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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>for <font color="Olive" title="b1">H</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">H</font>];<br/>
<a NAME="E1:110_1"/><i><font color="Green" title="E87">A5</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st ( for <font color="Olive" title="b2">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">n</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">k</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:110_1_1"/><i><font color="Green" title="E88">A6</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> holds <br/>for <font color="Olive" title="b2">H</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">H</font>]
 ;<br/>

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

<b>assume </b><a NAME="E2:110_1_1"/><i><font color="Green" title="E89">A7</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 ;<br/>

<a NAME="E3:110_1_1"/><i><font color="Green" title="E90">A8</font></i>: 
( <font color="Maroon" title="c2">H</font> is <a href="zf_lang.html#V7" title="ZF_LANG:attr.7">atomic</a> or <font color="Maroon" title="c2">H</font> is <a href="zf_lang.html#V4" title="ZF_LANG:attr.4">negative</a> or <font color="Maroon" title="c2">H</font> is <a href="zf_lang.html#V5" title="ZF_LANG:attr.5">conjunctive</a> or <font color="Maroon" title="c2">H</font> is <a href="zf_lang.html#V6" title="ZF_LANG:attr.6">universal</a> )
 
<b>by </b><i><a class="ref" href="zf_lang.html#T26" target="_self" title="ZF_LANG:th.26">Th26</a></i>;<br/>
<div><i><font color="Green" title="E91">A9</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:110_1_1_1"/><i><font color="Green" title="E92">A10</font></i>: 
<font color="Maroon" title="c2">H</font> is <a href="zf_lang.html#V4" title="ZF_LANG:attr.4">negative</a>
 ;<br/><a NAME="E2:110_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Maroon" title="c2">H</font></span>)</span>
 <b>by </b><i><a class="ref" href="zf_lang.html#D30" target="_self" title="ZF_LANG:def.30">Def30</a></i>;<br/><a NAME="E3:110_1_1_1"/><b>then </b>
 <a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Maroon" title="c2">H</font> <a href="zf_lang.html#R1" title="ZF_LANG:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c2">H</font>
 <b>by </b><i><a class="ref" href="zf_lang.html#T71" target="_self" title="ZF_LANG:th.71">Th71</a></i>;<br/><a NAME="E4:110_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Maroon" title="c2">H</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">H</font>
 <b>by </b><i><a class="ref" href="zf_lang.html#T81" target="_self" title="ZF_LANG:th.81">Th81</a></i>;<br/><a NAME="E5:110_1_1_1"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[ <a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Maroon" title="c2">H</font>]
 <b>by </b><i><a class="txt" href="zf_lang.html#E1:110_1_1"><i><font color="Green" title="E88">A6</font></i></a>, <a class="txt" href="zf_lang.html#E2:110_1_1"><i><font color="Green" title="E89">A7</font></i></a></i>;<br/><b>hence </b><a NAME="E6:110_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">H</font>]
 <b>by </b><i><a class="txt" href="zf_lang.html#E2:110"><i><font color="Green" title="E84">A2</font></i></a>, <a class="txt" href="zf_lang.html#E1:110_1_1_1"><i><font color="Green" title="E92">A10</font></i></a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E92">A11</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:110_1_1_2"/><i><font color="Green" title="E93">A12</font></i>: 
<font color="Maroon" title="c2">H</font> is <a href="zf_lang.html#V5" title="ZF_LANG:attr.5">conjunctive</a>
 ;<br/><a NAME="E2:110_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <font color="Maroon" title="c2">H</font></span>)</span> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <font color="Maroon" title="c2">H</font></span>)</span>
 <b>by </b><i><a class="ref" href="zf_lang.html#T58" target="_self" title="ZF_LANG:th.58">Th58</a></i>;<br/><a NAME="E3:110_1_1_2"/><b>then </b>
(  <a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <font color="Maroon" title="c2">H</font> <a href="zf_lang.html#R1" title="ZF_LANG:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c2">H</font> &amp;  <a href="zf_lang.html#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <font color="Maroon" title="c2">H</font> <a href="zf_lang.html#R1" title="ZF_LANG:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c2">H</font> )
 <b>by </b><i><a class="ref" href="zf_lang.html#T72" target="_self" title="ZF_LANG:th.72">Th72</a></i>;<br/><a NAME="E4:110_1_1_2"/><b>then </b>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <font color="Maroon" title="c2">H</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">H</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <font color="Maroon" title="c2">H</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">H</font> )
 <b>by </b><i><a class="ref" href="zf_lang.html#T81" target="_self" title="ZF_LANG:th.81">Th81</a></i>;<br/><a NAME="E5:110_1_1_2"/><b>then </b>
( <font color="Maroon">P<sub>1</sub></font>[ <a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <font color="Maroon" title="c2">H</font>] &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="zf_lang.html#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <font color="Maroon" title="c2">H</font>] )
 <b>by </b><i><a class="txt" href="zf_lang.html#E1:110_1_1"><i><font color="Green" title="E88">A6</font></i></a>, <a class="txt" href="zf_lang.html#E2:110_1_1"><i><font color="Green" title="E89">A7</font></i></a></i>;<br/><b>hence </b><a NAME="E6:110_1_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">H</font>]
 <b>by </b><i><a class="txt" href="zf_lang.html#E3:110"><i><font color="Green" title="E85">A3</font></i></a>, <a class="txt" href="zf_lang.html#E1:110_1_1_2"><i><font color="Green" title="E93">A12</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:110_1_1_3"/><i><font color="Green" title="E93">A13</font></i>: 
<font color="Maroon" title="c2">H</font> is <a href="zf_lang.html#V6" title="ZF_LANG:attr.6">universal</a>
 ;<br/><a NAME="E2:110_1_1_3"/><b>then </b>
<font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c2">H</font></span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c2">H</font></span>)</span>
 <b>by </b><i><a class="ref" href="zf_lang.html#T62" target="_self" title="ZF_LANG:th.62">Th62</a></i>;<br/><a NAME="E3:110_1_1_3"/><b>then </b>
 <a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c2">H</font> <a href="zf_lang.html#R1" title="ZF_LANG:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c2">H</font>
 <b>by </b><i><a class="ref" href="zf_lang.html#T73" target="_self" title="ZF_LANG:th.73">Th73</a></i>;<br/><a NAME="E4:110_1_1_3"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c2">H</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">H</font>
 <b>by </b><i><a class="ref" href="zf_lang.html#T81" target="_self" title="ZF_LANG:th.81">Th81</a></i>;<br/><a NAME="E5:110_1_1_3"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[ <a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c2">H</font>]
 <b>by </b><i><a class="txt" href="zf_lang.html#E1:110_1_1"><i><font color="Green" title="E88">A6</font></i></a>, <a class="txt" href="zf_lang.html#E2:110_1_1"><i><font color="Green" title="E89">A7</font></i></a></i>;<br/><b>hence </b><a NAME="E6:110_1_1_3"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">H</font>]
 <b>by </b><i><a class="txt" href="zf_lang.html#E4:110"><i><font color="Green" title="E86">A4</font></i></a>, <a class="txt" href="zf_lang.html#E1:110_1_1_3"><i><font color="Green" title="E93">A13</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:110_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">H</font>]
 <b>by </b><i><a class="txt" href="zf_lang.html#E1:110"><i><font color="Green" title="E83">A1</font></i></a>, <a class="txt" href="zf_lang.html#E3:110_1_1"><i><font color="Green" title="E90">A8</font></i></a>, <a class="txt" href="zf_lang.html#E4:110_1_1"><i><font color="Green" title="E91">A9</font></i></a>, <a class="txt" href="zf_lang.html#E5:110_1_1"><i><font color="Green" title="E92">A11</font></i></a></i>;<br/>


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

<a NAME="E2:110_1"/><i><font color="Green" title="E88">A14</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">H</font>
 
;<br/>
<a NAME="E3:110_1"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S4" title="NAT_1:sch.4">NAT_1:sch 4</a>(<a class="txt" href="zf_lang.html#E1:110_1"><i><font color="Green" title="E87">A5</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:110_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">H</font>]
 <b>by </b><i><a class="txt" href="zf_lang.html#E2:110_1"><i><font color="Green" title="E88">A14</font></i></a></i>;<br/>


</div>
