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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:29"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="yellow11.html#V1" target="_self">modular</a>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ( <font color="Maroon">a<sub>2</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>2</sub></font> ) );<br/>
<a NAME="E2:29"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2"><a href="yellow11.html#K3" target="_self">[#</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="yellow11.html#K3" target="_self">#]</a></span></span>)</span> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2"><a href="yellow11.html#K3" target="_self">[#</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span><a href="yellow11.html#K3" target="_self">#]</a></span></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:29_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2"><a href="yellow11.html#K3" target="_self">[#</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="yellow11.html#K3" target="_self">#]</a></span></span>)</span>
 ;<br/>

<a NAME="E2:29_1"/><b>then </b><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="yellow11.html#K3" target="_self">[#</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="yellow11.html#K3" target="_self">#]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>take </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>5</sub></font>;<br/>

<a NAME="E4:29_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="yellow11.html#E2:29_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="yellow11.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E5:29_1"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="yellow_5.html#T6">YELLOW_5:6</a></i>;<br/>
<a NAME="E6:29_1"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="lattice3.html#T18">LATTICE3:18</a></i>;<br/>
<a NAME="E7:29_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="yellow11.html#K3" target="_self">[#</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span><a href="yellow11.html#K3" target="_self">#]</a></span>
 
<b>by </b><i><a class="ref" href="yellow11.html#D4" target="_self">Def4</a></i>;<br/>
<b>hence </b><a NAME="E8:29_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2"><a href="yellow11.html#K3" target="_self">[#</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span><a href="yellow11.html#K3" target="_self">#]</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>

<b>thus </b><a NAME="E9:29_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2"><a href="yellow11.html#K3" target="_self">[#</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="yellow11.html#K3" target="_self">#]</a></span></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2"><a href="yellow11.html#K3" target="_self">[#</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span><a href="yellow11.html#K3" target="_self">#]</a></span></span>)</span><b> such that </b><br/><a NAME="E4:29"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2"><a href="yellow11.html#K3" target="_self">[#</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="yellow11.html#K3" target="_self">#]</a></span></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>(<a class="txt" href="yellow11.html#E2:29"><i><font color="Green">E18</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2"><a href="yellow11.html#K3" target="_self">[#</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="yellow11.html#K3" target="_self">#]</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2"><a href="yellow11.html#K3" target="_self">[#</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span><a href="yellow11.html#K3" target="_self">#]</a></span></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D8">WAYBEL_1:def 8</a><br/>

<b>thus </b><a NAME="E6:29"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="waybel_0.html#V23">isomorphic</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><b>proof </b></a><div class="add">

<a NAME="E1:29_2"/>
not  <a href="yellow_0.html#K5">subrelstr</a> <span class="p1"><a href="yellow11.html#K3" target="_self">[#</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="yellow11.html#K3" target="_self">#]</a></span> is <a href="struct_0.html#V3">empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:29_2_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/>
<a NAME="E2:29_2_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="yellow11.html#K3" target="_self">[#</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="yellow11.html#K3" target="_self">#]</a></span>
 
<b>by </b><i><a class="ref" href="yellow11.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E3:29_2_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2"><a href="yellow11.html#K3" target="_self">[#</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="yellow11.html#K3" target="_self">#]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<b>hence </b><a NAME="E4:29_2_1"/>
not  <a href="yellow_0.html#K5">subrelstr</a> <span class="p1"><a href="yellow11.html#K3" target="_self">[#</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="yellow11.html#K3" target="_self">#]</a></span> is <a href="struct_0.html#V3">empty</a>
 <b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="yellow_0.html#K5">subrelstr</a> <span class="p1"><a href="yellow11.html#K3" target="_self">[#</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="yellow11.html#K3" target="_self">#]</a></span> as  non <a href="struct_0.html#V3">empty</a> <a href="yellow_0.html#V4">full</a> <a href="lattice5.html#NM2">Sublattice</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:29_2"/>
not  <a href="yellow_0.html#K5">subrelstr</a> <span class="p1"><a href="yellow11.html#K3" target="_self">[#</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span><a href="yellow11.html#K3" target="_self">#]</a></span> is <a href="struct_0.html#V3">empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:29_2_2"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/>
<a NAME="E2:29_2_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="yellow11.html#K3" target="_self">[#</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span><a href="yellow11.html#K3" target="_self">#]</a></span>
 
<b>by </b><i><a class="ref" href="yellow11.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E3:29_2_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2"><a href="yellow11.html#K3" target="_self">[#</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span><a href="yellow11.html#K3" target="_self">#]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<b>hence </b><a NAME="E4:29_2_2"/>
not  <a href="yellow_0.html#K5">subrelstr</a> <span class="p1"><a href="yellow11.html#K3" target="_self">[#</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span><a href="yellow11.html#K3" target="_self">#]</a></span> is <a href="struct_0.html#V3">empty</a>
 <b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <a href="yellow_0.html#K5">subrelstr</a> <span class="p1"><a href="yellow11.html#K3" target="_self">[#</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span><a href="yellow11.html#K3" target="_self">#]</a></span> as  non <a href="struct_0.html#V3">empty</a> <a href="yellow_0.html#V4">full</a> <a href="lattice5.html#NM2">Sublattice</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> ;<br/>
<a NAME="E6:29_2"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2"><a href="yellow11.html#K3" target="_self">[#</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="yellow11.html#K3" target="_self">#]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:29_2"/><b>then </b><i><font color="Green">E20</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow11.html#K3" target="_self">[#</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="yellow11.html#K3" target="_self">#]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<a NAME="E8:29_2"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>8</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/>

<b>assume </b><a NAME="E1:29_2_3"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="yellow11.html#E7:29_2"><i><font color="Green">E20</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="yellow11.html#E7:29_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="yellow11.html#E1:29_2_3"><i><font color="Green">E22</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="yellow11.html#E4:29"><i><font color="Green">E19</font></i></a>, <a class="txt" href="yellow11.html#E1:29_2_3"><i><font color="Green">E22</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="yellow11.html#E4:29"><i><font color="Green">E19</font></i></a>, <a class="txt" href="yellow11.html#E1:29_2_3"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E6:29_2_3"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="yellow11.html#E7:29_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="yellow11.html#E1:29_2_3"><i><font color="Green">E22</font></i></a>, <a class="ref" href="yellow11.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E7:29_2_3"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="yellow11.html#E4:29"><i><font color="Green">E19</font></i></a>, <a class="txt" href="yellow11.html#E1:29_2_3"><i><font color="Green">E22</font></i></a></i>;<br/>
<font color="Maroon">c<sub>11</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="yellow11.html#E6:29_2_3"><i><font color="Green">E23</font></i></a>, <a class="ref" href="yellow_5.html#T10">YELLOW_5:10</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="yellow11.html#E1:29"><i><font color="Green">E17</font></i></a>, <a class="txt" href="yellow11.html#E1:29_2_3"><i><font color="Green">E22</font></i></a>, <a class="txt" href="yellow11.html#E6:29_2_3"><i><font color="Green">E23</font></i></a>, <a class="txt" href="yellow11.html#E7:29_2_3"><i><font color="Green">E24</font></i></a>, <a class="ref" href="yellow11.html#D3" target="_self">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="yellow11.html#E1:29"><i><font color="Green">E17</font></i></a>, <a class="txt" href="yellow11.html#E6:29_2_3"><i><font color="Green">E23</font></i></a>, <a class="ref" href="yellow11.html#D3" target="_self">Def3</a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="yellow11.html#E6:29_2_3"><i><font color="Green">E23</font></i></a>, <a class="ref" href="yellow_5.html#T10">YELLOW_5:10</a></i>
;<br/>
<b>hence </b><a NAME="E9:29_2_3"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E9:29_2"/><i><font color="Green">E22</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2"><a href="yellow11.html#K3" target="_self">[#</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span><a href="yellow11.html#K3" target="_self">#]</a></span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:29_2_4"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2"><a href="yellow11.html#K3" target="_self">[#</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span><a href="yellow11.html#K3" target="_self">#]</a></span></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>thus </b><a NAME="E2:29_2_4"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2"><a href="yellow11.html#K3" target="_self">[#</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span><a href="yellow11.html#K3" target="_self">#]</a></span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>8</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:29_2_4_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2"><a href="yellow11.html#K3" target="_self">[#</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span><a href="yellow11.html#K3" target="_self">#]</a></span></span>)</span>
 ;<br/>

<a NAME="E2:29_2_4_1"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="yellow11.html#K3" target="_self">[#</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span><a href="yellow11.html#K3" target="_self">#]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E4:29_2_4_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="yellow11.html#E2:29_2_4_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="yellow11.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E5:29_2_4_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="waybel_1.html#T3">WAYBEL_1:3</a></i>;<br/>
<a NAME="E6:29_2_4_1"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="lattice3.html#T17">LATTICE3:17</a></i>;<br/>
<a NAME="E7:29_2_4_1"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="yellow11.html#K3" target="_self">[#</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="yellow11.html#K3" target="_self">#]</a></span>
 
<b>by </b><i><a class="ref" href="yellow11.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E8:29_2_4_1"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2"><a href="yellow11.html#K3" target="_self">[#</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="yellow11.html#K3" target="_self">#]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="yellow11.html#E4:29"><i><font color="Green">E19</font></i></a></i>;<br/>
<font color="Maroon">c<sub>11</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="yellow11.html#E4:29"><i><font color="Green">E19</font></i></a>, <a class="txt" href="yellow11.html#E8:29_2_4_1"><i><font color="Green">E26</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="yellow11.html#E1:29"><i><font color="Green">E17</font></i></a>, <a class="txt" href="yellow11.html#E4:29_2_4_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="yellow11.html#D3" target="_self">Def3</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="yellow11.html#E4:29_2_4_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="yellow_5.html#T8">YELLOW_5:8</a></i>
;<br/>
<b>hence </b><a NAME="E11:29_2_4_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="yellow11.html#E7:29_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="yellow11.html#E7:29_2_4_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E10:29_2"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>6</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>2</sub></font> iff <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>6</sub></font>;<br/>

<a NAME="E1:29_2_5"/><i><font color="Green">E23</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow11.html#K3" target="_self">[#</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="yellow11.html#K3" target="_self">#]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<a NAME="E2:29_2_5"/><b>then </b><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="yellow11.html#K3" target="_self">[#</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="yellow11.html#K3" target="_self">#]</a></span> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="yellow11.html#K3" target="_self">[#</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="yellow11.html#K3" target="_self">#]</a></span> )
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="yellow11.html#E2:29_2_5"><i><font color="Green">E24</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="yellow11.html#E4:29"><i><font color="Green">E19</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="yellow11.html#E4:29"><i><font color="Green">E19</font></i></a></i>;<br/>
<b>thus </b><a NAME="E7:29_2_5"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> implies <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_5_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_2_5_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<a NAME="E2:29_2_5_1"/><b>then </b><i><font color="Green">E25</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E3:29_2_5_1"/>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D13">YELLOW_0:def 13</a></i>;<br/>
<a NAME="E4:29_2_5_1"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="yellow11.html#E2:29_2_5_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E5:29_2_5_1"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="yellow11.html#E4:29"><i><font color="Green">E19</font></i></a></i>;<br/>
<a NAME="E6:29_2_5_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="yellow11.html#E4:29_2_5_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="waybel_1.html#T2">WAYBEL_1:2</a></i>;<br/>
<b>hence </b><a NAME="E7:29_2_5_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T61">YELLOW_0:61</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E8:29_2_5"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font> implies <font color="Maroon">c<sub>9</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_5_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_2_5_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<a NAME="E2:29_2_5_2"/><b>then </b><i><font color="Green">E25</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E3:29_2_5_2"/>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D13">YELLOW_0:def 13</a></i>;<br/>
<a NAME="E4:29_2_5_2"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="yellow11.html#E2:29_2_5_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E5:29_2_5_2"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="yellow11.html#E4:29"><i><font color="Green">E19</font></i></a></i>;<br/>
<a NAME="E6:29_2_5_2"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow11.html#E4:29_2_5_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="waybel_1.html#T3">WAYBEL_1:3</a></i>;<br/>
<a NAME="E7:29_2_5_2"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="yellow11.html#E1:29_2_5"><i><font color="Green">E23</font></i></a>, <a class="ref" href="yellow11.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E8:29_2_5_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>11</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow11.html#E1:29"><i><font color="Green">E17</font></i></a>, <a class="txt" href="yellow11.html#E6:29_2_5_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="yellow11.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E9:29_2_5_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>11</sub></font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="yellow11.html#E1:29"><i><font color="Green">E17</font></i></a>, <a class="txt" href="yellow11.html#E7:29_2_5_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="yellow11.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E10:29_2_5_2"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="yellow11.html#E7:29_2_5_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="yellow_5.html#T10">YELLOW_5:10</a></i>;<br/>
<a NAME="E11:29_2_5_2"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="yellow11.html#E7:29_2_5_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="yellow_5.html#T10">YELLOW_5:10</a></i>;<br/>
<b>hence </b><a NAME="E12:29_2_5_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T61">YELLOW_0:61</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:29_2"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="waybel_0.html#V23">isomorphic</a>
 <b>by </b><i><a class="txt" href="yellow11.html#E8:29_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="yellow11.html#E9:29_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/>


</div><b>end;</b></div>


</div>
