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<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="zf_lang.html#NM2">ZF-formula</a>] <b>means </b> <a href="zf_lang1.html#K3" target="_self">variables_in</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> ;<br/>
<a NAME="E1:147_1_1"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="zf_lang.html#NM1">Variable</a> holds <br/> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font> <a href="zf_lang.html#K4">'='</a> <font color="Olive">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font> <a href="zf_lang.html#K5">'in'</a> <font color="Olive">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="147_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/>



<a NAME="E1:147_1_1_1"/>
(  <a href="zf_lang1.html#K3" target="_self">variables_in</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span> &amp;  <a href="zf_lang1.html#K3" target="_self">variables_in</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span> )
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T8" target="_self">Th8</a>, <a class="ref" href="zf_lang1.html#T9" target="_self">Th9</a></i>;<br/>
<b>hence </b><a NAME="E2:147_1_1_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">y</font>] )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:147_1_1"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">H</font> being  <a href="zf_lang.html#NM2">ZF-formula</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">H</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K6">'not'</a> <font color="Olive">H</font>]
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T16" target="_self">Th16</a></i>;<br/>
<a NAME="E3:147_1_1"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">H1</font>, <font color="Olive">H2</font> being  <a href="zf_lang.html#NM2">ZF-formula</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">H1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">H2</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Olive">H2</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="147_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">H1</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/><b>let </b><font color="Maroon">H2</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/>



<a NAME="E1:147_1_1_2"/>
 <a href="zf_lang1.html#K3" target="_self">variables_in</a> <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_lang1.html#K3" target="_self">variables_in</a> <font color="Maroon">H1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="zf_lang1.html#K3" target="_self">variables_in</a> <font color="Maroon">H2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="zf_lang1.html#E1:147_1_1"><i><font color="Green">E46</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:147_1_1_2"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">H1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">H2</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font>] )
 <b>by </b><i><a class="ref" href="xboole_1.html#T15">XBOOLE_1:15</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:147_1_1"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">H</font> being  <a href="zf_lang.html#NM2">ZF-formula</a><br/> for <font color="Olive">x</font> being  <a href="zf_lang.html#NM1">Variable</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">H</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K8">All</a> <font color="Olive">x</font>,<font color="Olive">H</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="147_1_1_3"><b>proof </b></a><div class="add">

<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">x</font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/>



<a NAME="E1:147_1_1_3"/>
(  <a href="zf_lang1.html#K3" target="_self">variables_in</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">x</font>,<font color="Maroon">H</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_lang1.html#K3" target="_self">variables_in</a> <font color="Maroon">H</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T20" target="_self">Th20</a></i>;<br/>
<b>hence </b><a NAME="E2:147_1_1_3"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">H</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K8">All</a> <font color="Maroon">x</font>,<font color="Maroon">H</font>] )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:147_1_1"/>
for <font color="Olive">H</font> being  <a href="zf_lang.html#NM2">ZF-formula</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">H</font>]
 
<b>from </b><i><a class="ref" href="zf_lang1.html#S1" target="_self">ZF_LANG1:sch 1</a>(<a class="txt" href="zf_lang1.html#E4:147_1_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="zf_lang1.html#T60" target="_self">Th60</a>, <a class="ref" href="zf_lang1.html#T63" target="_self">Th63</a>, <a class="ref" href="zf_lang1.html#T64" target="_self">Th64</a>);<br/></i>
<b>then reconsider </b><font color="Maroon">D</font> =  <a href="zf_lang1.html#K3" target="_self">variables_in</a> <font color="Maroon">H</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E7:147_1_1"/>
<font color="Maroon">D</font> <a href="tarski.html#R1">c=</a>  <a href="zf_lang.html#K1">VAR</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="147_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:147_1_1_4"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/>

<a NAME="E2:147_1_1_4"/><b>then </b><i><font color="Green">E65</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 2 &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 3 &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 4 )
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T6" target="_self">Th6</a>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">i</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E4:147_1_1_4"/>
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> &amp; 0 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T65" target="_self">Th65</a></i>;<br/>
<a NAME="E5:147_1_1_4"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E6:147_1_1_4"/><b>then </b>
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> &amp; 1 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 2 )
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T65" target="_self">Th65</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E7:147_1_1_4"/><b>then </b>
2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E8:147_1_1_4"/><b>then </b>
( 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> &amp; 2 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 3 )
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T65" target="_self">Th65</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E9:147_1_1_4"/><b>then </b>
3 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E10:147_1_1_4"/><b>then </b>
( 3 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> &amp; 3 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 4 )
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T65" target="_self">Th65</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E11:147_1_1_4"/><b>then </b>
4 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E12:147_1_1_4"/><b>then </b>
( 4 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> &amp; 4 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 5 )
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T65" target="_self">Th65</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E13:147_1_1_4"/><b>then </b>
5 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>hence </b><a NAME="E14:147_1_1_4"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="zf_lang.html#K1">VAR</a> 
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:147_1_1"/>
 <a href="zf_lang1.html#K3" target="_self">variables_in</a> <font color="Maroon">H</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="zf_lang.html#K1">VAR</a> 
 ;<br/>


</div>
