<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">H</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/> for <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <font color="Olive" title="b2">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Maroon" title="c1">H</font> holds <br/><font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> &amp;  not  for <font color="Olive" title="b1">x</font> being   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a> holds  <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Maroon" title="c1">H</font> )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>] <span class="kw">means </span> ex <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/> for <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <font color="Olive" title="b2">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> $1 holds <br/><font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font>;<br/>
<a NAME="E1:4"/><span class="lab"><font color="Green" title="E4">A1</font></span>: 
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a> holds <br/> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Olive" title="b2">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <font color="Olive" title="b2">y</font>] )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">x</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c3">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">x</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <font color="Maroon" title="c3">y</font>] )</span><br/>



<span class="kw">consider </span><font color="Maroon" title="c4">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E2:4_1"/><span class="lab"><font color="Green" title="E5">A2</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T77" onmouseover="rs('zf_lang1/T77')" onmouseout="rh()">ZF_LANG1:77</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E4:4_1"/><span class="lab"><font color="Green" title="E6">A3</font></span>: 
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <font color="Maroon" title="c5">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T77" onmouseover="rs('zf_lang1/T77')" onmouseout="rh()">ZF_LANG1:77</a></span>;<br/>
<a NAME="E5:4_1"/>
<font color="Maroon" title="c5">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E6:4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A4</font></span>: 
<font color="Maroon" title="c5">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E7:4_1"/>
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E8:4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E8">A5</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E9:4_1"/><span class="lab"><font color="Green" title="E9">A6</font></span>: 
 <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T138" onmouseover="rs('zf_lang1/T138')" onmouseout="rh()">ZF_LANG1:138</a></span>;<br/>
<span class="kw">thus </span><a NAME="E10:4_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">x</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c3">y</font>]
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">x</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <font color="Maroon" title="c3">y</font>]</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="4_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <font color="Olive" title="b1">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c3">y</font></span>)</span> holds <br/><font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <font color="Maroon" title="c6">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c3">y</font></span>)</span> implies <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )</span><br/>

<span class="kw">assume </span><a NAME="E1:4_1_1"/>
 <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <font color="Maroon" title="c6">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c3">y</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span><br/>

<a NAME="E2:4_1_1"/><span class="kw">then </span>
(  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <font color="Maroon" title="c4">i</font> or  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <font color="Maroon" title="c5">j</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:4_1"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="txt" href="#E4:4_1"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="txt" href="#E9:4_1"><span class="lab"><font color="Green" title="E9">A6</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:4_1_1"/>
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:4_1"><span class="lab"><font color="Green" title="E8">A5</font></span></a>, <a class="txt" href="#E6:4_1"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="ref" href="zf_lang1.html#T76" onmouseover="rs('zf_lang1/T76')" onmouseout="rh()">ZF_LANG1:76</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">take </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <font color="Olive" title="b1">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <font color="Maroon" title="c3">y</font></span>)</span> holds <br/><font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <font color="Maroon" title="c6">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <font color="Maroon" title="c3">y</font></span>)</span> implies <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )</span><br/>

<a NAME="E11:4_1"/><span class="lab"><font color="Green" title="E10">A7</font></span>: 
 <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T139" onmouseover="rs('zf_lang1/T139')" onmouseout="rh()">ZF_LANG1:139</a></span>;<br/>
<span class="kw">assume </span><a NAME="E12:4_1"/>
 <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <font color="Maroon" title="c6">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <font color="Maroon" title="c3">y</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span><br/>

<a NAME="E13:4_1"/><span class="kw">then </span>
(  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <font color="Maroon" title="c4">i</font> or  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <font color="Maroon" title="c5">j</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:4_1"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="txt" href="#E4:4_1"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="txt" href="#E11:4_1"><span class="lab"><font color="Green" title="E10">A7</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:4_1"/>
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:4_1"><span class="lab"><font color="Green" title="E8">A5</font></span></a>, <a class="txt" href="#E6:4_1"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="ref" href="zf_lang1.html#T76" onmouseover="rs('zf_lang1/T76')" onmouseout="rh()">ZF_LANG1:76</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>
<a NAME="E2:4"/><span class="lab"><font color="Green" title="E5">A8</font></span>: 
 for <font color="Olive" title="b1">H1</font>, <font color="Olive" title="b2">H2</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">H1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">H2</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Olive" title="b2">H2</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="4_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">H1</font>, <font color="Maroon" title="c3">H2</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">H1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">H2</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c3">H2</font>] )</span><br/>



<span class="kw">given </span><font color="Maroon" title="c4">i1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><a NAME="E2:4_2"/><span class="lab"><font color="Green" title="E6">A9</font></span>: 
 for <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <font color="Olive" title="b1">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Maroon" title="c2">H1</font> holds <br/><font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">H2</font>] or <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c3">H2</font>] )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c5">i2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><a NAME="E4:4_2"/><span class="lab"><font color="Green" title="E7">A10</font></span>: 
 for <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <font color="Olive" title="b1">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Maroon" title="c3">H2</font> holds <br/><font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">i2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c3">H2</font>]</span><br/>

<a NAME="E5:4_2"/>
( <font color="Maroon" title="c4">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i2</font> or <font color="Maroon" title="c4">i1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">i2</font> )
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E7:4_2"/><span class="lab"><font color="Green" title="E8">A11</font></span>: 
( ( <font color="Maroon" title="c4">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i2</font> &amp; <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i2</font> ) or ( <font color="Maroon" title="c4">i1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">i2</font> &amp; <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i1</font> ) )
 ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c6">i</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <font color="Olive" title="b1">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c3">H2</font></span>)</span> holds <br/><font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c7">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <font color="Maroon" title="c7">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c3">H2</font></span>)</span> implies <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font> )</span><br/>

<span class="kw">assume </span><a NAME="E8:4_2"/>
 <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <font color="Maroon" title="c7">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c3">H2</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font></span><br/>

<a NAME="E9:4_2"/><span class="kw">then </span>
 <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <font color="Maroon" title="c7">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Maroon" title="c2">H1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Maroon" title="c3">H2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T141" onmouseover="rs('zf_lang1/T141')" onmouseout="rh()">ZF_LANG1:141</a></span>;<br/>
<a NAME="E10:4_2"/><span class="kw">then </span>
(  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <font color="Maroon" title="c7">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Maroon" title="c2">H1</font> or  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <font color="Maroon" title="c7">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Maroon" title="c3">H2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E11:4_2"/><span class="kw">then </span>
( <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i1</font> or <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">i2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:4_2"><span class="lab"><font color="Green" title="E6">A9</font></span></a>, <a class="txt" href="#E4:4_2"><span class="lab"><font color="Green" title="E7">A10</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:4_2"/>
<font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:4_2"><span class="lab"><font color="Green" title="E8">A11</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>
<a NAME="E3:4"/><span class="lab"><font color="Green" title="E6">A12</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><br/>  for <font color="Olive" title="b2">x</font> being   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">H</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">H</font>)]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="4_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">H</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">H</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c2">H</font>)]</span><br/><span class="kw">let </span><font color="Maroon" title="c3">x</font> be   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">H</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">H</font>)] )</span><br/>



<span class="kw">given </span><font color="Maroon" title="c4">i1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><a NAME="E2:4_3"/><span class="lab"><font color="Green" title="E7">A13</font></span>: 
 for <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <font color="Olive" title="b1">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Maroon" title="c2">H</font> holds <br/><font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">H</font>)]</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">i2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E4:4_3"/><span class="lab"><font color="Green" title="E8">A14</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <font color="Maroon" title="c5">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T77" onmouseover="rs('zf_lang1/T77')" onmouseout="rh()">ZF_LANG1:77</a></span>;<br/>
<a NAME="E5:4_3"/>
( <font color="Maroon" title="c4">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c4">i1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E7:4_3"/><span class="lab"><font color="Green" title="E9">A15</font></span>: 
( ( <font color="Maroon" title="c4">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) or ( <font color="Maroon" title="c4">i1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i1</font> ) )
 ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c6">i</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <font color="Olive" title="b1">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">H</font>)</span>)</span> holds <br/><font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c7">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <font color="Maroon" title="c7">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">H</font>)</span>)</span> implies <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font> )</span><br/>

<span class="kw">assume </span><a NAME="E8:4_3"/>
 <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <font color="Maroon" title="c7">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">H</font>)</span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font></span><br/>

<a NAME="E9:4_3"/><span class="kw">then </span>
 <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <font color="Maroon" title="c7">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Maroon" title="c2">H</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T142" onmouseover="rs('zf_lang1/T142')" onmouseout="rh()">ZF_LANG1:142</a></span>;<br/>
<a NAME="E10:4_3"/><span class="kw">then </span>
(  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <font color="Maroon" title="c7">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Maroon" title="c2">H</font> or (  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <font color="Maroon" title="c7">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c5">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i2</font> &amp;  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E11:4_3"/><span class="kw">then </span>
( <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i1</font> or (  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <font color="Maroon" title="c5">i2</font> &amp; <font color="Maroon" title="c5">i2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:4_3"><span class="lab"><font color="Green" title="E7">A13</font></span></a>, <a class="txt" href="#E4:4_3"><span class="lab"><font color="Green" title="E8">A14</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/>
<a NAME="E12:4_3"/><span class="kw">then </span>
( <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i1</font> or <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T76" onmouseover="rs('zf_lang1/T76')" onmouseout="rh()">ZF_LANG1:76</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:4_3"/>
<font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:4_3"><span class="lab"><font color="Green" title="E9">A15</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>
<a NAME="E4:4"/><span class="lab"><font color="Green" title="E7">A16</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 <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">H</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'not'</a> <font color="Olive" title="b1">H</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="4_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">H</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">H</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'not'</a> <font color="Maroon" title="c2">H</font>] )</span><br/>

<a NAME="E1:4_4"/>
 <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K5" title="ZF_LANG:func.5">'not'</a> <font color="Maroon" title="c2">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Maroon" title="c2">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T140" onmouseover="rs('zf_lang1/T140')" onmouseout="rh()">ZF_LANG1:140</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:4_4"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">H</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'not'</a> <font color="Maroon" title="c2">H</font>] )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>
<a NAME="E5:4"/>
 for <font color="Olive" title="b1">H</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">H</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="zf_lang1.html#S1" onmouseover="rs('zf_lang1/S1')" onmouseout="rh()">ZF_LANG1:sch 1</a>(<a class="txt" href="#E1:4"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="txt" href="#E4:4"><span class="lab"><font color="Green" title="E7">A16</font></span></a>, <a class="txt" href="#E2:4"><span class="lab"><font color="Green" title="E5">A8</font></span></a>, <a class="txt" href="#E3:4"><span class="lab"><font color="Green" title="E6">A12</font></span></a>);<br/></span>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c2">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E7:4"/><span class="lab"><font color="Green" title="E8">A17</font></span>: 
 for <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <font color="Olive" title="b1">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Maroon" title="c1">H</font> holds <br/><font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">i</font>
 ;<br/>
<span class="kw">thus </span><a NAME="E8:4"/>
 ex <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/> for <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <font color="Olive" title="b2">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Maroon" title="c1">H</font> holds <br/><font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:4"><span class="lab"><font color="Green" title="E8">A17</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not  for <font color="Olive" title="b1">x</font> being   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a> holds  <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Maroon" title="c1">H</font></span><br/>

<span class="kw">take </span>
 <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <font color="Maroon" title="c2">i</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <font color="Maroon" title="c2">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Maroon" title="c1">H</font></span><br/>

<span class="kw">thus </span><a NAME="E9:4"/>
 not  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <font color="Maroon" title="c2">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Maroon" title="c1">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:4"><span class="lab"><font color="Green" title="E8">A17</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
