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<b>let </b><font color="Maroon" title="c1">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> ;<br/><b>let </b><font color="Maroon" title="c2">H</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>;<br/><b>let </b><font color="Maroon" title="c3">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="c1">M</font>;<br/>





<b>assume </b><a NAME="E1:11"/><i><font color="Green" title="E11">A1</font></i>: 
( not  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 0 <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="c2">H</font> &amp; <font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c3">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> <span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K13" title="ZF_LANG:func.13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">H</font> <a href="zf_lang.html#K12" title="ZF_LANG:func.12">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> )
 ;<br/>

<b>let </b><font color="Maroon" title="c4">m1</font>, <font color="Maroon" title="c5">m2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font>;<br/>



<a NAME="E2:11"/>
 <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T86" title="ZF_LANG1:th.86">ZF_LANG1:86</a></i>;<br/>
<a NAME="E3:11"/><b>then </b><i><font color="Green" title="E12">A2</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c4">m1</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<font color="Maroon" title="c5">m2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">m2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c4">m1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c4">m1</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<font color="Maroon" title="c5">m2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c4">m1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span> )
 
<b>by </b><i><a class="ref" href="zf_lang1.html#D1" title="ZF_LANG1:def.1">ZF_LANG1:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">y</font> be   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>;<br/><b>assume </b><a NAME="E1:11_1"/><i><font color="Green" title="E13">A3</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c4">m1</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<font color="Maroon" title="c5">m2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font>
 ;<br/><b>assume </b><a NAME="E2:11_1"/>
(  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 0 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">y</font> &amp;  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">y</font> &amp;  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">y</font> )
 ;<br/><a NAME="E3:11_1"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c4">m1</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<font color="Maroon" title="c5">m2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c4">m1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c4">m1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font> )
 <b>by </b><i><a class="ref" href="zf_lang1.html#D1" title="ZF_LANG1:def.1">ZF_LANG1:def 1</a></i>;<br/><b>hence </b><a NAME="E4:11_1"/>
contradiction
 <b>by </b><i><a class="txt" href="zfmodel2.html#E1:11_1"><i><font color="Green" title="E13">A3</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:11"/>
( <span class="p1">(<span class="default"><a href="zfmodel1.html#K1" title="ZFMODEL1:func.1">def_func'</a> <font color="Maroon" title="c2">H</font>,<font color="Maroon" title="c3">v</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">m2</font> iff <font color="Maroon" title="c1">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c4">m1</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<font color="Maroon" title="c5">m2</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c2">H</font> )
 <b>by </b><i><a class="txt" href="zfmodel2.html#E1:11"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="zfmodel2.html#E3:11"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="zfmodel1.html#D1" title="ZFMODEL1:def.1">ZFMODEL1:def 1</a></i>;<br/>


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