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<div class="add">

<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="waybel_0.html#V24">up-complete</a> <a href="orders_2.html#NM1">Poset</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="E1:33"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="waybel_0.html#V23">isomorphic</a>
 ;<br/>

<a NAME="E2:33"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="waybel_0.html#D38">WAYBEL_0:def 38</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</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#E1:33"><i><font color="Green">E28</font></i></a>, <a class="ref" href="waybel_0.html#T67">WAYBEL_0:67</a></i>;<br/>
<a NAME="E4:33"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="waybel_0.html#V23">isomorphic</a>
 
<b>by </b><i><a class="txt" href="waybel13.html#E1:33"><i><font color="Green">E28</font></i></a>, <a class="ref" href="waybel_0.html#T68">WAYBEL_0:68</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E5:33"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E1:33_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:33_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>7</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="txt" href="waybel13.html#E1:33"><i><font color="Green">E28</font></i></a>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:33_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:33_1_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E4:33_1_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E5:33_1_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E6:33_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel13.html#E3:33_1_1_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="waybel13.html#E4:33_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="waybel13.html#E5:33_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E7:33_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel13.html#E2:33"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_1.html#T155">FUNCT_1:155</a></i>;<br/><b>hence </b><a NAME="E8:33_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="waybel13.html#E2:33_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#T147">FUNCT_1:147</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>8</sub></font> as  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E4:33_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E5:33_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T1">WAYBEL_0:1</a></i>;<br/><b>take </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font>;<br/><a NAME="E6:33_1_1"/>
( not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> )
 ;<br/><a NAME="E7:33_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>hence </b><a NAME="E8:33_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="waybel13.html#E4:33_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E9:33_1_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <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> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E10:33_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E11:33_1_1"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T146">FUNCT_1:146</a></i>;<br/><a NAME="E12:33_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="waybel13.html#E2:33"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_1.html#T152">FUNCT_1:152</a></i>;<br/><font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>9</sub></font> = 
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel13.html#E2:33"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_1.html#T155">FUNCT_1:155</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="waybel13.html#E11:33_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="waybel13.html#E12:33_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>
;<br/><b>hence </b><a NAME="E14:33_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="waybel13.html#E4:33"><i><font color="Green">E30</font></i></a>, <a class="txt" href="waybel13.html#E5:33_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="waybel13.html#T19" target="_self">Th19</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>7</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="waybel_0.html#T1">WAYBEL_0:1</a></i>;<br/><a NAME="E5:33_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <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="E6:33_1"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E7:33_1"/><i><font color="Green">E34</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T75">WAYBEL_0:75</a></i>;<br/><a NAME="E8:33_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="waybel_0.html#V18">sups-preserving</a>
 <b>by </b><i><a class="txt" href="waybel13.html#E4:33"><i><font color="Green">E30</font></i></a>, <a class="ref" href="waybel13.html#T20" target="_self">Th20</a></i>;<br/><a NAME="E9:33_1"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D33">WAYBEL_0:def 33</a></i>;<br/><b>assume </b><a NAME="E10:33_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><a NAME="E11:33_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel13.html#E4:33"><i><font color="Green">E30</font></i></a>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/><a NAME="E12:33_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel13.html#E2:33"><i><font color="Green">E29</font></i></a>, <a class="txt" href="waybel13.html#E6:33_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_1.html#T56">FUNCT_1:56</a></i>;<br/><a NAME="E13:33_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="waybel13.html#E7:33_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="waybel13.html#E9:33_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E15:33_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E16:33_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="waybel13.html#E5:33"><i><font color="Green">E31</font></i></a>, <a class="ref" href="waybel_3.html#D1">WAYBEL_3:def 1</a></i>;<br/><b>take </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font>;<br/><a NAME="E17:33_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <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="E18:33_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E19:33_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="waybel13.html#E15:33_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E20:33_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel13.html#E2:33"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_1.html#T155">FUNCT_1:155</a></i>;<br/><b>hence </b><a NAME="E21:33_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="waybel13.html#E2:33_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#T147">FUNCT_1:147</a></i>;<br/><b>thus </b><a NAME="E22:33_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="waybel13.html#E1:33"><i><font color="Green">E28</font></i></a>, <a class="txt" href="waybel13.html#E16:33_1"><i><font color="Green">E37</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="E7:33"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="waybel_3.html#D1">WAYBEL_3:def 1</a></i>;<br/>


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