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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:10"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> holds <br/>for <font color="Olive">x</font> being  <a href="zf_lang.html#NM1">Variable</a><br/> for <font color="Olive">E</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">H</font> being  <a href="zf_lang.html#NM2">ZF-formula</a><br/> for <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Olive">E</font>  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">H</font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Olive">H</font> &amp; <font color="Olive">E</font>,<font color="Olive">f</font> <a href="zf_model.html#R1">|=</a> <font color="Olive">H</font> holds <br/><font color="Olive">E</font>,<font color="Olive">f</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <font color="Olive">x</font>,<font color="Olive">H</font>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/><b>let </b><font color="Maroon">E</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">H</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">E</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E2:10"/><i><font color="Green">E38</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E3:10"/><i><font color="Green">E39</font></i>: 
( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font> &amp; <font color="Maroon">E</font>,<font color="Maroon">f</font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H</font> )
 ;<br/>

<div><i><font color="Green">E41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_1"/>
<font color="Maroon">H</font> <a href="zf_lang.html#NR1" target="_self">is_equality</a> 
 ;<br/><a NAME="E2:10_1"/><b>then </b><i><font color="Green">E42</font></i>: 
( <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">H</font></span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">H</font></span>)</span> &amp;  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">H</font></span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">H</font></span>)</span></span><a href="tarski.html#K2">}</a></span> )
 <b>by </b><i><a class="ref" href="zf_lang.html#T53">ZF_LANG:53</a>, <a class="ref" href="zf_model.html#T1">ZF_MODEL:1</a></i>;<br/><a NAME="E3:10_1"/><b>then </b><i><font color="Green">E43</font></i>: 
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">H</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">H</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K18">Var1</a> <font color="Maroon">H</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K19">Var2</a> <font color="Maroon">H</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a>, <a class="ref" href="zf_model.html#T12">ZF_MODEL:12</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">g</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">E</font>;<br/><b>assume </b><a NAME="E1:10_1_1"/>
for <font color="Olive">y</font> being  <a href="zf_lang.html#NM1">Variable</a>  st <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> holds <br/><font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font>
 ;<br/><a NAME="E2:10_1_1"/><b>then </b>
( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">H</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">H</font></span>)</span> &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">H</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">H</font></span>)</span> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:10_1_1"/>
<font color="Maroon">E</font>,<font color="Maroon">g</font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H</font>
 <b>by </b><i>, , <a class="ref" href="zf_model.html#T12">ZF_MODEL:12</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:10_1"/>
<font color="Maroon">E</font>,<font color="Maroon">f</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <font color="Maroon">x</font>,<font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="zf_model.html#T16">ZF_MODEL:16</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E44</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_2"/>
<font color="Maroon">H</font> <a href="zf_lang.html#NR2" target="_self">is_membership</a> 
 ;<br/><a NAME="E2:10_2"/><b>then </b><i><font color="Green">E45</font></i>: 
( <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">H</font></span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">H</font></span>)</span> &amp;  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">H</font></span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">H</font></span>)</span></span><a href="tarski.html#K2">}</a></span> )
 <b>by </b><i><a class="ref" href="zf_lang.html#T54">ZF_LANG:54</a>, <a class="ref" href="zf_model.html#T1">ZF_MODEL:1</a></i>;<br/><a NAME="E3:10_2"/><b>then </b><i><font color="Green">E48</font></i>: 
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">H</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">H</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K18">Var1</a> <font color="Maroon">H</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K19">Var2</a> <font color="Maroon">H</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a>, <a class="ref" href="zf_model.html#T13">ZF_MODEL:13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">g</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">E</font>;<br/><b>assume </b><a NAME="E1:10_2_1"/>
for <font color="Olive">y</font> being  <a href="zf_lang.html#NM1">Variable</a>  st <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> holds <br/><font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font>
 ;<br/><a NAME="E2:10_2_1"/><b>then </b>
( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">H</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">H</font></span>)</span> &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">H</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">H</font></span>)</span> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:10_2_1"/>
<font color="Maroon">E</font>,<font color="Maroon">g</font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H</font>
 <b>by </b><i>, , <a class="ref" href="zf_model.html#T13">ZF_MODEL:13</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:10_2"/>
<font color="Maroon">E</font>,<font color="Maroon">f</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <font color="Maroon">x</font>,<font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="zf_model.html#T16">ZF_MODEL:16</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E49</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_3"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">H</font> is <a href="zf_lang.html#V4">negative</a>
 ;<br/><a NAME="E2:10_3"/><b>then </b><i><font color="Green">E52</font></i>: 
( <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">H</font></span>)</span> &amp;  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">H</font></span>)</span> )
 <b>by </b><i><a class="ref" href="zf_lang.html#D30">ZF_LANG:def 30</a>, <a class="ref" href="zf_model.html#T1">ZF_MODEL:1</a></i>;<br/><b>assume </b><a NAME="E3:10_3"/>
not <font color="Maroon">E</font>,<font color="Maroon">f</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <font color="Maroon">x</font>,<font color="Maroon">H</font>
 ;<br/><b>then consider </b><font color="Maroon">g</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">E</font><b> such that </b><br/><a NAME="E5:10_3"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">y</font> being  <a href="zf_lang.html#NM1">Variable</a>  st <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> holds <br/><font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font>
 <b>and </b><br/><a NAME="E6:10_3"/><i><font color="Green">E54</font></i>: 
not <font color="Maroon">E</font>,<font color="Maroon">g</font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="zf_model.html#T16">ZF_MODEL:16</a></i>;<br/><a NAME="E7:10_3"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">E</font>,<font color="Maroon">g</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">H</font>
 <b>by </b><i>, , <a class="ref" href="zf_model.html#T14">ZF_MODEL:14</a></i>;<br/><a NAME="E8:10_3"/>
 <a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">H</font> <a href="zf_lang.html#R1">is_immediate_constituent_of</a> <font color="Maroon">H</font>
 <b>by </b><i>, <a class="ref" href="zf_lang.html#D39">ZF_LANG:def 39</a></i>;<br/><a NAME="E9:10_3"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">H</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">H</font></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="zf_lang.html#T81">ZF_LANG:81</a>, <a class="ref" href="zf_model.html#T1">ZF_MODEL:1</a></i>;<br/><a NAME="E10:10_3"/><b>then </b>
<font color="Maroon">E</font>,<font color="Maroon">g</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">H</font></span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E11:10_3"/><b>then </b>
( <font color="Maroon">E</font>,<font color="Maroon">f</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">H</font> &amp; not <font color="Maroon">E</font>,<font color="Maroon">f</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">H</font> )
 <b>by </b><i>, , , <a class="ref" href="zf_model.html#T14">ZF_MODEL:14</a>, <a class="ref" href="zf_model.html#T16">ZF_MODEL:16</a></i>;<br/><b>hence </b><a NAME="E12:10_3"/>
contradiction
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green">E56</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_4"/>
<font color="Maroon">H</font> is <a href="zf_lang.html#V5">conjunctive</a>
 ;<br/><a NAME="E2:10_4"/><b>then </b><i><font color="Green">E57</font></i>: 
( <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">H</font></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">H</font></span>)</span> &amp;  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2">Free</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2">Free</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="zf_lang.html#T58">ZF_LANG:58</a>, <a class="ref" href="zf_model.html#T1">ZF_MODEL:1</a></i>;<br/><a NAME="E3:10_4"/><b>then </b><i><font color="Green">E58</font></i>: 
( <font color="Maroon">E</font>,<font color="Maroon">f</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">H</font> &amp; <font color="Maroon">E</font>,<font color="Maroon">f</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">H</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">H</font></span>)</span> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">H</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, <a class="ref" href="zf_model.html#T15">ZF_MODEL:15</a></i>;<br/><a NAME="E4:10_4"/>
(  <a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">H</font> <a href="zf_lang.html#R1">is_immediate_constituent_of</a> <font color="Maroon">H</font> &amp;  <a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">H</font> <a href="zf_lang.html#R1">is_immediate_constituent_of</a> <font color="Maroon">H</font> )
 <b>by </b><i>, <a class="ref" href="zf_lang.html#D39">ZF_LANG:def 39</a></i>;<br/><a NAME="E5:10_4"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">H</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">H</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> )
 <b>by </b><i><a class="ref" href="zf_lang.html#T81">ZF_LANG:81</a></i>;<br/><a NAME="E6:10_4"/><b>then </b><i><font color="Green">E60</font></i>: 
( <font color="Maroon">E</font>,<font color="Maroon">f</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">H</font></span>)</span> &amp; <font color="Maroon">E</font>,<font color="Maroon">f</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">H</font></span>)</span> )
 <b>by </b><i>, , </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">g</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">E</font>;<br/><b>assume </b><a NAME="E1:10_4_1"/>
for <font color="Olive">y</font> being  <a href="zf_lang.html#NM1">Variable</a>  st <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> holds <br/><font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font>
 ;<br/><a NAME="E2:10_4_1"/><b>then </b>
( <font color="Maroon">E</font>,<font color="Maroon">g</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">H</font> &amp; <font color="Maroon">E</font>,<font color="Maroon">g</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">H</font> )
 <b>by </b><i>, <a class="ref" href="zf_model.html#T16">ZF_MODEL:16</a></i>;<br/><b>hence </b><a NAME="E3:10_4_1"/>
<font color="Maroon">E</font>,<font color="Maroon">g</font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H</font>
 <b>by </b><i>, <a class="ref" href="zf_model.html#T15">ZF_MODEL:15</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:10_4"/>
<font color="Maroon">E</font>,<font color="Maroon">f</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <font color="Maroon">x</font>,<font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="zf_model.html#T16">ZF_MODEL:16</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_5"/>
<font color="Maroon">H</font> is <a href="zf_lang.html#V6">universal</a>
 ;<br/><a NAME="E2:10_5"/><b>then </b><i><font color="Green">E61</font></i>: 
( <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">H</font></span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">H</font></span>)</span> &amp;  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2">Free</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">H</font></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="zf_lang.html#T62">ZF_LANG:62</a>, <a class="ref" href="zf_model.html#T1">ZF_MODEL:1</a></i>;<br/><div><i><font color="Green">E64</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_5_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">H</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_5_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">g</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">E</font>;<br/><b>assume </b><a NAME="E1:10_5_1_1"/>
for <font color="Olive">y</font> being  <a href="zf_lang.html#NM1">Variable</a>  holds <br/>(  not <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> or  <a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> )
 ;<br/><a NAME="E2:10_5_1_1"/><b>then </b>
for <font color="Olive">y</font> being  <a href="zf_lang.html#NM1">Variable</a>  st <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> holds <br/> <a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:10_5_1_1"/>
<font color="Maroon">E</font>,<font color="Maroon">g</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">H</font>
 <b>by </b><i>, , <a class="ref" href="zf_model.html#T16">ZF_MODEL:16</a></i>;<br/></div><b>end;</b></div><a NAME="E3:10_5_1"/><b>then </b>
<font color="Maroon">E</font>,<font color="Maroon">f</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K14">All</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">H</font></span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">H</font></span>)</span>
 <b>by </b><i><a class="ref" href="zf_model.html#T22">ZF_MODEL:22</a></i>;<br/><b>hence </b><a NAME="E4:10_5_1"/>
<font color="Maroon">E</font>,<font color="Maroon">f</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <font color="Maroon">x</font>,<font color="Maroon">H</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><a NAME="E4:10_5"/><i><font color="Green">E66</font></i>: 
(  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">H</font></span>)</span> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">H</font></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_5_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_5_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">H</font>
 ;<br/><b>assume </b><a NAME="E2:10_5_2"/>
not <font color="Maroon">E</font>,<font color="Maroon">f</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <font color="Maroon">x</font>,<font color="Maroon">H</font>
 ;<br/><b>then consider </b><font color="Maroon">g</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">E</font><b> such that </b><br/><a NAME="E4:10_5_2"/><i><font color="Green">E71</font></i>: 
for <font color="Olive">y</font> being  <a href="zf_lang.html#NM1">Variable</a>  st <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> holds <br/><font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font>
 <b>and </b><br/><a NAME="E5:10_5_2"/><i><font color="Green">E72</font></i>: 
not <font color="Maroon">E</font>,<font color="Maroon">g</font> <a href="zf_model.html#R1">|=</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="zf_model.html#T16">ZF_MODEL:16</a></i>;<br/><b>consider </b><font color="Maroon">h</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">E</font><b> such that </b><br/><a NAME="E7:10_5_2"/><i><font color="Green">E73</font></i>: 
for <font color="Olive">y</font> being  <a href="zf_lang.html#NM1">Variable</a>  st <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> holds <br/> <a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font>
 <b>and </b><br/><a NAME="E8:10_5_2"/><i><font color="Green">E78</font></i>: 
not <font color="Maroon">E</font>,<font color="Maroon">h</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">H</font>
 <b>by </b><i>, , <a class="ref" href="zf_model.html#T16">ZF_MODEL:16</a></i>;<br/><b>consider </b><font color="Maroon">m</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">E</font><b> such that </b><br/><a NAME="E10:10_5_2"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">m</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">H</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">H</font></span>)</span> &amp; ( for <font color="Olive">y</font> being  <a href="zf_lang.html#NM1">Variable</a>  st <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">H</font> holds <br/><font color="Maroon">m</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> ) )
 <b>by </b><i><a class="ref" href="zf_model.html#T21">ZF_MODEL:21</a></i>;<br/><a NAME="E11:10_5_2"/>
( ( for <font color="Olive">y</font> being  <a href="zf_lang.html#NM1">Variable</a>  st <font color="Maroon">m</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> holds <br/> <a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> ) &amp;  <a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">H</font> <a href="zf_lang.html#R1">is_immediate_constituent_of</a> <font color="Maroon">H</font> )
 <b>by </b><i>, , <a class="ref" href="zf_lang.html#D39">ZF_LANG:def 39</a></i>;<br/><a NAME="E12:10_5_2"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">H</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> &amp; <font color="Maroon">E</font>,<font color="Maroon">m</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">H</font> )
 <b>by </b><i>, , <a class="ref" href="zf_lang.html#T81">ZF_LANG:81</a>, <a class="ref" href="zf_model.html#T16">ZF_MODEL:16</a></i>;<br/><a NAME="E13:10_5_2"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">E</font>,<font color="Maroon">m</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">H</font></span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_5_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/><b>assume </b><a NAME="E1:10_5_2_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">m</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E2:10_5_2_1"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">H</font>
 <b>by </b><i/>;<br/><b>assume </b><a NAME="E3:10_5_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E4:10_5_2_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> &amp; <font color="Maroon">m</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> &amp; <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> )
 <b>by </b><i>, , , </i>;<br/><b>hence </b><a NAME="E5:10_5_2_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E15:10_5_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="zf_model.html#T16">ZF_MODEL:16</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:10_5"/>
<font color="Maroon">E</font>,<font color="Maroon">f</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <font color="Maroon">x</font>,<font color="Maroon">H</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:10"/>
<font color="Maroon">E</font>,<font color="Maroon">f</font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <font color="Maroon">x</font>,<font color="Maroon">H</font>
 <b>by </b><i>, , , , <a class="ref" href="zf_lang.html#T25">ZF_LANG:25</a></i>;<br/>


</div>
