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<span class="kw">let </span><font color="Maroon" title="c1">M1</font>, <font color="Maroon" title="c2">M2</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">M1</font> <a href="zfrefle1.html#R2" title="ZFREFLE1:pred.2">&lt;==&gt;</a> <font color="Maroon" title="c2">M2</font> iff  for <font color="Olive" title="b1">H</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a> holds <br/> ( <font color="Maroon" title="c1">M1</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a> <font color="Olive" title="b1">H</font> iff <font color="Maroon" title="c2">M2</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a> <font color="Olive" title="b1">H</font> ) )</span><br/>



<span class="kw">thus </span><a NAME="E1:11"/>
( <font color="Maroon" title="c1">M1</font> <a href="zfrefle1.html#R2" title="ZFREFLE1:pred.2">&lt;==&gt;</a> <font color="Maroon" title="c2">M2</font> implies  for <font color="Olive" title="b1">H</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a> holds <br/> ( <font color="Maroon" title="c1">M1</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a> <font color="Olive" title="b1">H</font> iff <font color="Maroon" title="c2">M2</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a> <font color="Olive" title="b1">H</font> ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">H</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a> holds <br/> ( <font color="Maroon" title="c1">M1</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a> <font color="Olive" title="b1">H</font> iff <font color="Maroon" title="c2">M2</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a> <font color="Olive" title="b1">H</font> ) ) implies <font color="Maroon" title="c1">M1</font> <a href="zfrefle1.html#R2" title="ZFREFLE1:pred.2">&lt;==&gt;</a> <font color="Maroon" title="c2">M2</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:11_1"/><span class="lab"><font color="Green" title="E8">A1</font></span>: 
 for <font color="Olive" title="b1">H</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>  st  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/>( <font color="Maroon" title="c1">M1</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a> <font color="Olive" title="b1">H</font> iff <font color="Maroon" title="c2">M2</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a> <font color="Olive" title="b1">H</font> )
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="zfrefle1.html#D2" target="_self" title="ZFREFLE1:def.2">ZFREFLE1:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">H</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a> holds <br/> ( <font color="Maroon" title="c1">M1</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a> <font color="Olive" title="b1">H</font> iff <font color="Maroon" title="c2">M2</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a> <font color="Olive" title="b1">H</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">H</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">M1</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a> <font color="Maroon" title="c3">H</font> iff <font color="Maroon" title="c2">M2</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a> <font color="Maroon" title="c3">H</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c4">S</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a><span class="kw"> such that </span><br/><a NAME="E3:11_1"/><span class="lab"><font color="Green" title="E9">A2</font></span>: 
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c4">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">and </span><br/><a NAME="E4:11_1"/><span class="lab"><font color="Green" title="E10">A3</font></span>: 
 for <font color="Olive" title="b1">M</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">M</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a> <font color="Maroon" title="c4">S</font> iff <font color="Olive" title="b1">M</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a> <font color="Maroon" title="c3">H</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfrefle1.html#T6" target="_self" title="ZFREFLE1:th.6">Th6</a></span>;<br/>
<a NAME="E5:11_1"/>
( <font color="Maroon" title="c1">M1</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a> <font color="Maroon" title="c4">S</font> iff <font color="Maroon" title="c2">M2</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a> <font color="Maroon" title="c4">S</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11_1"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="txt" href="#E3:11_1"><span class="lab"><font color="Green" title="E9">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:11_1"/>
( <font color="Maroon" title="c1">M1</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a> <font color="Maroon" title="c3">H</font> iff <font color="Maroon" title="c2">M2</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a> <font color="Maroon" title="c3">H</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:11_1"><span class="lab"><font color="Green" title="E10">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:11"/><span class="lab"><font color="Green" title="E8">A4</font></span>: 
 for <font color="Olive" title="b1">H</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a> holds <br/> ( <font color="Maroon" title="c1">M1</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a> <font color="Olive" title="b1">H</font> iff <font color="Maroon" title="c2">M2</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a> <font color="Olive" title="b1">H</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">M1</font> <a href="zfrefle1.html#R2" title="ZFREFLE1:pred.2">&lt;==&gt;</a> <font color="Maroon" title="c2">M2</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">H</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="zfrefle1.html#D2" target="_self" title="ZFREFLE1:def.2">ZFREFLE1:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c3">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies ( <font color="Maroon" title="c1">M1</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a> <font color="Maroon" title="c3">H</font> iff <font color="Maroon" title="c2">M2</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a> <font color="Maroon" title="c3">H</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E3:11"/>
(  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c3">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies ( <font color="Maroon" title="c1">M1</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a> <font color="Maroon" title="c3">H</font> iff <font color="Maroon" title="c2">M2</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a> <font color="Maroon" title="c3">H</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:11"><span class="lab"><font color="Green" title="E8">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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