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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <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#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</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="E1:20"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="waybel_0.html#V23">isomorphic</a>
 ;<br/>

<a NAME="E2:20"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> is <a href="orders_3.html#V5">monotone</a>
 
<b>by </b><i><a class="ref" href="waybel_0.html#D38">WAYBEL_0:def 38</a></i>;<br/>
<b>hence </b><a NAME="E3:20"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_2.html#T16">YELLOW_2:16</a></i>;<br/>

<a NAME="E4:20"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="waybel13.html#E1:20"><i><font color="Green">E17</font></i></a>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/>
<b>thus </b><a NAME="E5:20"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>3</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K2">"</a>  as   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel13.html#E1:20"><i><font color="Green">E17</font></i></a>, <a class="ref" href="waybel_0.html#T67">WAYBEL_0:67</a></i>;<br/>
<a NAME="E2:20_1"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="waybel_0.html#V23">isomorphic</a>
 
<b>by </b><i><a class="txt" href="waybel13.html#E1:20"><i><font color="Green">E17</font></i></a>, <a class="ref" href="waybel_0.html#T68">WAYBEL_0:68</a></i>;<br/>
<a NAME="E3:20_1"/><b>then </b><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="orders_3.html#V5">monotone</a>
 
<b>by </b><i><a class="ref" href="waybel_0.html#D38">WAYBEL_0:def 38</a></i>;<br/>
<a NAME="E4:20_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="waybel13.html#E4:20"><i><font color="Green">E18</font></i></a>, <a class="ref" href="funct_1.html#T152">FUNCT_1:152</a></i>;<br/>
<a NAME="E5:20_1"/>
( not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> )
 
;<br/>
<a NAME="E6:20_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:20_1"/><b>then </b><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T146">FUNCT_1:146</a></i>;<br/>
<i><font color="Green">E22</font></i>: <font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
<font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel13.html#E4:20"><i><font color="Green">E18</font></i></a>, <a class="ref" href="funct_1.html#T155">FUNCT_1:155</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="waybel13.html#E4:20_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="waybel13.html#E7:20_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>
;<br/>
<a NAME="E9:20_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a> )
 
;<br/>
<a NAME="E10:20_1"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>assume </b><a NAME="E11:20_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E12:20_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel13.html#E3:20_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="yellow_2.html#T16">YELLOW_2:16</a></i>;<br/>
<b>hence </b><a NAME="E13:20_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="waybel13.html#E4:20"><i><font color="Green">E18</font></i></a>, <a class="txt" href="waybel13.html#E8:20_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="waybel13.html#E10:20_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="funct_1.html#T56">FUNCT_1:56</a></i>;<br/>


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


</div>
