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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="funct_1.html#NM1">Function</a>;<br/>







<b>given </b><font color="Maroon">c<sub>5</sub></font> being   <a href="zf_lang.html#NM2">ZF-formula</a><b> such that </b><a NAME="E1:21"/><i><font color="Green">E18</font></i>: 
 <a href="zf_model.html#K2">Free</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span><a href="tarski.html#K2">}</a></span>
 <b>and </b><br/><a NAME="E2:21"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="zf_model.html#R2">|=</a>  <a href="zf_lang.html#K8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E3:21"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="zfmodel1.html#K2">def_func</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="zfmodel1.html#D3">ZFMODEL1:def 3</a><br/>

<b>given </b><font color="Maroon">c<sub>6</sub></font> being   <a href="zf_lang.html#NM2">ZF-formula</a><b> such that </b><a NAME="E4:21"/><i><font color="Green">E21</font></i>: 
 <a href="zf_model.html#K2">Free</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span><a href="tarski.html#K2">}</a></span>
 <b>and </b><br/><a NAME="E5:21"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="zf_model.html#R2">|=</a>  <a href="zf_lang.html#K8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E6:21"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="zfmodel1.html#K2">def_func</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="zfmodel1.html#D3">ZFMODEL1:def 3</a><br/>

<b>assume </b><a NAME="E7:21"/><i><font color="Green">E24</font></i>: 
 <a href="zf_model.html#K2">Free</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E8:21"/><b>then </b><i><font color="Green">E25</font></i>: 
not  <a href="zf_lang.html#K2">x.</a> 4 <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="zfmodel2.html#E12"><i><font color="Green">Lemma11</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="funct_1.html#NM1">Function</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E9:21"/><i><font color="Green">E26</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E10:21"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>1</sub></font> holds <br/>( ( <font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span></span>)</span> ) &amp; ( <font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span></span>)</span> ) )
 ;<br/>

<b>take </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="zf_lang.html#K10">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="zfmodel1.html#D3">ZFMODEL1:def 3</a><br/>

<a NAME="E11:21"/><i><font color="Green">E28</font></i>: 
 <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T65">ZF_LANG1:65</a></i>;<br/>
<a NAME="E12:21"/>
(  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2">Free</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2">Free</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp;  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2">Free</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2">Free</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span><a href="enumset1.html#K1">}</a></span> &amp; <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span><a href="enumset1.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span><a href="tarski.html#K2">}</a></span> )
 
<b>by </b><i><a class="ref" href="enumset1.html#T42">ENUMSET1:42</a>, <a class="ref" href="enumset1.html#T70">ENUMSET1:70</a>, <a class="ref" href="zf_lang1.html#T66">ZF_LANG1:66</a></i>;<br/>
<a NAME="E13:21"/><b>then </b>
(  <a href="zf_model.html#K2">Free</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2">Free</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2">Free</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> &amp;  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span><a href="tarski.html#K2">}</a></span> &amp;  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span><a href="tarski.html#K2">}</a></span> )
 
<b>by </b><i><a class="txt" href="zfmodel2.html#E1:21"><i><font color="Green">E18</font></i></a>, <a class="txt" href="zfmodel2.html#E4:21"><i><font color="Green">E21</font></i></a>, <a class="txt" href="zfmodel2.html#E7:21"><i><font color="Green">E24</font></i></a>, <a class="txt" href="zfmodel2.html#E11:21"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xboole_1.html#T13">XBOOLE_1:13</a>, <a class="ref" href="zf_lang1.html#T68">ZF_LANG1:68</a></i>;<br/>
<b>hence </b><a NAME="E14:21"/><i><font color="Green">E29</font></i>: 
 <a href="zf_model.html#K2">Free</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>

<a NAME="E15:21"/><b>then </b><i><font color="Green">E30</font></i>: 
( not  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">c<sub>8</sub></font> &amp; not  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">c<sub>5</sub></font> &amp; not  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="zfmodel2.html#E1:21"><i><font color="Green">E18</font></i></a>, <a class="txt" href="zfmodel2.html#E4:21"><i><font color="Green">E21</font></i></a>, <a class="txt" href="zfmodel2.html#E12"><i><font color="Green">Lemma11</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>1</sub></font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:21_1_1"/>
<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="zfmodel2.html#E2:21"><i><font color="Green">E19</font></i></a>, <a class="ref" href="zf_model.html#D5">ZF_MODEL:def 5</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:21_1_1"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<font color="Olive">b<sub>1</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>5</sub></font> iff <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="zfmodel2.html#E15:21"><i><font color="Green">E30</font></i></a>, <a class="ref" href="zfmodel2.html#T20" target="_self">Th20</a></i>;<br/><a NAME="E4:21_1_1"/>
<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="zfmodel2.html#E5:21"><i><font color="Green">E22</font></i></a>, <a class="ref" href="zf_model.html#D5">ZF_MODEL:def 5</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:21_1_1"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<font color="Olive">b<sub>1</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>6</sub></font> iff <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="zfmodel2.html#E15:21"><i><font color="Green">E30</font></i></a>, <a class="ref" href="zfmodel2.html#T20" target="_self">Th20</a></i>;<br/><a NAME="E7:21_1_1"/>
( ( not <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>10</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>10</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>2</sub></font> ) or ( <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>10</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font> &amp; not <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>10</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>2</sub></font> ) )
 <b>by </b><i><a class="ref" href="zf_model.html#T14">ZF_MODEL:14</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E9:21_1_1"/><i><font color="Green">E33</font></i>: 
( ( not <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>10</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>10</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> ) or ( <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>10</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; not <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>10</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>2</sub></font> ) )
 ;<br/><b>take </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>thus </b><a NAME="E10:21_1_1"/>
( <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<font color="Maroon">c<sub>15</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>8</sub></font> implies <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<font color="Maroon">c<sub>15</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<a NAME="E2:21_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<font color="Maroon">c<sub>15</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<font color="Maroon">c<sub>15</sub></font> <a href="zf_model.html#R1">|=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="ref" href="zf_model.html#T17">ZF_MODEL:17</a></i>;<br/>
<a NAME="E3:21_1_1_1"/><b>then </b>
( ( <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<font color="Maroon">c<sub>15</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<font color="Maroon">c<sub>15</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>5</sub></font> ) or ( <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<font color="Maroon">c<sub>15</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<font color="Maroon">c<sub>15</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>6</sub></font> ) )
 
<b>by </b><i><a class="ref" href="zf_model.html#T15">ZF_MODEL:15</a></i>;<br/>
<b>hence </b><a NAME="E4:21_1_1_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="zfmodel2.html#E8:21"><i><font color="Green">E25</font></i></a>, <a class="txt" href="zfmodel2.html#E11:21"><i><font color="Green">E28</font></i></a>, <a class="txt" href="zfmodel2.html#E3:21_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="zfmodel2.html#E6:21_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="zfmodel2.html#E9:21_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="zfmodel2.html#T10" target="_self">Th10</a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E11:21_1_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/><a NAME="E12:21_1_1"/><b>then </b>
( ( <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<font color="Maroon">c<sub>15</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<font color="Maroon">c<sub>15</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>5</sub></font> ) or ( <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<font color="Maroon">c<sub>15</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<font color="Maroon">c<sub>15</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>6</sub></font> ) )
 <b>by </b><i><a class="txt" href="zfmodel2.html#E8:21"><i><font color="Green">E25</font></i></a>, <a class="txt" href="zfmodel2.html#E11:21"><i><font color="Green">E28</font></i></a>, <a class="txt" href="zfmodel2.html#E3:21_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="zfmodel2.html#E6:21_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="zfmodel2.html#E9:21_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="zfmodel2.html#T10" target="_self">Th10</a></i>;<br/><a NAME="E13:21_1_1"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<font color="Maroon">c<sub>15</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<font color="Maroon">c<sub>15</sub></font> <a href="zf_model.html#R1">|=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="zf_model.html#T15">ZF_MODEL:15</a></i>;<br/><b>hence </b><a NAME="E14:21_1_1"/>
<font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<font color="Maroon">c<sub>15</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="zf_model.html#T17">ZF_MODEL:17</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:21_1"/>
<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="zfmodel2.html#E15:21"><i><font color="Green">E30</font></i></a>, <a class="ref" href="zfmodel2.html#T20" target="_self">Th20</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E17:21"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="zf_model.html#R2">|=</a>  <a href="zf_lang.html#K8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="zf_model.html#D5">ZF_MODEL:def 5</a></i>;<br/>

<a NAME="E18:21"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:21_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</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="E3:21_2"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="zfmodel2.html#E9:21"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="zfmodel2.html#E3:21_2"><i><font color="Green">E32</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E6:21_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; ( <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>11</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>11</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font> ) )
 
<b>by </b><i><a class="ref" href="zf_lang1.html#D1">ZF_LANG1:def 1</a>, <a class="ref" href="zf_model.html#T14">ZF_MODEL:14</a></i>;<br/>
<a NAME="E7:21_2"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zfmodel1.html#K2">def_func</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> or <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zfmodel1.html#K2">def_func</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> )
 
<b>by </b><i><a class="txt" href="zfmodel2.html#E3:21"><i><font color="Green">E20</font></i></a>, <a class="txt" href="zfmodel2.html#E6:21"><i><font color="Green">E23</font></i></a>, <a class="txt" href="zfmodel2.html#E10:21"><i><font color="Green">E27</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:21_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="zfmodel2.html#E3:21_2"><i><font color="Green">E32</font></i></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>7</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="zfmodel2.html#E9:21"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:21_3"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>set </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1">(<span class="default"><a href="zfmodel1.html#K2">def_func</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E4:21_3"/>
( <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> &amp;  <a href="zfmodel1.html#K2">def_func</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#R2">=</a>  <a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="zfmodel2.html#E14:21"><i><font color="Green">E29</font></i></a>, <a class="txt" href="zfmodel2.html#E17:21"><i><font color="Green">E31</font></i></a>, <a class="ref" href="zfmodel2.html#T12" target="_self">Th12</a>, <a class="ref" href="zf_model.html#D5">ZF_MODEL:def 5</a></i>;<br/><a NAME="E5:21_3"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zfmodel1.html#K2">def_func</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="zfmodel2.html#E15:21"><i><font color="Green">E30</font></i></a>, <a class="ref" href="zfmodel2.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E6:21_3"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zfmodel1.html#K2">def_func</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>2</sub></font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zfmodel1.html#K2">def_func</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="zf_model.html#R1">|=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="zf_model.html#T17">ZF_MODEL:17</a></i>;<br/><a NAME="E7:21_3"/><b>then </b>
( ( <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zfmodel1.html#K2">def_func</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zfmodel1.html#K2">def_func</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> &amp;  <a href="zfmodel1.html#K2">def_func</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#R2">=</a>  <a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>13</sub></font> ) or ( <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zfmodel1.html#K2">def_func</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zfmodel1.html#K2">def_func</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> &amp;  <a href="zfmodel1.html#K2">def_func</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#R2">=</a>  <a href="zfmodel1.html#K1">def_func'</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>13</sub></font> ) )
 <b>by </b><i><a class="txt" href="zfmodel2.html#E1:21"><i><font color="Green">E18</font></i></a>, <a class="txt" href="zfmodel2.html#E2:21"><i><font color="Green">E19</font></i></a>, <a class="txt" href="zfmodel2.html#E4:21"><i><font color="Green">E21</font></i></a>, <a class="txt" href="zfmodel2.html#E5:21"><i><font color="Green">E22</font></i></a>, <a class="ref" href="zfmodel2.html#T12" target="_self">Th12</a>, <a class="ref" href="zf_model.html#T15">ZF_MODEL:15</a>, <a class="ref" href="zf_model.html#D5">ZF_MODEL:def 5</a></i>;<br/><a NAME="E8:21_3"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; ( ( <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>11</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><a href="zfmodel1.html#K2">def_func</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</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>11</sub></font> ) or ( <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="zf_lang1.html#K1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">c<sub>11</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><a href="zfmodel1.html#K2">def_func</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> ) ) )
 <b>by </b><i><a class="txt" href="zfmodel2.html#E3:21"><i><font color="Green">E20</font></i></a>, <a class="txt" href="zfmodel2.html#E6:21"><i><font color="Green">E23</font></i></a>, <a class="txt" href="zfmodel2.html#E8:21"><i><font color="Green">E25</font></i></a>, <a class="txt" href="zfmodel2.html#E11:21"><i><font color="Green">E28</font></i></a>, <a class="txt" href="zfmodel2.html#E15:21"><i><font color="Green">E30</font></i></a>, <a class="ref" href="zfmodel2.html#T10" target="_self">Th10</a>, <a class="ref" href="zfmodel2.html#T11" target="_self">Th11</a>, <a class="ref" href="zf_lang1.html#D1">ZF_LANG1:def 1</a></i>;<br/><b>hence </b><a NAME="E9:21_3"/>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zfmodel1.html#K2">def_func</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="zfmodel2.html#E10:21"><i><font color="Green">E27</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E21:21"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="zfmodel1.html#K2">def_func</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#T18">FUNCT_2:18</a></i>;<br/>


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