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

<span class="kw">let </span><font color="Maroon" title="c1">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <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#K5" title="NUMBERS:func.5">NAT</a> <br/>  for <font color="Olive" title="b2">f</font> being   <a href="sppol_2.html#NM1" title="SPPOL_2:NM.1">S-Sequence_in_R2</a>  st <font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Olive" title="b2">f</font>,<font color="Olive" title="b1">i</font>) &amp;  not <font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b2">f</font> holds <br/> <a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Olive" title="b2">f</font>,<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c1">p</font>) is  <a href="topreal1.html#V4" title="TOPREAL1:attr.4">being_S-Seq</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">i</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="sppol_2.html#NM1" title="SPPOL_2:NM.1">S-Sequence_in_R2</a>  st <font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Olive" title="b1">f</font>,<font color="Maroon" title="c2">i</font>) &amp;  not <font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f</font> holds <br/> <a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Olive" title="b1">f</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">p</font>) is  <a href="topreal1.html#V4" title="TOPREAL1:attr.4">being_S-Seq</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="sppol_2.html#NM1" title="SPPOL_2:NM.1">S-Sequence_in_R2</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">i</font>) &amp;  not <font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">f</font> implies  <a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">p</font>) is  <a href="topreal1.html#V4" title="TOPREAL1:attr.4">being_S-Seq</a>  )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:73"/><span class="lab"><font color="Green" title="E58">A1</font></span>: 
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">i</font>)
 <span class="kw">and </span><br/><a NAME="E2:73"/><span class="lab"><font color="Green" title="E59">A2</font></span>: 
 not <font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">p</font>) is  <a href="topreal1.html#V4" title="TOPREAL1:attr.4">being_S-Seq</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">g</font> =  <a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">p</font>);<br/>
<span class="kw">thus </span><a NAME="E3:73"/>
 <a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">p</font>) is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="ref" href="finseq_5.html#T76" title="FINSEQ_5:th.76">FINSEQ_5:76</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="topreal1.html#D8" title="TOPREAL1:def.8">TOPREAL1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">p</font>)</span>)</span> &amp;  <a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">p</font>) is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp;  <a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">p</font>) is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp;  <a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">p</font>) is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  )</span><br/>

<a NAME="E4:73"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">p</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T69" title="FINSEQ_5:th.69">FINSEQ_5:69</a></span>;<br/>
<a NAME="E5:73"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A3</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">p</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E6:73"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal1.html#D8" title="TOPREAL1:def.8">TOPREAL1:def 8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:73"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">p</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73"><span class="lab"><font color="Green" title="E60">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">p</font>) is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp;  <a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">p</font>) is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp;  <a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">p</font>) is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E8:73"/>
 <a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">p</font>) is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="ref" href="sppol_2.html#T32" target="_self" title="SPPOL_2:th.32">Th33</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">p</font>) is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp;  <a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">p</font>) is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E9:73"/>
 <a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">p</font>) is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="txt" href="#E2:73"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="ref" href="sppol_2.html#T37" target="_self" title="SPPOL_2:th.37">Th38</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">p</font>) is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> </span><br/>

<span class="kw">thus </span><a NAME="E10:73"/>
 <a href="finseq_5.html#K5" title="FINSEQ_5:func.5">Ins</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">p</font>) is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="ref" href="sppol_2.html#T41" target="_self" title="SPPOL_2:th.41">Th44</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
