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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">H</font> being  <a href="zf_lang.html#NM2">ZF-formula</a>  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">H</font>];<br/>
<a NAME="E1:110_1"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st ( 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="Olive">n</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">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">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="E1:110_1_1"/><i><font color="Green">E41</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">H</font> being  <a href="zf_lang.html#NM2">ZF-formula</a>  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> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">H</font>]
 ;<br/>

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

<b>assume </b><a NAME="E2:110_1_1"/><i><font color="Green">E42</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>
 ;<br/>

<a NAME="E3:110_1_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">H</font> is <a href="zf_lang.html#V7" target="_self">atomic</a> or <font color="Maroon">H</font> is <a href="zf_lang.html#V4" target="_self">negative</a> or <font color="Maroon">H</font> is <a href="zf_lang.html#V5" target="_self">conjunctive</a> or <font color="Maroon">H</font> is <a href="zf_lang.html#V6" target="_self">universal</a> )
 
<b>by </b><i/>;<br/>
<div><i><font color="Green">E44</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">E45</font></i>: 
<font color="Maroon">H</font> is <a href="zf_lang.html#V4" target="_self">negative</a>
 ;<br/><a NAME="E2:110_1_1_1"/><b>then </b>
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K6" target="_self">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K20" target="_self">the_argument_of</a> <font color="Maroon">H</font></span>)</span>
 <b>by </b><i><a class="txt" href="zf_lang.html#E1:110_1"><i><font color="Green">E40</font></i></a></i>;<br/><a NAME="E3:110_1_1_1"/><b>then </b>
 <a href="zf_lang.html#K20" target="_self">the_argument_of</a> <font color="Maroon">H</font> <a href="zf_lang.html#R1" target="_self">is_immediate_constituent_of</a> <font color="Maroon">H</font>
 <b>by </b><i/>;<br/><a NAME="E4:110_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K20" target="_self">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>
 <b>by </b><i><a class="ref" href="zf_lang.html#D15" target="_self">Def15</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" target="_self">the_argument_of</a> <font color="Maroon">H</font>]
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E6:110_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">H</font>]
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E46</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">E47</font></i>: 
<font color="Maroon">H</font> is <a href="zf_lang.html#V5" target="_self">conjunctive</a>
 ;<br/><a NAME="E2:110_1_1_2"/><b>then </b>
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K21" target="_self">the_left_argument_of</a> <font color="Maroon">H</font></span>)</span> <a href="zf_lang.html#K7" target="_self">'&amp;'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K22" target="_self">the_right_argument_of</a> <font color="Maroon">H</font></span>)</span>
 <b>by </b><i><a class="txt" href="zf_lang.html#E4:110_1_1"><i><font color="Green">E44</font></i></a></i>;<br/><a NAME="E3:110_1_1_2"/><b>then </b>
(  <a href="zf_lang.html#K21" target="_self">the_left_argument_of</a> <font color="Maroon">H</font> <a href="zf_lang.html#R1" target="_self">is_immediate_constituent_of</a> <font color="Maroon">H</font> &amp;  <a href="zf_lang.html#K22" target="_self">the_right_argument_of</a> <font color="Maroon">H</font> <a href="zf_lang.html#R1" target="_self">is_immediate_constituent_of</a> <font color="Maroon">H</font> )
 <b>by </b><i/>;<br/><a NAME="E4:110_1_1_2"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K21" target="_self">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" target="_self">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#D15" target="_self">Def15</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" target="_self">the_left_argument_of</a> <font color="Maroon">H</font>] &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="zf_lang.html#K22" target="_self">the_right_argument_of</a> <font color="Maroon">H</font>] )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E6:110_1_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">H</font>]
 <b>by </b><i><a class="ref" href="zf_lang.html#T46" target="_self">Th46</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">E48</font></i>: 
<font color="Maroon">H</font> is <a href="zf_lang.html#V6" target="_self">universal</a>
 ;<br/><a NAME="E2:110_1_1_3"/><b>then </b>
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K8" target="_self">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" target="_self">bound_in</a> <font color="Maroon">H</font></span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K24" target="_self">the_scope_of</a> <font color="Maroon">H</font></span>)</span>
 <b>by </b><i><a class="txt" href="zf_lang.html#E1:110_1_1_3"><i><font color="Green">E48</font></i></a></i>;<br/><a NAME="E3:110_1_1_3"/><b>then </b>
 <a href="zf_lang.html#K24" target="_self">the_scope_of</a> <font color="Maroon">H</font> <a href="zf_lang.html#R1" target="_self">is_immediate_constituent_of</a> <font color="Maroon">H</font>
 <b>by </b><i/>;<br/><a NAME="E4:110_1_1_3"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" target="_self">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>
 <b>by </b><i><a class="ref" href="zf_lang.html#D15" target="_self">Def15</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" target="_self">the_scope_of</a> <font color="Maroon">H</font>]
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E6:110_1_1_3"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">H</font>]
 <b>by </b><i>, </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">H</font>]
 <b>by </b><i>, , , </i>;<br/>


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

<a NAME="E2:110_1"/><i><font color="Green">E49</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font>
 
;<br/>
<a NAME="E3:110_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S4">NAT_1:sch 4</a>();<br/></i>
<b>hence </b><a NAME="E4:110_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">H</font>]
 <b>by </b><i/>;<br/>


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