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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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#K4" title="TOPREAL1:func.4">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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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#K4" title="TOPREAL1:func.4">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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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#K4" title="TOPREAL1:func.4">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/>





<b>assume </b><b>that </b><br/><a NAME="E1:74"/><i><font color="Green" title="E61">A1</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c2">i</font>
 <b>and </b><br/><a NAME="E2:74"/><i><font color="Green" title="E62">A2</font></i>: 
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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>set </b><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/>
<b>thus </b><a NAME="E3:74"/>
 <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>
 <b>by </b><i><a class="txt" href="#E2:74"><i><font color="Green" title="E62">A2</font></i></a>, <a class="ref" href="finseq_5.html#T79" title="FINSEQ_5:th.79">FINSEQ_5:79</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="topreal1.html#D10" title="TOPREAL1:def.10">TOPREAL1:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="E4:74"/><i><font color="Green" title="E63">A3</font></i>: 
 <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
 
<b>by </b><i><a class="ref" href="topreal1.html#D10" title="TOPREAL1:def.10">TOPREAL1:def 10</a></i>;<br/>
<a NAME="E5:74"/>
 <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
 
<b>by </b><i><a class="ref" href="finseq_5.html#T72" title="FINSEQ_5:th.72">FINSEQ_5:72</a></i>;<br/>
<a NAME="E6:74"/><b>then </b>
 <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>
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<b>hence </b><a NAME="E7:74"/>
 <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
 <b>by </b><i><a class="txt" href="#E4:74"><i><font color="Green" title="E63">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>thus </b><a NAME="E8:74"/>
 <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>
 <b>by </b><i><a class="txt" href="#E1:74"><i><font color="Green" title="E61">A1</font></i></a>, <a class="ref" href="sppol_2.html#T33" target="_self" title="SPPOL_2:th.33">Th33</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>thus </b><a NAME="E9:74"/>
 <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>
 <b>by </b><i><a class="txt" href="#E1:74"><i><font color="Green" title="E61">A1</font></i></a>, <a class="txt" href="#E2:74"><i><font color="Green" title="E62">A2</font></i></a>, <a class="ref" href="sppol_2.html#T38" target="_self" title="SPPOL_2:th.38">Th38</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>thus </b><a NAME="E10:74"/>
 <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>
 <b>by </b><i><a class="txt" href="#E1:74"><i><font color="Green" title="E61">A1</font></i></a>, <a class="ref" href="sppol_2.html#T44" target="_self" title="SPPOL_2:th.44">Th44</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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