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<span class="kw">let </span><font color="Maroon" title="c1">f</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">p</font> being  <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>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Maroon" title="c1">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  holds <br/> <a href="jordan3.html#K4" title="JORDAN3:func.4">B_Cut</a> <font color="Maroon" title="c1">f</font>,<font color="Olive" title="b1">p</font>,<font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b1">p</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Maroon" title="c1">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  implies  <a href="jordan3.html#K4" title="JORDAN3:func.4">B_Cut</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:26"/><span class="lab"><font color="Green" title="E10">A1</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font>
 <span class="kw">and </span><br/><a NAME="E2:26"/><span class="lab"><font color="Green" title="E11">A2</font></span>: 
<font color="Maroon" title="c1">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="jordan3.html#K4" title="JORDAN3:func.4">B_Cut</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><br/>

<a NAME="E3:26"/><span class="lab"><font color="Green" title="E12">A3</font></span>: 
 <a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
;<br/>
<a NAME="E4:26"/><span class="lab"><font color="Green" title="E13">A4</font></span>: 
 <a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">f</font> <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="c1">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="jordan3.html#T41" title="JORDAN3:th.41">JORDAN3:41</a></span>;<br/>
<a NAME="E5:26"/><span class="lab"><font color="Green" title="E14">A5</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="jordan3.html#T41" title="JORDAN3:th.41">JORDAN3:41</a></span>;<br/>
<a NAME="E6:26"/><span class="lab"><font color="Green" title="E15">A6</font></span>: 
<span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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> <font color="Maroon" title="c1">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E7:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A7</font></span>: 
 <a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="ref" href="seq_4.html#T151" title="SEQ_4:th.151">SEQ_4:151</a></span>;<br/>
<a NAME="E8:26"/><span class="lab"><font color="Green" title="E17">A8</font></span>: 
<span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="ref" href="seq_4.html#T151" title="SEQ_4:th.151">SEQ_4:151</a></span>;<br/>
<a NAME="E9:26"/>
<font color="Maroon" title="c2">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="c1">f</font>,<span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="jordan3.html#T42" title="JORDAN3:th.42">JORDAN3:42</a></span>;<br/>
<a NAME="E10:26"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></span>;<br/>
<a NAME="E11:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A9</font></span>: 
 <a href="jordan3.html#R2" title="JORDAN3:pred.2">LE</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">f</font></span>)</span>,<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">Index</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="txt" href="#E7:26"><span class="lab"><font color="Green" title="E16">A7</font></span></a>, <a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E17">A8</font></span></a>, <a class="ref" href="jordan5b.html#T9" target="_self" title="JORDAN5B:th.9">Th9</a>, <a class="ref" href="partfun2.html#T17" title="PARTFUN2:th.17">PARTFUN2:17</a></span>;<br/>
<a NAME="E12:26"/>
<span class="p1">(<span class="default"><a href="jordan3.html#K2" title="JORDAN3:func.2">L_Cut</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="jordan3.html#T58" title="JORDAN3:th.58">JORDAN3:58</a></span>;<br/>
<a NAME="E13:26"/><span class="kw">then </span>
 <a href="jordan3.html#K3" title="JORDAN3:func.3">R_Cut</a> <span class="p1">(<span class="default"><a href="jordan3.html#K2" title="JORDAN3:func.2">L_Cut</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan3.html#D5" title="JORDAN3:def.5">JORDAN3:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:26"/>
 <a href="jordan3.html#K4" title="JORDAN3:func.4">B_Cut</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:26"><span class="lab"><font color="Green" title="E18">A9</font></span></a>, <a class="ref" href="jordan3.html#D8" title="JORDAN3:def.8">JORDAN3:def 8</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/>


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