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

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </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  <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="E1:131_1"/><span class="lab"><font color="Green" title="E87">A5</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st ( for <font color="Olive" title="b2">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">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="131_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">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/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">n</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:131_1_1"/><span class="lab"><font color="Green" title="E88">A6</font></span>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">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>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">n</font>]</span><br/>

<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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <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> implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">H</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E2:131_1_1"/><span class="lab"><font color="Green" title="E89">A7</font></span>: 
 <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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">H</font>]</span><br/>

<div><span class="lab"><font color="Green" title="E90">A8</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="131_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:131_1_1_1"/><span class="lab"><font color="Green" title="E91">A9</font></span>: 
<font color="Maroon" title="c2">H</font> is  <a href="zf_lang.html#V5" title="ZF_LANG:attr.5">conjunctive</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">H</font>]</span><br/><a NAME="E2:131_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A10</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T58" target="_self" title="ZF_LANG:th.58">Th58</a></span>;<br/><a NAME="E3:131_1_1_1"/><span class="kw">then </span>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T72" target="_self" title="ZF_LANG:th.72">Th72</a></span>;<br/><a NAME="E4:131_1_1_1"/><span class="kw">then </span>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T81" target="_self" title="ZF_LANG:th.81">Th81</a></span>;<br/><a NAME="E5:131_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A11</font></span>: 
<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>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:131_1_1"><span class="lab"><font color="Green" title="E88">A6</font></span></a>, <a class="txt" href="#E2:131_1_1"><span class="lab"><font color="Green" title="E89">A7</font></span></a></span>;<br/><a NAME="E6:131_1_1_1"/>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:131_1_1_1"><span class="lab"><font color="Green" title="E92">A10</font></span></a>, <a class="ref" href="zf_lang.html#T72" target="_self" title="ZF_LANG:th.72">Th72</a></span>;<br/><a NAME="E7:131_1_1_1"/><span class="kw">then </span>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T81" target="_self" title="ZF_LANG:th.81">Th81</a></span>;<br/><a NAME="E8:131_1_1_1"/><span class="kw">then </span>
<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>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:131_1_1"><span class="lab"><font color="Green" title="E88">A6</font></span></a>, <a class="txt" href="#E2:131_1_1"><span class="lab"><font color="Green" title="E89">A7</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E9:131_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">H</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:131"><span class="lab"><font color="Green" title="E85">A3</font></span></a>, <a class="txt" href="#E1:131_1_1_1"><span class="lab"><font color="Green" title="E91">A9</font></span></a>, <a class="txt" href="#E5:131_1_1_1"><span class="lab"><font color="Green" title="E93">A11</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E91">A12</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="131_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:131_1_1_2"/><span class="lab"><font color="Green" title="E92">A13</font></span>: 
<font color="Maroon" title="c2">H</font> is  <a href="zf_lang.html#V6" title="ZF_LANG:attr.6">universal</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">H</font>]</span><br/><a NAME="E2:131_1_1_2"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T62" target="_self" title="ZF_LANG:th.62">Th62</a></span>;<br/><a NAME="E3:131_1_1_2"/><span class="kw">then </span>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T73" target="_self" title="ZF_LANG:th.73">Th73</a></span>;<br/><a NAME="E4:131_1_1_2"/><span class="kw">then </span>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T81" target="_self" title="ZF_LANG:th.81">Th81</a></span>;<br/><span class="kw">hence </span><a NAME="E5:131_1_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">H</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:131"><span class="lab"><font color="Green" title="E86">A4</font></span></a>, <a class="txt" href="#E1:131_1_1"><span class="lab"><font color="Green" title="E88">A6</font></span></a>, <a class="txt" href="#E2:131_1_1"><span class="lab"><font color="Green" title="E89">A7</font></span></a>, <a class="txt" href="#E1:131_1_1_2"><span class="lab"><font color="Green" title="E92">A13</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="131_1_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:131_1_1_3"/><span class="lab"><font color="Green" title="E92">A14</font></span>: 
<font color="Maroon" title="c2">H</font> is  <a href="zf_lang.html#V4" title="ZF_LANG:attr.4">negative</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">H</font>]</span><br/><a NAME="E2:131_1_1_3"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#D30" target="_self" title="ZF_LANG:def.30">Def30</a></span>;<br/><a NAME="E3:131_1_1_3"/><span class="kw">then </span>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T71" target="_self" title="ZF_LANG:th.71">Th71</a></span>;<br/><a NAME="E4:131_1_1_3"/><span class="kw">then </span>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T81" target="_self" title="ZF_LANG:th.81">Th81</a></span>;<br/><span class="kw">hence </span><a NAME="E5:131_1_1_3"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">H</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:131"><span class="lab"><font color="Green" title="E84">A2</font></span></a>, <a class="txt" href="#E1:131_1_1"><span class="lab"><font color="Green" title="E88">A6</font></span></a>, <a class="txt" href="#E2:131_1_1"><span class="lab"><font color="Green" title="E89">A7</font></span></a>, <a class="txt" href="#E1:131_1_1_3"><span class="lab"><font color="Green" title="E92">A14</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:131_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">H</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:131"><span class="lab"><font color="Green" title="E83">A1</font></span></a>, <a class="txt" href="#E3:131_1_1"><span class="lab"><font color="Green" title="E90">A8</font></span></a>, <a class="txt" href="#E4:131_1_1"><span class="lab"><font color="Green" title="E91">A12</font></span></a>, <a class="ref" href="zf_lang.html#T26" target="_self" title="ZF_LANG:th.26">Th26</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:131_1"/><span class="lab"><font color="Green" title="E88">A15</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S4" title="NAT_1:sch.4">NAT_1:sch 4</a>(<a class="txt" href="#E1:131_1"><span class="lab"><font color="Green" title="E87">A5</font></span></a>);<br/></span>
<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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">H</font>]</span><br/>

<a NAME="E3:131_1"/>
 <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/>
<span class="kw">hence </span><a NAME="E4:131_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">H</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:131_1"><span class="lab"><font color="Green" title="E88">A15</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
