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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:13"/><i><font color="Green">E9</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM2">ZF-formula</a><br/>for <font color="Olive">b<sub>2</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>2</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Olive">b<sub>1</sub></font> iff <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="zf_lang1.html#K2">!</a> <font color="Olive">b<sub>2</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Olive">b<sub>1</sub></font> ) ) )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="zfrefle1.html#D3" target="_self">ZFREFLE1:def 3</a><br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="zf_lang.html#NM2">ZF-formula</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="zfrefle1.html#D2" target="_self">ZFREFLE1:def 2</a><br/>

<b>assume </b><a NAME="E2:13"/><i><font color="Green">E10</font></i>: 
 <a href="zf_model.html#K2">Free</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>thus </b><a NAME="E3:13"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="zf_model.html#R2">|=</a> <font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">c<sub>2</sub></font> <a href="zf_model.html#R2">|=</a> <font color="Maroon">c<sub>3</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_1"/><i><font color="Green">E11</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 <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>3</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_model.html#D5">ZF_MODEL:def 5</a><br/>

<b>let </b><font color="Maroon">c<sub>4</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>2</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_model.html#D5">ZF_MODEL:def 5</a><br/>

<b>consider </b><font color="Maroon">c<sub>5</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="E3:13_1"/>
<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E1:13_1"><i><font color="Green">E11</font></i></a></i>;<br/>
<a NAME="E4:13_1"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>5</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>3</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E1:13"><i><font color="Green">E9</font></i></a>, <a class="txt" href="zfrefle1.html#E2:13"><i><font color="Green">E10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:13_1"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="zf_lang1.html#T84">ZF_LANG1:84</a></i>;<br/>


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

<b>assume </b><a NAME="E4:13"/><i><font color="Green">E11</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>2</sub></font> holds <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>3</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_model.html#D5">ZF_MODEL:def 5</a><br/>

<b>let </b><font color="Maroon">c<sub>4</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>; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_model.html#D5">ZF_MODEL:def 5</a><br/>

<a NAME="E5:13"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="zf_lang1.html#K2">!</a> <font color="Maroon">c<sub>4</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E4:13"><i><font color="Green">E11</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:13"/>
<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="zfrefle1.html#E1:13"><i><font color="Green">E9</font></i></a></i>;<br/>


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