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<b>let </b><font color="Maroon" title="c1">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">f</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:49"/><i><font color="Green" title="E35">A1</font></i>: 
<font color="Maroon" title="c2">f</font> is <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>
 <b>and </b><br/><a NAME="E2:49"/><i><font color="Green" title="E36">A2</font></i>: 
<font color="Maroon" title="c2">f</font> is <a href="sppol_1.html#V3" title="SPPOL_1:attr.3">alternating</a>
 <b>and </b><br/><a NAME="E3:49"/><i><font color="Green" title="E37">A3</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i</font>
 <b>and </b><br/><a NAME="E4:49"/><i><font color="Green" title="E38">A4</font></i>: 
<font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 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> <font color="Maroon" title="c2">f</font>
 ;<br/>

<a NAME="E5:49"/><i><font color="Green" title="E39">A5</font></i>: 
(  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">i</font> is <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</a> &amp; not <!--HUMANRECFAILED-->(  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">i</font> is <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</a> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="sppol_1.html#V1" title="SPPOL_1:attr.1">horizontal</a> ) implies (  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">i</font> is <a href="sppol_1.html#V1" title="SPPOL_1:attr.1">horizontal</a> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</a> ) )
 
<b>by </b><i><a class="txt" href="sppol_1.html#E1:49"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="sppol_1.html#E2:49"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="sppol_1.html#E3:49"><i><font color="Green" title="E37">A3</font></i></a>, <a class="txt" href="sppol_1.html#E4:49"><i><font color="Green" title="E38">A4</font></i></a>, <a class="ref" href="sppol_1.html#T55" target="_self" title="SPPOL_1:th.55">Th55</a></i>;<br/>
<a NAME="E6:49"/>
(  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">i</font> is <a href="sppol_1.html#V1" title="SPPOL_1:attr.1">horizontal</a> &amp; not <!--HUMANRECFAILED-->(  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">i</font> is <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</a> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="sppol_1.html#V1" title="SPPOL_1:attr.1">horizontal</a> ) implies (  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">i</font> is <a href="sppol_1.html#V1" title="SPPOL_1:attr.1">horizontal</a> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</a> ) )
 
<b>by </b><i><a class="txt" href="sppol_1.html#E1:49"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="sppol_1.html#E2:49"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="sppol_1.html#E3:49"><i><font color="Green" title="E37">A3</font></i></a>, <a class="txt" href="sppol_1.html#E4:49"><i><font color="Green" title="E38">A4</font></i></a>, <a class="ref" href="sppol_1.html#T56" target="_self" title="SPPOL_1:th.56">Th56</a></i>;<br/>
<b>hence </b><a NAME="E7:49"/>
( (  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">i</font> is <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</a> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="sppol_1.html#V1" title="SPPOL_1:attr.1">horizontal</a> ) or (  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">i</font> is <a href="sppol_1.html#V1" title="SPPOL_1:attr.1">horizontal</a> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</a> ) )
 <b>by </b><i><a class="txt" href="sppol_1.html#E1:49"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="sppol_1.html#E5:49"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="sppol_1.html#T41" target="_self" title="SPPOL_1:th.41">Th41</a></i>;<br/>


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