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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="struct_0.html#NM8">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:32"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="topreal1.html#V1">special</a> &amp; 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 ;<br/>

<a NAME="E2:32"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="struct_0.html#NM8">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E4:32"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="jgraph_1.html#R1" target="_self">is_Shortcut_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="jgraph_1.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E5:32"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="jgraph_1.html#K4" target="_self">PairF</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="jgraph_1.html#K4" target="_self">PairF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E1:32"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jgraph_1.html#E4:32"><i><font color="Green">E28</font></i></a>, <a class="ref" href="jgraph_1.html#T15" target="_self">Th15</a>, <a class="ref" href="jgraph_1.html#D3" target="_self">Def3</a>, <a class="ref" href="jgraph_1.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E6:32"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="topreal1.html#V1">special</a>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E1:32"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jgraph_1.html#E4:32"><i><font color="Green">E28</font></i></a>, <a class="ref" href="jgraph_1.html#T28" target="_self">Th28</a></i>;<br/>
<a NAME="E7:32"/><i><font color="Green">E31</font></i>: 
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E4:32"><i><font color="Green">E28</font></i></a>, <a class="ref" href="jgraph_1.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E8:32"/><i><font color="Green">E32</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E4:32"><i><font color="Green">E28</font></i></a>, <a class="ref" href="jgraph_1.html#T26" target="_self">Th26</a></i>;<br/>
<a NAME="E9:32"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E4:32"><i><font color="Green">E28</font></i></a>, <a class="ref" href="jgraph_1.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E10:32"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E1:32"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jgraph_1.html#E5:32"><i><font color="Green">E29</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E11:32"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>hence </b><a NAME="E12:32"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM8">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="topreal1.html#V1">special</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="topreal1.html#K5">L~</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font></span>)</span> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="jgraph_1.html#E5:32"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jgraph_1.html#E6:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jgraph_1.html#E7:32"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jgraph_1.html#E8:32"><i><font color="Green">E32</font></i></a></i>;<br/>


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