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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>thus </b><a NAME="E1:11"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="zfrefle1.html#R2" target="_self">&lt;==&gt;</a> <font color="Maroon">c<sub>2</sub></font> implies for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM2">ZF-formula</a> holds <br/>( <font color="Maroon">c<sub>1</sub></font> <a href="zf_model.html#R2">|=</a> <font color="Olive">b<sub>1</sub></font> iff <font color="Maroon">c<sub>2</sub></font> <a href="zf_model.html#R2">|=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:11_1"/><i><font color="Green">E7</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM2">ZF-formula</a> st  <a href="zf_model.html#K2">Free</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/>( <font color="Maroon">c<sub>1</sub></font> <a href="zf_model.html#R2">|=</a> <font color="Olive">b<sub>1</sub></font> iff <font color="Maroon">c<sub>2</sub></font> <a href="zf_model.html#R2">|=</a> <font color="Olive">b<sub>1</sub></font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="zfrefle1.html#D2" target="_self">ZFREFLE1:def 2</a><br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/>

<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="zf_lang.html#NM2">ZF-formula</a><b> such that </b><br/><a NAME="E3:11_1"/><i><font color="Green">E8</font></i>: 
(  <a href="zf_model.html#K2">Free</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( for <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="zf_model.html#R2">|=</a> <font color="Maroon">c<sub>4</sub></font> iff <font color="Olive">b<sub>1</sub></font> <a href="zf_model.html#R2">|=</a> <font color="Maroon">c<sub>3</sub></font> ) ) )
 <b>by </b><i><a class="ref" href="zfrefle1.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E4:11_1"/>
( ( <font color="Maroon">c<sub>1</sub></font> <a href="zf_model.html#R2">|=</a> <font color="Maroon">c<sub>4</sub></font> implies <font color="Maroon">c<sub>2</sub></font> <a href="zf_model.html#R2">|=</a> <font color="Maroon">c<sub>4</sub></font> ) &amp; ( <font color="Maroon">c<sub>2</sub></font> <a href="zf_model.html#R2">|=</a> <font color="Maroon">c<sub>4</sub></font> implies <font color="Maroon">c<sub>1</sub></font> <a href="zf_model.html#R2">|=</a> <font color="Maroon">c<sub>4</sub></font> ) &amp; ( <font color="Maroon">c<sub>1</sub></font> <a href="zf_model.html#R2">|=</a> <font color="Maroon">c<sub>4</sub></font> implies <font color="Maroon">c<sub>1</sub></font> <a href="zf_model.html#R2">|=</a> <font color="Maroon">c<sub>3</sub></font> ) &amp; ( <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>1</sub></font> <a href="zf_model.html#R2">|=</a> <font color="Maroon">c<sub>4</sub></font> ) &amp; ( <font color="Maroon">c<sub>2</sub></font> <a href="zf_model.html#R2">|=</a> <font color="Maroon">c<sub>4</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> ) &amp; ( <font color="Maroon">c<sub>2</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>4</sub></font> ) )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E1:11_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="zfrefle1.html#E3:11_1"><i><font color="Green">E8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:11_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="zf_model.html#R2">|=</a> <font color="Maroon">c<sub>3</sub></font> iff <font color="Maroon">c<sub>2</sub></font> <a href="zf_model.html#R2">|=</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>


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

<b>assume </b><a NAME="E2:11"/><i><font color="Green">E7</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM2">ZF-formula</a> holds <br/>( <font color="Maroon">c<sub>1</sub></font> <a href="zf_model.html#R2">|=</a> <font color="Olive">b<sub>1</sub></font> iff <font color="Maroon">c<sub>2</sub></font> <a href="zf_model.html#R2">|=</a> <font color="Olive">b<sub>1</sub></font> )
 ;<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>thus </b><a NAME="E3:11"/>
(  <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>  implies ( <font color="Maroon">c<sub>1</sub></font> <a href="zf_model.html#R2">|=</a> <font color="Maroon">c<sub>3</sub></font> iff <font color="Maroon">c<sub>2</sub></font> <a href="zf_model.html#R2">|=</a> <font color="Maroon">c<sub>3</sub></font> ) )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E2:11"><i><font color="Green">E7</font></i></a></i>;<br/>


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