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

<b>let </b><font color="Maroon">L1</font> be  <a href="orders_2.html#V4">antisymmetric</a>  <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">L2</font> be  <a href="orders_2.html#V4">antisymmetric</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">A1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L1</font>, <font color="Maroon">B1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L1</font>;<br/>



<b>assume </b><a NAME="E1:16"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">A1</font>,<font color="Maroon">B1</font> <a href="orders_4.html#R1" target="_self">form_upper_lower_partition_of</a> <font color="Maroon">L1</font>
 ;<br/>

<b>let </b><font color="Maroon">A2</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L2</font>, <font color="Maroon">B2</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L2</font>;<br/>

<b>assume </b><a NAME="E2:16"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">A2</font>,<font color="Maroon">B2</font> <a href="orders_4.html#R1" target="_self">form_upper_lower_partition_of</a> <font color="Maroon">L2</font>
 ;<br/>

<b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span>;<br/>

<b>assume </b><a NAME="E3:16"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">f</font> is <a href="waybel_0.html#V23">isomorphic</a>
 ;<br/>

<b>let </b><font color="Maroon">g</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B1</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B2</font></span>)</span>;<br/>

<b>assume </b><a NAME="E4:16"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">g</font> is <a href="waybel_0.html#V23">isomorphic</a>
 ;<br/>

<a NAME="E5:16"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L1</font> &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> holds <br/><font color="Olive">a</font> <a href="orders_2.html#R2">&lt;</a> <font color="Olive">b</font> ) )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:16"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">A1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B1</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:16"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">A2</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B2</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:16"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">A2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L2</font> &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font> holds <br/><font color="Olive">a</font> <a href="orders_2.html#R2">&lt;</a> <font color="Olive">b</font> ) )
 
<b>by </b><i>, </i>;<br/>
<b>set </b><font color="Maroon">h</font> = <font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1"/>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1"><b>suppose </b></a><a NAME="E1:16_1_1"/><i><font color="Green">E60</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:16_1_1"/><b>then </b><i><font color="Green">E61</font></i>: 
( <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T15">XBOOLE_1:15</a></i>;<br/><a NAME="E3:16_1_1"/><b>then </b><i><font color="Green">E62</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E4:16_1_1"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</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> <font color="Maroon">A1</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E5:16_1_1"/><b>then </b><i><font color="Green">E63</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E6:16_1_1"/><i><font color="Green">E64</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E7:16_1_1"/>
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E8:16_1_1"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</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> <font color="Maroon">B1</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E9:16_1_1"/><b>then </b><i><font color="Green">E65</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B1</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E10:16_1_1"/><i><font color="Green">E66</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/><a NAME="E11:16_1_1"/><i><font color="Green">E67</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L1</font>
 <b>by </b><i>, , , <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/><a NAME="E12:16_1_1"/><i><font color="Green">E68</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:16_1_1_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, , , , <a class="ref" href="relat_1.html#T65">RELAT_1:65</a></i>;<br/>
<b>hence </b><a NAME="E2:16_1_1_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E13:16_1_1"/><i><font color="Green">E69</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E14:16_1_1"/><b>then </b>
 <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font> 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/><a NAME="E15:16_1_1"/><b>then </b><i><font color="Green">E70</font></i>: 
 <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font> is <a href="struct_0.html#V3">empty</a>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D38">WAYBEL_0:def 38</a></i>;<br/><a NAME="E16:16_1_1"/>
 <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B1</font> 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/><a NAME="E17:16_1_1"/><b>then </b><i><font color="Green">E71</font></i>: 
 <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B2</font> is <a href="struct_0.html#V3">empty</a>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D38">WAYBEL_0:def 38</a></i>;<br/><a NAME="E18:16_1_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L1</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:16_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L1</font>
 ;<br/>

<a NAME="E2:16_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E3:16_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L2</font>
 <b>by </b><i><a class="ref" href="orders_4.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">h</font> = <font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L1</font>,<font color="Maroon">L2</font> <b>by </b><i>, <a class="ref" href="funct_2.html#T5">FUNCT_2:5</a></i>;<br/><a NAME="E20:16_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><a NAME="E21:16_1_1"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">A2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E22:16_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B2</font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="orders_4.html#E1:16"><i><font color="Green">E28</font></i></a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><a NAME="E23:16_1_1"/><b>then </b>
<font color="Maroon">B2</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E24:16_1_1"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">L2</font> 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/><a NAME="E25:16_1_1"/>
<font color="Maroon">L1</font> 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/><a NAME="E26:16_1_1"/><b>then </b>
<font color="Maroon">h</font> is <a href="waybel_0.html#V23">isomorphic</a>
 <b>by </b><i><a class="txt" href="orders_4.html#E2:16"><i><font color="Green">E30</font></i></a>, <a class="ref" href="waybel_0.html#D38">WAYBEL_0:def 38</a></i>;<br/><b>hence </b><a NAME="E27:16_1_1"/>
 ex <font color="Olive">h</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L1</font>,<font color="Maroon">L2</font> st <br/>( <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font> &amp; <font color="Olive">h</font> is <a href="waybel_0.html#V23">isomorphic</a> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2"><b>suppose </b></a><a NAME="E1:16_1_2"/><i><font color="Green">E74</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:16_1_2"/><b>then </b>
( <font color="Maroon">A1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">B1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i/>;<br/><a NAME="E3:16_1_2"/><b>then </b>
(  not the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span> is <a href="xboole_0.html#V1">empty</a> or  not the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B1</font></span>)</span> is <a href="xboole_0.html#V1">empty</a> )
 <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E4:16_1_2"/><b>then </b>
(  not  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font> is <a href="struct_0.html#V3">empty</a> or  not  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B1</font> 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/><a NAME="E5:16_1_2"/><b>then </b><i><font color="Green">E75</font></i>: 
(  not  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font> is <a href="struct_0.html#V3">empty</a> or  not  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B2</font> is <a href="struct_0.html#V3">empty</a> )
 <b>by </b><i>, , <a class="ref" href="waybel_0.html#D38">WAYBEL_0:def 38</a></i>;<br/><a NAME="E6:16_1_2"/><b>then </b><i><font color="Green">E76</font></i>: 
(  not the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span> is <a href="xboole_0.html#V1">empty</a> or  not the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B2</font></span>)</span> is <a href="xboole_0.html#V1">empty</a> )
 <b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><a NAME="E7:16_1_2"/><b>then </b>
( <font color="Maroon">A2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">B2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E8:16_1_2"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">A2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T15">XBOOLE_1:15</a></i>;<br/><a NAME="E9:16_1_2"/>
( ( not <font color="Maroon">A1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; not <font color="Maroon">B1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) or <font color="Maroon">A2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_1_2_1"/>
( <font color="Maroon">A1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">B1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<a NAME="E2:16_1_2_1"/>
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:16_1_2_1"/>
( <font color="Maroon">A2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>


</div><b>end;</b></div><a NAME="E10:16_1_2"/><b>then </b>
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E11:16_1_2"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</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> <font color="Maroon">A1</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E12:16_1_2"/><b>then </b><i><font color="Green">E78</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E13:16_1_2"/>
( ( not <font color="Maroon">A2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; not <font color="Maroon">B2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) or <font color="Maroon">B2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_1_2_2"/>
( <font color="Maroon">A2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">B2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<a NAME="E2:16_1_2_2"/>
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:16_1_2_2"/>
( <font color="Maroon">B2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>


</div><b>end;</b></div><a NAME="E14:16_1_2"/><b>then </b><i><font color="Green">E79</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E15:16_1_2"/><b>then </b><i><font color="Green">E80</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</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> <font color="Maroon">B1</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E16:16_1_2"/><b>then </b><i><font color="Green">E81</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B1</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E17:16_1_2"/><i><font color="Green">E82</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/><a NAME="E18:16_1_2"/><i><font color="Green">E83</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> implies  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_1_2_3"/><i><font color="Green">E84</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 ;<br/>

<a NAME="E2:16_1_2_3"/><i><font color="Green">E85</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T18">FUNCT_4:18</a></i>;<br/>
<a NAME="E3:16_1_2_3"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:16_1_2_3_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_2_3_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_3_1_1_1"><b>suppose </b></a><a NAME="E1:16_1_2_3_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:16_1_2_3_1_1_1"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:16_1_2_3_1_1_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="orders_4.html#D1" target="_self">Def1</a>, <a class="txt" href="orders_4.html#E2:16"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E5:16_1_2_3_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2_3_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:16_1_2_3_1_1_1_1"/>
not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i>, <a class="txt" href="orders_4.html#E2:16"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>hence </b><a NAME="E2:16_1_2_3_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="txt" href="orders_4.html#E2:16"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:16_1_2_3_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_3_1_1_2"><b>suppose </b></a><a NAME="E1:16_1_2_3_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:16_1_2_3_1_1_2"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:16_1_2_3_1_1_2"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="orders_4.html#D1" target="_self">Def1</a>, <a class="txt" href="orders_4.html#E2:16"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E5:16_1_2_3_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="txt" href="orders_4.html#E2:16"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>;<br/><b>hence </b><a NAME="E6:16_1_2_3_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="orders_4.html#E2:16"><i><font color="Green">E30</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>

</div><b>end;</b></div>
<b>hence </b><a NAME="E4:16_1_2_3"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="orders_4.html#E1:16"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div><a NAME="E19:16_1_2"/><i><font color="Green">E91</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2_4"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_2_4_1"/>
( ( <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) or ( <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">A1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) or ( <font color="Maroon">A2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) or ( <font color="Maroon">A2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">A1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_4_1_1"><b>suppose </b></a><a NAME="E1:16_1_2_4_1_1"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><a NAME="E2:16_1_2_4_1_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B1</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, , <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E3:16_1_2_4_1_1"/><b>then </b><i><font color="Green">E93</font></i>: 
( not  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B2</font> is <a href="struct_0.html#V3">empty</a> &amp; not  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B1</font> is <a href="struct_0.html#V3">empty</a> )
 <b>by </b><i>, <a class="txt" href="orders_4.html#E1:16"><i><font color="Green">E28</font></i></a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E4:16_1_2_4_1_1"/><i><font color="Green">E94</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="txt" href="orders_4.html#E1:16"><i><font color="Green">E28</font></i></a>, <a class="ref" href="relat_1.html#T65">RELAT_1:65</a></i>;<br/><a NAME="E5:16_1_2_4_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</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> <font color="Maroon">B2</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/><b>hence </b><a NAME="E6:16_1_2_4_1_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L2</font>
 <b>by </b><i>, , , <a class="txt" href="orders_4.html#E1:16"><i><font color="Green">E28</font></i></a>, <a class="txt" href="orders_4.html#E2:16"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xboole_1.html#T65">XBOOLE_1:65</a>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_4_1_2"><b>suppose </b></a><a NAME="E1:16_1_2_4_1_2"/>
( <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">A1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><a NAME="E2:16_1_2_4_1_2"/><b>then </b>
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><b>hence </b><a NAME="E3:16_1_2_4_1_2"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L2</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_4_1_3"><b>suppose </b></a><a NAME="E1:16_1_2_4_1_3"/>
( <font color="Maroon">A2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><a NAME="E2:16_1_2_4_1_3"/><b>then </b>
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><b>hence </b><a NAME="E3:16_1_2_4_1_3"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L2</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_4_1_4"><b>suppose </b></a><a NAME="E1:16_1_2_4_1_4"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">A2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">A1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><a NAME="E2:16_1_2_4_1_4"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2_4_1_4_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_4_1_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_2_4_1_4_1_1"/>
( <font color="Maroon">B2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">B2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_4_1_4_1_1_1"><b>suppose </b></a><a NAME="E1:16_1_2_4_1_4_1_1_1"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">B2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:16_1_2_4_1_4_1_1_1"/>
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="txt" href="orders_4.html#E1:16"><i><font color="Green">E28</font></i></a>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E3:16_1_2_4_1_4_1_1_1"/><b>then </b>
( not  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font> is <a href="struct_0.html#V3">empty</a> &amp; not  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font> 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/><a NAME="E4:16_1_2_4_1_4_1_1_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</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> <font color="Maroon">A2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/><a NAME="E5:16_1_2_4_1_4_1_1_1"/><b>then </b><i><font color="Green">E97</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E6:16_1_2_4_1_4_1_1_1"/><i><font color="Green">E98</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E7:16_1_2_4_1_4_1_1_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B1</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E8:16_1_2_4_1_4_1_1_1"/><b>then </b>
( not  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B2</font> is <a href="struct_0.html#V3">empty</a> &amp; not  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B1</font> 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/><a NAME="E9:16_1_2_4_1_4_1_1_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</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> <font color="Maroon">B2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/><b>hence </b><a NAME="E10:16_1_2_4_1_4_1_1_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L2</font>
 <b>by </b><i>, , , , , <a class="txt" href="orders_4.html#E2:16"><i><font color="Green">E30</font></i></a>, , <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_4_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:16_1_2_4_1_4_1_1_2"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">B2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:16_1_2_4_1_4_1_1_2"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E3:16_1_2_4_1_4_1_1_2"/><b>then </b>
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   holds not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T20">RELAT_1:20</a></i>;<br/><a NAME="E4:16_1_2_4_1_4_1_1_2"/><b>then </b><i><font color="Green">E100</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="relat_1.html#T56">RELAT_1:56</a></i>;<br/><a NAME="E5:16_1_2_4_1_4_1_1_2"/>
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="txt" href="orders_4.html#E1:16"><i><font color="Green">E28</font></i></a>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E6:16_1_2_4_1_4_1_1_2"/><b>then </b>
( not  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font> is <a href="struct_0.html#V3">empty</a> &amp; not  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font> 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/><a NAME="E7:16_1_2_4_1_4_1_1_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</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> <font color="Maroon">A2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/><b>hence </b><a NAME="E8:16_1_2_4_1_4_1_1_2"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L2</font>
 <b>by </b><i>, , , <a class="txt" href="orders_4.html#E2:16"><i><font color="Green">E30</font></i></a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a>, <a class="ref" href="xboole_1.html#T65">XBOOLE_1:65</a>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E3:16_1_2_4_1_4"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L2</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E20:16_1_2"/><i><font color="Green">E101</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L1</font>
 <b>by </b><i>, , , <a class="ref" href="orders_4.html#D1" target="_self">Def1</a>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E21:16_1_2"/><b>then </b><i><font color="Green">E102</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L1</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L2</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">L1</font> = <font color="Maroon">L1</font> as  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a>  <b>by </b><i>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">L2</font> = <font color="Maroon">L2</font> as  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a>  <b>by </b><i>, , <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">h</font> = <font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L1</font>,<font color="Maroon">L2</font> <b>by </b><i><a class="txt" href="orders_4.html#E1:16"><i><font color="Green">E28</font></i></a>, , <a class="ref" href="funct_2.html#T5">FUNCT_2:5</a></i>;<br/><a NAME="E25:16_1_2"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">h</font> is <a href="funct_1.html#V2">one-to-one</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>, <font color="Maroon">x2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D1">WAYBEL_1:def 1</a><br/>

<b>assume </b><a NAME="E1:16_1_2_5"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_2_5_1"/>
( ( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> ) or ( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> ) or ( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> ) or ( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> ) )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_5_1_1"><b>suppose </b></a><a NAME="E1:16_1_2_5_1_1"/><i><font color="Green">E107</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> )
 ;<br/><div class="add"><a NAME="E2:16_1_2_5_1_1"/><b>then </b><i><font color="Green">E108</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E3:16_1_2_5_1_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E4:16_1_2_5_1_1"/><b>then </b><i><font color="Green">E109</font></i>: 
( not  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font> is <a href="struct_0.html#V3">empty</a> &amp; not  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font> is <a href="struct_0.html#V3">empty</a> )
 <b>by </b><i><a class="txt" href="orders_4.html#E3:16"><i><font color="Green">E33</font></i></a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><a NAME="E5:16_1_2_5_1_1"/>
not <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E6:16_1_2_5_1_1"/><b>then </b><i><font color="Green">E110</font></i>: 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font>
 <b>by </b><i>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/><a NAME="E7:16_1_2_5_1_1"/>
not <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E8:16_1_2_5_1_1"/><b>then </b><i><font color="Green">E111</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
 <b>by </b><i>, , <a class="ref" href="orders_4.html#D2" target="_self">Def2</a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/><a NAME="E9:16_1_2_5_1_1"/>
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i>, , <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/><b>hence </b><a NAME="E10:16_1_2_5_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_5_1_2"><b>suppose </b></a><a NAME="E1:16_1_2_5_1_2"/><i><font color="Green">E112</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> )
 ;<br/><div class="add"><a NAME="E2:16_1_2_5_1_2"/><b>then </b>
not <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E3:16_1_2_5_1_2"/><b>then </b>
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font>
 <b>by </b><i>, , <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/><a NAME="E4:16_1_2_5_1_2"/><b>then </b><i><font color="Green">E113</font></i>: 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:16_1_2_5_1_2"/><i><font color="Green">E114</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E6:16_1_2_5_1_2"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E7:16_1_2_5_1_2"/><b>then </b>
( not  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font> is <a href="struct_0.html#V3">empty</a> &amp; not  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font> 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/><a NAME="E8:16_1_2_5_1_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</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> <font color="Maroon">A2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/><a NAME="E9:16_1_2_5_1_2"/><b>then </b><i><font color="Green">E115</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E10:16_1_2_5_1_2"/><i><font color="Green">E116</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2_5_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:16_1_2_5_1_2_1"/><i><font color="Green">E117</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B1</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<a NAME="E2:16_1_2_5_1_2_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:16_1_2_5_1_2_1"/><b>then </b>
( not  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B2</font> is <a href="struct_0.html#V3">empty</a> &amp; not  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B1</font> 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/>
<a NAME="E4:16_1_2_5_1_2_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</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> <font color="Maroon">B2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/>
<b>hence </b><a NAME="E5:16_1_2_5_1_2_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="orders_4.html#D2" target="_self">Def2</a>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>


</div><b>end;</b></div><a NAME="E11:16_1_2_5_1_2"/>
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
 <b>by </b><i>, , <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>;<br/><a NAME="E12:16_1_2_5_1_2"/><b>then </b>
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E13:16_1_2_5_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i><a class="txt" href="orders_4.html#E3:16"><i><font color="Green">E33</font></i></a>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_5_1_3"><b>suppose </b></a><a NAME="E1:16_1_2_5_1_3"/><i><font color="Green">E118</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> )
 ;<br/><div class="add"><a NAME="E2:16_1_2_5_1_3"/><b>then </b>
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font>
 <b>by </b><i>, , <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>;<br/><a NAME="E3:16_1_2_5_1_3"/><b>then </b><i><font color="Green">E119</font></i>: 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:16_1_2_5_1_3"/><i><font color="Green">E120</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E5:16_1_2_5_1_3"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E6:16_1_2_5_1_3"/><b>then </b>
( not  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font> is <a href="struct_0.html#V3">empty</a> &amp; not  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font> 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/><b>then </b><i><font color="Green">E121</font></i>:  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> = 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span>
<b>by </b><i>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>
<br/>.= 
<font color="Maroon">A2</font>
<b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>
;<br/><a NAME="E8:16_1_2_5_1_3"/><i><font color="Green">E122</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B1</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E9:16_1_2_5_1_3"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E10:16_1_2_5_1_3"/><b>then </b>
( not  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B2</font> is <a href="struct_0.html#V3">empty</a> &amp; not  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B1</font> 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/><a NAME="E11:16_1_2_5_1_3"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</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> <font color="Maroon">B2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/><a NAME="E12:16_1_2_5_1_3"/><b>then </b><i><font color="Green">E123</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="orders_4.html#D2" target="_self">Def2</a>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E13:16_1_2_5_1_3"/>
not <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E14:16_1_2_5_1_3"/><b>then </b>
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/><a NAME="E15:16_1_2_5_1_3"/><b>then </b>
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E16:16_1_2_5_1_3"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i><a class="txt" href="orders_4.html#E3:16"><i><font color="Green">E33</font></i></a>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_5_1_4"><b>suppose </b></a><a NAME="E1:16_1_2_5_1_4"/><i><font color="Green">E124</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> )
 ;<br/><div class="add"><a NAME="E2:16_1_2_5_1_4"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B1</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E3:16_1_2_5_1_4"/><b>then </b>
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B1</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i>, </i>;<br/><a NAME="E4:16_1_2_5_1_4"/><b>then </b><i><font color="Green">E125</font></i>: 
( not  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B1</font> is <a href="struct_0.html#V3">empty</a> &amp; not  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B2</font> 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/><a NAME="E5:16_1_2_5_1_4"/>
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font>
 <b>by </b><i>, , <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>;<br/><a NAME="E6:16_1_2_5_1_4"/><b>then </b><i><font color="Green">E126</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
 <b>by </b><i>, , , <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>;<br/><a NAME="E7:16_1_2_5_1_4"/>
<font color="Maroon">g</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i>, <a class="txt" href="orders_4.html#E3:16"><i><font color="Green">E33</font></i></a>, <a class="ref" href="waybel_0.html#D38">WAYBEL_0:def 38</a></i>;<br/><b>hence </b><a NAME="E8:16_1_2_5_1_4"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E26:16_1_2"/>
<font color="Maroon">h</font> is <a href="waybel_0.html#V23">isomorphic</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2_6"><b>proof </b></a><div class="add">

<a NAME="E1:16_1_2_6"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font> holds <br/> ( <font color="Olive">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">y</font> iff <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2_6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>;<br/>

<a NAME="E1:16_1_2_6_1"/><i><font color="Green">E127</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i>, , , <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_6_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_2_6_1_1"/>
( ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> ) or ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> ) or ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> ) or ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> ) )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_6_1_1_1"><b>suppose </b></a><a NAME="E1:16_1_2_6_1_1_1"/><i><font color="Green">E128</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> )
 ;<br/><div class="add"><a NAME="E2:16_1_2_6_1_1_1"/><b>then </b><i><font color="Green">E129</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><b>reconsider </b><font color="Maroon">A1'</font> = <font color="Maroon">A1</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L1</font> <b>by </b><i/>;<br/><a NAME="E4:16_1_2_6_1_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="txt" href="orders_4.html#E3:16"><i><font color="Green">E33</font></i></a>, </i>;<br/><b>then reconsider </b><font color="Maroon">A2'</font> = <font color="Maroon">A2</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L2</font> <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><b>reconsider </b><font color="Maroon">ax</font> = <font color="Maroon">x</font>, <font color="Maroon">ay</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1'</font></span>)</span> <b>by </b><i>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1'</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2'</font></span>)</span> ;<br/><a NAME="E8:16_1_2_6_1_1_1"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , , , <a class="ref" href="funct_4.html#T17">FUNCT_4:17</a></i>;<br/><a NAME="E9:16_1_2_6_1_1_1"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 <b>by </b><i>, , , , <a class="ref" href="funct_4.html#T17">FUNCT_4:17</a></i>;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:16_1_2_6_1_1_1_1"/>
<font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E2:16_1_2_6_1_1_1_1"/><b>then </b>
<font color="Maroon">ax</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">ay</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T61">YELLOW_0:61</a></i>;<br/><a NAME="E3:16_1_2_6_1_1_1_1"/><b>then </b>
<font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">ax</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">ay</font>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/><b>hence </b><a NAME="E4:16_1_2_6_1_1_1_1"/>
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="orders_4.html#D2" target="_self">Def2</a>, <a class="ref" href="yellow_0.html#T60">YELLOW_0:60</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E11:16_1_2_6_1_1_1"/>
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E12:16_1_2_6_1_1_1"/><b>then </b>
<font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">ax</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">ay</font>
 <b>by </b><i>, <a class="ref" href="orders_4.html#D2" target="_self">Def2</a>, <a class="ref" href="yellow_0.html#T61">YELLOW_0:61</a></i>;<br/><a NAME="E13:16_1_2_6_1_1_1"/><b>then </b>
<font color="Maroon">ax</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">ay</font>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/><b>hence </b><a NAME="E14:16_1_2_6_1_1_1"/>
<font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T60">YELLOW_0:60</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_6_1_1_2"><b>suppose </b></a><a NAME="E1:16_1_2_6_1_1_2"/><i><font color="Green">E137</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> )
 ;<br/><div class="add"><a NAME="E2:16_1_2_6_1_1_2"/><b>then </b><i><font color="Green">E138</font></i>: 
<font color="Maroon">x</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">y</font>
 <b>by </b><i>, </i>;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:16_1_2_6_1_1_2_1"/>
<font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E2:16_1_2_6_1_1_2_1"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , , <a class="ref" href="funct_4.html#T17">FUNCT_4:17</a></i>;<br/><a NAME="E3:16_1_2_6_1_1_2_1"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 <b>by </b><i>, , <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>;<br/><a NAME="E4:16_1_2_6_1_1_2_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E5:16_1_2_6_1_1_2_1"/><b>then </b><i><font color="Green">E141</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E6:16_1_2_6_1_1_2_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B1</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E7:16_1_2_6_1_1_2_1"/><b>then </b><i><font color="Green">E142</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">A1'</font> = <font color="Maroon">A1</font>, <font color="Maroon">B1'</font> = <font color="Maroon">B1</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L1</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">A2'</font> = <font color="Maroon">A2</font>, <font color="Maroon">B2'</font> = <font color="Maroon">B2</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L2</font> <b>by </b><i>, , <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><b>reconsider </b><font color="Maroon">ax</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1'</font></span>)</span> <b>by </b><i>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><b>reconsider </b><font color="Maroon">ay</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B1'</font></span>)</span> <b>by </b><i>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1'</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2'</font></span>)</span> ;<br/><b>reconsider </b><font color="Maroon">g'</font> = <font color="Maroon">g</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B1'</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B2'</font></span>)</span> ;<br/><a NAME="E14:16_1_2_6_1_1_2_1"/>
( <font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">ax</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> <font color="Maroon">A2'</font></span>)</span> &amp; <font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">ay</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> <font color="Maroon">B2'</font></span>)</span> )
 ;<br/><a NAME="E15:16_1_2_6_1_1_2_1"/><b>then </b>
( <font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">ax</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2'</font> &amp; <font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">ay</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2'</font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E16:16_1_2_6_1_1_2_1"/><b>then </b>
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 <b>by </b><i>, , <a class="ref" href="orders_4.html#D2" target="_self">Def2</a>, </i>;<br/><b>hence </b><a NAME="E17:16_1_2_6_1_1_2_1"/>
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D10">ORDERS_2:def 10</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E4:16_1_2_6_1_1_2"/>
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 ;<br/><b>thus </b><a NAME="E5:16_1_2_6_1_1_2"/>
<font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="orders_4.html#E3:16"><i><font color="Green">E33</font></i></a>, <a class="ref" href="orders_2.html#D10">ORDERS_2:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_6_1_1_3"><b>suppose </b></a><a NAME="E1:16_1_2_6_1_1_3"/><i><font color="Green">E146</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> )
 ;<br/><div class="add"><a NAME="E2:16_1_2_6_1_1_3"/><b>then </b>
<font color="Maroon">y</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">x</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:16_1_2_6_1_1_3"/>
( <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font> implies <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="orders_2.html#T30">ORDERS_2:30</a></i>;<br/><b>assume </b><a NAME="E4:16_1_2_6_1_1_3"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E5:16_1_2_6_1_1_3"/><i><font color="Green">E148</font></i>: 
( <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> )
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E6:16_1_2_6_1_1_3"/><i><font color="Green">E149</font></i>: 
( not the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span> is <a href="xboole_0.html#V1">empty</a> &amp; not the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B1</font></span>)</span> is <a href="xboole_0.html#V1">empty</a> )
 <b>by </b><i>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E7:16_1_2_6_1_1_3"/><b>then </b><i><font color="Green">E150</font></i>: 
( not  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font> is <a href="struct_0.html#V3">empty</a> &amp; not  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B1</font> 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/><a NAME="E8:16_1_2_6_1_1_3"/>
( not the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span> is <a href="xboole_0.html#V1">empty</a> &amp; not the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B2</font></span>)</span> is <a href="xboole_0.html#V1">empty</a> )
 <b>by </b><i>, , <a class="ref" href="orders_4.html#D2" target="_self">Def2</a>, </i>;<br/><a NAME="E9:16_1_2_6_1_1_3"/><b>then </b>
( not  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font> is <a href="struct_0.html#V3">empty</a> &amp; not  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B2</font> 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/><a NAME="E10:16_1_2_6_1_1_3"/><b>then </b>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</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> <font color="Maroon">A2</font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</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> <font color="Maroon">B2</font></span>)</span> )
 <b>by </b><i>, , , <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/><a NAME="E11:16_1_2_6_1_1_3"/><b>then </b><i><font color="Green">E151</font></i>: 
( <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> )
 <b>by </b><i>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E12:16_1_2_6_1_1_3"/>
( <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> )
 <b>by </b><i>, , , , <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a>, <a class="ref" href="funct_4.html#T17">FUNCT_4:17</a></i>;<br/><a NAME="E13:16_1_2_6_1_1_3"/><b>then </b>
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="orders_2.html#NR4" target="_self">&gt;</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E14:16_1_2_6_1_1_3"/>
<font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="orders_4.html#E3:16"><i><font color="Green">E33</font></i></a>, <a class="ref" href="orders_2.html#T30">ORDERS_2:30</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_6_1_1_4"><b>suppose </b></a><a NAME="E1:16_1_2_6_1_1_4"/><i><font color="Green">E152</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> )
 ;<br/><div class="add"><a NAME="E2:16_1_2_6_1_1_4"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B1</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E3:16_1_2_6_1_1_4"/><b>then </b><i><font color="Green">E153</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">B1'</font> = <font color="Maroon">B1</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L1</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">B2'</font> = <font color="Maroon">B2</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L2</font> <b>by </b><i><a class="txt" href="orders_4.html#E3:16"><i><font color="Green">E33</font></i></a>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><b>reconsider </b><font color="Maroon">ax</font> = <font color="Maroon">x</font>, <font color="Maroon">ay</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B1'</font></span>)</span> <b>by </b><i>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><b>reconsider </b><font color="Maroon">g'</font> = <font color="Maroon">g</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B1'</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B2'</font></span>)</span> ;<br/><a NAME="E8:16_1_2_6_1_1_4"/><i><font color="Green">E154</font></i>: 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>;<br/><a NAME="E9:16_1_2_6_1_1_4"/><i><font color="Green">E155</font></i>: 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 <b>by </b><i>, , <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:16_1_2_6_1_1_4_1"/>
<font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E2:16_1_2_6_1_1_4_1"/><b>then </b>
<font color="Maroon">ax</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">ay</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T61">YELLOW_0:61</a></i>;<br/><a NAME="E3:16_1_2_6_1_1_4_1"/><b>then </b>
<font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">ax</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">ay</font>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/><b>hence </b><a NAME="E4:16_1_2_6_1_1_4_1"/>
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="orders_4.html#D2" target="_self">Def2</a>, <a class="ref" href="yellow_0.html#T60">YELLOW_0:60</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E11:16_1_2_6_1_1_4"/>
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E12:16_1_2_6_1_1_4"/><b>then </b>
<font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">ax</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">ay</font>
 <b>by </b><i>, <a class="ref" href="orders_4.html#D2" target="_self">Def2</a>, <a class="ref" href="yellow_0.html#T61">YELLOW_0:61</a></i>;<br/><a NAME="E13:16_1_2_6_1_1_4"/><b>then </b>
<font color="Maroon">ax</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">ay</font>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/><b>hence </b><a NAME="E14:16_1_2_6_1_1_4"/>
<font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T60">YELLOW_0:60</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E2:16_1_2_6"/>
<font color="Maroon">h</font> is <a href="waybel_0.html#V23">isomorphic</a>
 <b>by </b><i>, <a class="txt" href="orders_4.html#E2:16"><i><font color="Green">E30</font></i></a>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E27:16_1_2"/>
 ex <font color="Olive">h</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L1</font>,<font color="Maroon">L2</font> st <br/>( <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g</font> &amp; <font color="Olive">h</font> is <a href="waybel_0.html#V23">isomorphic</a> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
