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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>





<b>assume </b><a NAME="E1:40"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="waybel_0.html#V17">infs-preserving</a>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="yellow_0.html#K5">subrelstr</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>
<a NAME="E2:40"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> st  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> holds <br/> <a href="yellow_0.html#K2">"/\"</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>;<br/>

<b>assume </b><a NAME="E1:40_1"/>
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:40_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="waybel_0.html#R3">preserves_inf_of</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="yellow_2.html#E1:40"><i><font color="Green">E20</font></i></a>, <a class="ref" href="waybel_0.html#D32">WAYBEL_0:def 32</a></i>;<br/>
<a NAME="E3:40_1"/><i><font color="Green">E22</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E4:40_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E5:40_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<b>then </b> <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> = 
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T147">FUNCT_1:147</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="yellow_2.html#E2:40_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="yellow_2.html#E3:40_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="waybel_0.html#D30">WAYBEL_0:def 30</a></i>
;<br/>
<a NAME="E7:40_1"/><b>then </b>
 <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T6">FUNCT_2:6</a></i>;<br/>
<b>hence </b><a NAME="E8:40_1"/>
 <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <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="E3:40"/><b>then </b>
 <a href="yellow_0.html#K5">subrelstr</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> is <a href="yellow_0.html#V7">infs-inheriting</a>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D18">YELLOW_0:def 18</a></i>;<br/>
<b>hence </b><a NAME="E4:40"/>
 <a href="yellow_2.html#K1" target="_self">Image</a> <font color="Maroon">c<sub>3</sub></font> is <a href="yellow_0.html#V7">infs-inheriting</a>
 ;<br/>


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