title="c24">h</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c3">FS</font> <b>by </b><i><a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c25">h</font>
;<br/>

<b>thus </b><a NAME="E44:55"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 <b>by </b><i><a class="txt" href="lattice8.html#E35:55"><i><font color="Green" title="E65">A27</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>

<a NAME="E45:55"/><i><font color="Green" title="E69">A32</font></i>: 
<font color="Maroon" title="c25">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E36:55"><i><font color="Green" title="E66">A28</font></i></a>, <a class="txt" href="lattice8.html#E38:55"><i><font color="Green" title="E67">A29</font></i></a></i>;<br/>
<i><font color="Green" title="E70">A33</font></i>: <font color="Maroon" title="c25">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">h</font></span>)</span> = 
<font color="Maroon" title="c25">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 4
<b>by </b><i><a class="txt" href="lattice8.html#E35:55"><i><font color="Green" title="E65">A27</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c9">y</font>
<b>by </b><i><a class="txt" href="lattice8.html#E36:55"><i><font color="Green" title="E66">A28</font></i></a>, <a class="txt" href="lattice8.html#E38:55"><i><font color="Green" title="E67">A29</font></i></a></i>
;<br/>
<a NAME="E47:55"/>
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">h</font> holds <br/>( ( not <font color="Olive" title="b1">j</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c25">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">e1</font> ) &amp; ( <font color="Olive" title="b1">j</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c25">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">e2</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c26">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:55_5"/><i><font color="Green" title="E71">A34</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">j</font> &amp; <font color="Maroon" title="c26">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">h</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_5_1"/>
( <font color="Maroon" title="c26">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c26">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c26">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 </span><b>by </b><i><a class="txt" href="lattice8.html#E39:55"><i><font color="Green" title="E68">A30</font></i></a>, <a class="txt" href="lattice8.html#E1:55_5"><i><font color="Green" title="E71">A34</font></i></a>, <a class="txt" href="lattice8.html#E50"><i><font color="Green" title="E42">Lm4</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_5_1_1"><b>suppose </b></a><a NAME="E1:55_5_1_1"/><i><font color="Green" title="E72">A35</font></i>: 
<font color="Maroon" title="c26">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:55_5_1_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">u</font>,<font color="Maroon" title="c21">z1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">e1'</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E18:55"><i><font color="Green" title="E54">A11</font></i></a>, <a class="txt" href="lattice8.html#E28:55"><i><font color="Green" title="E60">A17</font></i></a></i>;<br/><a NAME="E3:55_5_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<font color="Maroon" title="c21">z1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">e1'</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E36:55"><i><font color="Green" title="E66">A28</font></i></a>, <a class="txt" href="lattice8.html#E38:55"><i><font color="Green" title="E67">A29</font></i></a></i>;<br/><b>hence </b><a NAME="E4:55_5_1_1"/>
( ( not <font color="Maroon" title="c26">j</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c25">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">e1</font> ) &amp; ( <font color="Maroon" title="c26">j</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c25">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">e2</font> ) )
 <b>by </b><i><a class="txt" href="lattice8.html#E8:55"><i><font color="Green" title="E50">A7</font></i></a>, <a class="txt" href="lattice8.html#E17:55"><i><font color="Green" title="E53">A10</font></i></a>, <a class="txt" href="lattice8.html#E36:55"><i><font color="Green" title="E66">A28</font></i></a>, <a class="txt" href="lattice8.html#E38:55"><i><font color="Green" title="E67">A29</font></i></a>, <a class="txt" href="lattice8.html#E1:55_5_1_1"><i><font color="Green" title="E72">A35</font></i></a>, <a class="txt" href="lattice8.html#E4"><i><font color="Green" title="E3">Lm1</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_5_1_2"><b>suppose </b></a><a NAME="E1:55_5_1_2"/><i><font color="Green" title="E72">A36</font></i>: 
<font color="Maroon" title="c26">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c27">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:55_5_1_2"/><i><font color="Green" title="E73">A37</font></i>: 
<font color="Maroon" title="c27">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><a NAME="E4:55_5_1_2"/><i><font color="Green" title="E74">A38</font></i>: 
2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c27">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:55_5_1_2"><i><font color="Green" title="E72">A36</font></i></a>, <a class="txt" href="lattice8.html#E3:55_5_1_2"><i><font color="Green" title="E73">A37</font></i></a></i>;<br/><a NAME="E5:55_5_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u</font>,<font color="Maroon" title="c15">v</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c16">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c17">b</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c16">a</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E26:55"><i><font color="Green" title="E59">A16</font></i></a>, <a class="ref" href="waybel_1.html#T3" title="WAYBEL_1:th.3">WAYBEL_1:3</a></i>;<br/><a NAME="E6:55_5_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u</font>,<font color="Maroon" title="c15">v</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c16">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c17">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c16">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="lattice3.html#T14" title="LATTICE3:th.14">LATTICE3:14</a></i>;<br/><a NAME="E7:55_5_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u</font>,<font color="Maroon" title="c15">v</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c16">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c17">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c16">a</font>
 <b>by </b><i><a class="ref" href="yellow_5.html#T1" title="YELLOW_5:th.1">YELLOW_5:1</a></i>;<br/><a NAME="E8:55_5_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u</font>,<font color="Maroon" title="c15">v</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c17">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c17">b</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c16">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_1.html#T2" title="WAYBEL_1:th.2">WAYBEL_1:2</a></i>;<br/><a NAME="E9:55_5_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">FD</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u</font>,<font color="Maroon" title="c15">v</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c17">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c17">b</font>
 <b>by </b><i><a class="ref" href="lattice3.html#T18" title="LATTICE3:th.18">LATTICE3:18</a></i>;<br/><a NAME="E10:55_5_1_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c21">z1</font>,<font color="Maroon" title="c22">z2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">e2'</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E21:55"><i><font color="Green" title="E56">A13</font></i></a>, <a class="txt" href="lattice8.html#E29:55"><i><font color="Green" title="E61">A18</font></i></a></i>;<br/><a NAME="E11:55_5_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span>,<font color="Maroon" title="c22">z2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">e2'</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E36:55"><i><font color="Green" title="E66">A28</font></i></a>, <a class="txt" href="lattice8.html#E38:55"><i><font color="Green" title="E67">A29</font></i></a></i>;<br/><b>hence </b><a NAME="E12:55_5_1_2"/>
( ( not <font color="Maroon" title="c26">j</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c25">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">e1</font> ) &amp; ( <font color="Maroon" title="c26">j</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c25">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">e2</font> ) )
 <b>by </b><i><a class="txt" href="lattice8.html#E11:55"><i><font color="Green" title="E52">A9</font></i></a>, <a class="txt" href="lattice8.html#E20:55"><i><font color="Green" title="E55">A12</font></i></a>, <a class="txt" href="lattice8.html#E36:55"><i><font color="Green" title="E66">A28</font></i></a>, <a class="txt" href="lattice8.html#E38:55"><i><font color="Green" title="E67">A29</font></i></a>, <a class="txt" href="lattice8.html#E1:55_5_1_2"><i><font color="Green" title="E72">A36</font></i></a>, <a class="txt" href="lattice8.html#E4:55_5_1_2"><i><font color="Green" title="E74">A38</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_5_1_3"><b>suppose </b></a><a NAME="E1:55_5_1_3"/><i><font color="Green" title="E72">A39</font></i>: 
<font color="Maroon" title="c26">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c27">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:55_5_1_3"/><i><font color="Green" title="E73">A40</font></i>: 
<font color="Maroon" title="c27">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><a NAME="E4:55_5_1_3"/><i><font color="Green" title="E74">A41</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c27">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:55_5_1_3"><i><font color="Green" title="E72">A39</font></i></a>, <a class="txt" href="lattice8.html#E3:55_5_1_3"><i><font color="Green" title="E73">A40</font></i></a></i>;<br/><a NAME="E5:55_5_1_3"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c22">z2</font>,<font color="Maroon" title="c15">v</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">e1'</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E18:55"><i><font color="Green" title="E54">A11</font></i></a>, <a class="txt" href="lattice8.html#E30:55"><i><font color="Green" title="E62">A19</font></i></a></i>;<br/><a NAME="E6:55_5_1_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3</span>)</span>,<font color="Maroon" title="c15">v</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">e1'</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E36:55"><i><font color="Green" title="E66">A28</font></i></a>, <a class="txt" href="lattice8.html#E38:55"><i><font color="Green" title="E67">A29</font></i></a></i>;<br/><b>hence </b><a NAME="E7:55_5_1_3"/>
( ( not <font color="Maroon" title="c26">j</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c25">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">e1</font> ) &amp; ( <font color="Maroon" title="c26">j</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c25">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">e2</font> ) )
 <b>by </b><i><a class="txt" href="lattice8.html#E8:55"><i><font color="Green" title="E50">A7</font></i></a>, <a class="txt" href="lattice8.html#E17:55"><i><font color="Green" title="E53">A10</font></i></a>, <a class="txt" href="lattice8.html#E36:55"><i><font color="Green" title="E66">A28</font></i></a>, <a class="txt" href="lattice8.html#E38:55"><i><font color="Green" title="E67">A29</font></i></a>, <a class="txt" href="lattice8.html#E1:55_5_1_3"><i><font color="Green" title="E72">A39</font></i></a>, <a class="txt" href="lattice8.html#E4:55_5_1_3"><i><font color="Green" title="E74">A41</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E48:55"/>
<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font> <a href="lattice5.html#R1" title="LATTICE5:pred.1">are_joint_by</a> <font color="Maroon" title="c25">h</font>,<font color="Maroon" title="c6">e1</font>,<font color="Maroon" title="c7">e2</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E45:55"><i><font color="Green" title="E69">A32</font></i></a>, <a class="txt" href="lattice8.html#E46:55"><i><font color="Green" title="E70">A33</font></i></a>, <a class="ref" href="lattice5.html#D3" title="LATTICE5:def.3">LATTICE5:def 3</a></i>;<br/>


</div>
