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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <span class="p1">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>





<a NAME="E1:32"/><i><font color="Green">E23</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="substlat.html#K4" target="_self">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="substlat.html#K4" target="_self">^</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="substlat.html#K4" target="_self">^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="substlat.html#K4" target="_self">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="substlat.html#K4" target="_self">^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="substlat.html#K4" target="_self">^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="substlat.html#K4" target="_self">^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="substlat.html#K4" target="_self">^</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="substlat.html#K4" target="_self">^</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="substlat.html#K4" target="_self">^</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="substlat.html#K4" target="_self">^</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="substlat.html#K4" target="_self">^</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="substlat.html#T3" target="_self">Th3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <span class="p1">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/><a NAME="E1:32_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a>  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="finsub_1.html#D5">FINSUB_1:def 5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_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="E1:32_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="substlat.html#K4" target="_self">^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="substlat.html#K4" target="_self">^</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:32_1_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E4:32_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="substlat.html#K4" target="_self">^</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E5:32_1_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="substlat.html#T15" target="_self">Th15</a></i>;<br/><a NAME="E6:32_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="substlat.html#K4" target="_self">^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="substlat.html#K4" target="_self">^</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="finsub_1.html#D5">FINSUB_1:def 5</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E8:32_1_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="substlat.html#E1:32_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="partfun1.html#D5">PARTFUN1:def 5</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:32_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E11:32_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E12:32_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="substlat.html#E4:32_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="substlat.html#T15" target="_self">Th15</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>14</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="substlat.html#E1:32_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="substlat.html#E3:32_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="substlat.html#E10:32_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="substlat.html#E11:32_1_1"><i><font color="Green">E31</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="substlat.html#E1:32_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="substlat.html#E3:32_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="substlat.html#E10:32_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="partfun1.html#T120">PARTFUN1:120</a></i>;<br/><a NAME="E15:32_1_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E16:32_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="substlat.html#E12:32_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E17:32_1_1"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="substlat.html#E15:32_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E18:32_1_1"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="partfun1.html#R1">tolerates</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="substlat.html#E5:32_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="substlat.html#E8:32_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="substlat.html#E15:32_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="partfun1.html#T139">PARTFUN1:139</a></i>;<br/><a NAME="E19:32_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/><a NAME="E20:32_1_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="substlat.html#E18:32_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_4.html#T31">FUNCT_4:31</a></i>;<br/><a NAME="E21:32_1_1"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font> <a href="relset_1.html#K1">\/</a> <font color="Maroon">c<sub>19</sub></font> is   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>15</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>16</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="partfun1.html#T119">PARTFUN1:119</a></i>;<br/><a NAME="E23:32_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="substlat.html#K4" target="_self">^</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="substlat.html#E3:32_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="substlat.html#E10:32_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="substlat.html#E18:32_1_1"><i><font color="Green">E35</font></i></a></i>;<br/><a NAME="E24:32_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="substlat.html#E12:32_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E25:32_1_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="substlat.html#E15:32_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E26:32_1_1"/>
<font color="Maroon">c<sub>20</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="substlat.html#E15:32_1_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="substlat.html#E17:32_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><a NAME="E27:32_1_1"/><b>then </b>
<font color="Maroon">c<sub>20</sub></font> <a href="partfun1.html#R1">tolerates</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="substlat.html#E5:32_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="substlat.html#E8:32_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="substlat.html#E25:32_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="partfun1.html#T139">PARTFUN1:139</a></i>;<br/><b>hence </b><a NAME="E28:32_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="substlat.html#K4" target="_self">^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="substlat.html#K4" target="_self">^</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="substlat.html#E5:32_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="substlat.html#E11:32_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="substlat.html#E12:32_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="substlat.html#E19:32_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="substlat.html#E23:32_1_1"><i><font color="Green">E37</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:32_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="substlat.html#K4" target="_self">^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="substlat.html#K4" target="_self">^</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="substlat.html#K4" target="_self">^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="substlat.html#K4" target="_self">^</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="substlat.html#E11"><i><font color="Green">Lemma8</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:32"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="substlat.html#K4" target="_self">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="substlat.html#K4" target="_self">^</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="substlat.html#K4" target="_self">^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="substlat.html#K4" target="_self">^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="substlat.html#K4" target="_self">^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="substlat.html#K4" target="_self">^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="substlat.html#K4" target="_self">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="substlat.html#K4" target="_self">^</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="substlat.html#E1:32"><i><font color="Green">E23</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:32"/>
<font color="Maroon">c<sub>3</sub></font> <a href="substlat.html#K4" target="_self">^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="substlat.html#K4" target="_self">^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="substlat.html#K4" target="_self">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="substlat.html#K4" target="_self">^</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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