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<div class="add">

<b>let </b><font color="Maroon" title="c1">f</font> be  non <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> <a href="goboard5.html#V2" title="GOBOARD5:attr.2">standard</a> <a href="goboard5.html#NM1" title="GOBOARD5:NM.1">special_circular_sequence</a>;<br/><b>let </b><font color="Maroon" title="c2">g</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>;<br/>



<a NAME="E1:59"/><i><font color="Green" title="E29">A1</font></i>: 
(  <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> &amp;  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> &amp;  <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> &amp;  <a href="pscomp_1.html#K17" title="PSCOMP_1:func.17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K17" title="PSCOMP_1:func.17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="sprect_1.html#T66" title="SPRECT_1:th.66">SPRECT_1:66</a>, <a class="ref" href="sprect_1.html#T67" title="SPRECT_1:th.67">SPRECT_1:67</a>, <a class="ref" href="sprect_1.html#T68" title="SPRECT_1:th.68">SPRECT_1:68</a>, <a class="ref" href="sprect_1.html#T69" title="SPRECT_1:th.69">SPRECT_1:69</a></i>;<br/>
<b>thus </b><a NAME="E2:59"/>
( <font color="Maroon" title="c2">g</font> <a href="sprect_2.html#R1" title="SPRECT_2:pred.1">is_in_the_area_of</a> <font color="Maroon" title="c1">f</font> implies <font color="Maroon" title="c2">g</font> <a href="sprect_2.html#R1" title="SPRECT_2:pred.1">is_in_the_area_of</a>  <a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:59_1"/><i><font color="Green" title="E30">A2</font></i>: 
<font color="Maroon" title="c2">g</font> <a href="sprect_2.html#R1" title="SPRECT_2:pred.1">is_in_the_area_of</a> <font color="Maroon" title="c1">f</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">n</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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="sprect_2.html#D1" title="SPRECT_2:def.1">SPRECT_2:def 1</a><br/>

<b>thus </b><a NAME="E2:59_1"/>
(  not <font color="Maroon" title="c3">n</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="c2">g</font> or ( not <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span></span>)</span> &amp; not  <a href="pscomp_1.html#K17" title="PSCOMP_1:func.17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; not <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span></span>)</span> &amp; not  <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) )
 <b>by </b><i><a class="txt" href="sprect_3.html#E1:59"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="sprect_3.html#E1:59_1"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="sprect_2.html#D1" title="SPRECT_2:def.1">SPRECT_2:def 1</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:59"/><i><font color="Green" title="E30">A3</font></i>: 
<font color="Maroon" title="c2">g</font> <a href="sprect_2.html#R1" title="SPRECT_2:pred.1">is_in_the_area_of</a>  <a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">n</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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="sprect_2.html#D1" title="SPRECT_2:def.1">SPRECT_2:def 1</a><br/>

<b>thus </b><a NAME="E4:59"/>
(  not <font color="Maroon" title="c3">n</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="c2">g</font> or ( not <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> &amp; not  <a href="pscomp_1.html#K17" title="PSCOMP_1:func.17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; not <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> &amp; not  <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) )
 <b>by </b><i><a class="txt" href="sprect_3.html#E1:59"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="sprect_3.html#E3:59"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="sprect_2.html#D1" title="SPRECT_2:def.1">SPRECT_2:def 1</a></i>;<br/>


</div>
