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<b>let </b><font color="Maroon" title="c1">H2</font>, <font color="Maroon" title="c2">H1</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a><br/> for <font color="Olive" title="b2">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>   st not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c1">H2</font> holds <br/><font color="Olive" title="b2">M</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Olive" title="b1">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K11" title="ZF_LANG:func.11">=&gt;</a> <font color="Maroon" title="c1">H2</font></span>)</span></span>)</span> <a href="zf_lang.html#K11" title="ZF_LANG:func.11">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K13" title="ZF_LANG:func.13">Ex</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c2">H1</font></span>)</span> <a href="zf_lang.html#K11" title="ZF_LANG:func.11">=&gt;</a> <font color="Maroon" title="c1">H2</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">x</font> be   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 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>   st not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c1">H2</font> holds <br/><font color="Olive" title="b1">M</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c3">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K11" title="ZF_LANG:func.11">=&gt;</a> <font color="Maroon" title="c1">H2</font></span>)</span></span>)</span> <a href="zf_lang.html#K11" title="ZF_LANG:func.11">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K13" title="ZF_LANG:func.13">Ex</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">H1</font></span>)</span> <a href="zf_lang.html#K11" title="ZF_LANG:func.11">=&gt;</a> <font color="Maroon" title="c1">H2</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">M</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c1">H2</font> implies <font color="Maroon" title="c4">M</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c3">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K11" title="ZF_LANG:func.11">=&gt;</a> <font color="Maroon" title="c1">H2</font></span>)</span></span>)</span> <a href="zf_lang.html#K11" title="ZF_LANG:func.11">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K13" title="ZF_LANG:func.13">Ex</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">H1</font></span>)</span> <a href="zf_lang.html#K11" title="ZF_LANG:func.11">=&gt;</a> <font color="Maroon" title="c1">H2</font></span>)</span> )</span><br/>







<b>assume </b><a NAME="E1:123"/><i><font color="Green" title="E40">A1</font></i>: 
not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c1">H2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">M</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c3">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K11" title="ZF_LANG:func.11">=&gt;</a> <font color="Maroon" title="c1">H2</font></span>)</span></span>)</span> <a href="zf_lang.html#K11" title="ZF_LANG:func.11">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K13" title="ZF_LANG:func.13">Ex</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">H1</font></span>)</span> <a href="zf_lang.html#K11" title="ZF_LANG:func.11">=&gt;</a> <font color="Maroon" title="c1">H2</font></span>)</span></span><br/>

<b>let </b><font color="Maroon" title="c5">v</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<font color="Maroon" title="c4">M</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_model.html#D5" title="ZF_MODEL:def.5">ZF_MODEL:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c3">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K11" title="ZF_LANG:func.11">=&gt;</a> <font color="Maroon" title="c1">H2</font></span>)</span></span>)</span> <a href="zf_lang.html#K11" title="ZF_LANG:func.11">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K13" title="ZF_LANG:func.13">Ex</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">H1</font></span>)</span> <a href="zf_lang.html#K11" title="ZF_LANG:func.11">=&gt;</a> <font color="Maroon" title="c1">H2</font></span>)</span></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="123_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:123_1"/><i><font color="Green" title="E41">A2</font></i>: 
<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K11" title="ZF_LANG:func.11">=&gt;</a> <font color="Maroon" title="c1">H2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K13" title="ZF_LANG:func.13">Ex</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">H1</font></span>)</span> <a href="zf_lang.html#K11" title="ZF_LANG:func.11">=&gt;</a> <font color="Maroon" title="c1">H2</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="123_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:123_1_1"/>
<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K13" title="ZF_LANG:func.13">Ex</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">H1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c1">H2</font></span><br/><b>then consider </b><font color="Maroon" title="c6">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">M</font><b> such that </b><br/><a NAME="E3:123_1_1"/><i><font color="Green" title="E42">A3</font></i>: 
<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c6">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c2">H1</font>
 <b>by </b><i><a class="ref" href="zf_lang1.html#T82" target="_self" title="ZF_LANG1:th.82">Th82</a></i>;<br/><a NAME="E4:123_1_1"/>
<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c6">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K11" title="ZF_LANG:func.11">=&gt;</a> <font color="Maroon" title="c1">H2</font>
 <b>by </b><i><a class="txt" href="zf_lang1.html#E1:123_1"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="zf_lang1.html#T80" target="_self" title="ZF_LANG1:th.80">Th80</a></i>;<br/><a NAME="E5:123_1_1"/><b>then </b>
<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c6">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c1">H2</font>
 <b>by </b><i><a class="txt" href="zf_lang1.html#E3:123_1_1"><i><font color="Green" title="E42">A3</font></i></a>, <a class="ref" href="zf_model.html#T18" title="ZF_MODEL:th.18">ZF_MODEL:18</a></i>;<br/><a NAME="E6:123_1_1"/><b>then </b>
<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c6">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c1">H2</font>
 <b>by </b><i><a class="txt" href="zf_lang1.html#E1:123"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="zfmodel1.html#T10" title="ZFMODEL1:th.10">ZFMODEL1:10</a></i>;<br/><a NAME="E7:123_1_1"/><b>then </b>
<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c1">H2</font>
 <b>by </b><i><a class="ref" href="zf_lang1.html#T81" target="_self" title="ZF_LANG1:th.81">Th81</a></i>;<br/><a NAME="E8:123_1_1"/><b>then </b>
<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c1">H2</font>
 <b>by </b><i><a class="ref" href="zf_lang1.html#T80" target="_self" title="ZF_LANG1:th.80">Th80</a></i>;<br/><b>hence </b><a NAME="E9:123_1_1"/>
<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c1">H2</font>
 <b>by </b><i><a class="ref" href="zf_lang1.html#T78" target="_self" title="ZF_LANG1:th.78">Th78</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E3:123_1"/>
<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K13" title="ZF_LANG:func.13">Ex</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">H1</font></span>)</span> <a href="zf_lang.html#K11" title="ZF_LANG:func.11">=&gt;</a> <font color="Maroon" title="c1">H2</font>
 <b>by </b><i><a class="ref" href="zf_model.html#T18" title="ZF_MODEL:th.18">ZF_MODEL:18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:123"/>
<font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c5">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c3">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K11" title="ZF_LANG:func.11">=&gt;</a> <font color="Maroon" title="c1">H2</font></span>)</span></span>)</span> <a href="zf_lang.html#K11" title="ZF_LANG:func.11">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K13" title="ZF_LANG:func.13">Ex</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">H1</font></span>)</span> <a href="zf_lang.html#K11" title="ZF_LANG:func.11">=&gt;</a> <font color="Maroon" title="c1">H2</font></span>)</span>
 <b>by </b><i><a class="ref" href="zf_model.html#T18" title="ZF_MODEL:th.18">ZF_MODEL:18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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