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<div class="add">

<b>let </b><font color="Maroon">R</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font>;<br/>



<a NAME="E1:19"/><i><font color="Green">E32</font></i>: 
(  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="yellow_9.html#K3" target="_self">opp+id</a> </span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="yellow_9.html#K3" target="_self">opp+id</a> </span>)</span> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">X</font></span>)</span> <a href="waybel_9.html#K4">opp+id</a> </span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">X</font></span>)</span> <a href="waybel_9.html#K4">opp+id</a> </span>)</span> #) &amp; the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="yellow_9.html#K3" target="_self">opp+id</a> </span>)</span> <a href="funct_2.html#R1">=</a> <span class="p1">(<span class="default"><a href="yellow_9.html#K1" target="_self">incl</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">X</font></span>)</span>,<font color="Maroon">R</font></span>)</span> <a href="partfun1.html#K1">*</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">X</font></span>)</span> <a href="waybel_9.html#K4">opp+id</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="waybel_9.html#D8">WAYBEL_9:def 8</a></i>;<br/>
<a NAME="E2:19"/><i><font color="Green">E33</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">X</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<a NAME="E3:19"/><i><font color="Green">E35</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">X</font></span>)</span> <a href="waybel_9.html#K4">opp+id</a> </span>)</span> <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">X</font></span>)</span> &amp;  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">X</font></span>)</span> <a href="waybel_9.html#K4">opp+id</a> </span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">X</font></span>)</span> <a href="waybel_9.html#K4">opp+id</a> </span>)</span> #) <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">X</font></span>)</span> <a href="yellow_7.html#NK1" target="_self">opp</a>  &amp; the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">X</font></span>)</span> <a href="waybel_9.html#K4">opp+id</a> </span>)</span> <a href="hidden.html#R1">=</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">X</font></span>)</span> <a href="relset_1.html#K6">~</a>  )
 
<b>by </b><i><a class="ref" href="waybel_9.html#T11">WAYBEL_9:11</a>, <a class="ref" href="waybel_9.html#D6">WAYBEL_9:def 6</a></i>;<br/>
<b>thus </b><a NAME="E4:19"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="yellow_9.html#K3" target="_self">opp+id</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 <b>by </b><i>, , <a class="ref" href="waybel_9.html#D6">WAYBEL_9:def 6</a></i>;<br/>

<a NAME="E5:19"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">R</font> <a href="yellow_7.html#NK1" target="_self">opp</a>  <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,<span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="relset_1.html#K6">~</a> </span>)</span> #)
 
<b>by </b><i><a class="ref" href="lattice3.html#D5">LATTICE3:def 5</a></i>;<br/>
<a NAME="E6:19"/>
the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#D14">YELLOW_0:def 14</a></i>;<br/>
<a NAME="E7:19"/><b>then </b><i><font color="Green">E38</font></i>: 
the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">X</font></span>)</span> <a href="waybel_9.html#K4">opp+id</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="relset_1.html#K6">~</a> </span>)</span> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="orders_1.html#T193">ORDERS_1:193</a></i>;<br/>
<a NAME="E8:19"/><b>then </b>
the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">X</font></span>)</span> <a href="waybel_9.html#K4">opp+id</a> </span>)</span> <a href="tarski.html#R1">c=</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">S</font> = <font color="Maroon">X</font> <a href="yellow_9.html#K3" target="_self">opp+id</a>  as    <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">R</font> <a href="yellow_7.html#NK1" target="_self">opp</a>  <b>by </b><i>, , , , <a class="ref" href="yellow_0.html#D13">YELLOW_0:def 13</a></i>;<br/>
<a NAME="E10:19"/>
<font color="Maroon">S</font> is <a href="yellow_0.html#V4">full</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:19_1"/>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span> <a href="wellord1.html#K2">|_2</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 <b>by </b><i>, , , , <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D14">YELLOW_0:def 14</a><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:19"/>
<font color="Maroon">X</font> <a href="yellow_9.html#K3" target="_self">opp+id</a>  is  <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">R</font> <a href="yellow_7.html#NK1" target="_self">opp</a> 
 ;<br/>

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="yellow_9.html#K3" target="_self">opp+id</a> </span>)</span>;<br/>

<a NAME="E12:19"/>
 <a href="grcat_1.html#K7">id</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">X</font></span>)</span> <a href="funct_2.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="grcat_1.html#D11">GRCAT_1:def 11</a></i>;<br/>
<a NAME="E13:19"/><b>then </b>
( the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">X</font></span>)</span> <a href="waybel_9.html#K4">opp+id</a> </span>)</span> <a href="funct_2.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <font color="Maroon">X</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp;  <a href="yellow_9.html#K1" target="_self">incl</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">X</font></span>)</span>,<font color="Maroon">R</font> <a href="funct_2.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <font color="Maroon">X</font> )
 
<b>by </b><i>, , <a class="ref" href="relat_1.html#T71">RELAT_1:71</a>, <a class="ref" href="waybel_9.html#D6">WAYBEL_9:def 6</a></i>;<br/>
<a NAME="E14:19"/><b>then </b>
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="yellow_9.html#K3" target="_self">opp+id</a> </span>)</span> <a href="funct_2.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T78">RELAT_1:78</a></i>;<br/>
<b>hence </b><a NAME="E15:19"/>
<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="yellow_9.html#K3" target="_self">opp+id</a> </span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/>


</div>
