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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>



<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="euclid.html#K15">TOP-REAL</a> 2;<br/>
<b>assume </b><a NAME="E1:11"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="jordan2c.html#V1">Bounded</a> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>1</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <b>by </b><i><a class="txt" href="jordan1c.html#E1:11"><i><font color="Green">E12</font></i></a>, <a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E4:11"/><i><font color="Green">E13</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<i><font color="Green">E14</font></i>: <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="pscomp_1.html#K12">||</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="pscomp_1.html#D26">PSCOMP_1:def 26</a></i>
<br/>.= 
<a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="extens_1.html#T1">EXTENS_1:1</a></i>
;<br/>
<a NAME="E6:11"/><i><font color="Green">E15</font></i>: 
<a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font> is <a href="seq_4.html#V3">bounded</a>
 
<b>by </b><i><a class="txt" href="jordan1c.html#E1:11"><i><font color="Green">E12</font></i></a>, <a class="ref" href="jct_misc.html#T23">JCT_MISC:23</a></i>;<br/>
<a NAME="E7:11"/><b>then </b><i><font color="Green">E16</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="pscomp_1.html#K12">||</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font> is <a href="seq_4.html#V1">bounded_above</a>
 
<b>by </b><i><a class="txt" href="jordan1c.html#E5:11"><i><font color="Green">E14</font></i></a>, <a class="ref" href="seq_4.html#D3">SEQ_4:def 3</a></i>;<br/>
<a NAME="E8:11"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="pscomp_1.html#K12">||</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font> is <a href="seq_4.html#V2">bounded_below</a>
 
<b>by </b><i><a class="txt" href="jordan1c.html#E5:11"><i><font color="Green">E14</font></i></a>, <a class="txt" href="jordan1c.html#E6:11"><i><font color="Green">E15</font></i></a>, <a class="ref" href="seq_4.html#D3">SEQ_4:def 3</a></i>;<br/>
<a NAME="E9:11"/><b>then </b><i><font color="Green">E17</font></i>: 
<a href="pscomp_1.html#K14">proj2</a>  <a href="pscomp_1.html#K12">||</a> <font color="Maroon">c<sub>4</sub></font> is <a href="seq_2.html#V2">bounded_below</a>
 
<b>by </b><i><a class="txt" href="jordan1c.html#E4:11"><i><font color="Green">E13</font></i></a>, <a class="ref" href="pscomp_1.html#D11">PSCOMP_1:def 11</a></i>;<br/>
<a NAME="E10:11"/>
<a href="pscomp_1.html#K14">proj2</a>  <a href="pscomp_1.html#K12">||</a> <font color="Maroon">c<sub>4</sub></font> is <a href="seq_2.html#V1">bounded_above</a>
 
<b>by </b><i><a class="txt" href="jordan1c.html#E4:11"><i><font color="Green">E13</font></i></a>, <a class="txt" href="jordan1c.html#E7:11"><i><font color="Green">E16</font></i></a>, <a class="ref" href="pscomp_1.html#D12">PSCOMP_1:def 12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <a href="pscomp_1.html#K14">proj2</a>  <a href="pscomp_1.html#K12">||</a> <font color="Maroon">c<sub>4</sub></font> as  <a href="seq_2.html#V3">bounded</a> <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <b>by </b><i><a class="txt" href="jordan1c.html#E9:11"><i><font color="Green">E17</font></i></a>, <a class="ref" href="seq_2.html#D5">SEQ_2:def 5</a></i>;<br/>
<a NAME="E12:11"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="pscomp_1.html#K12">||</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="txt" href="jordan1c.html#E1:11"><i><font color="Green">E12</font></i></a>, <a class="ref" href="pscomp_1.html#T70">PSCOMP_1:70</a></i>;<br/>
<a NAME="E13:11"/><b>then </b>
(  <a href="pscomp_1.html#K10">inf</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K11">sup</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T47">PSCOMP_1:47</a>, <a class="ref" href="pscomp_1.html#T50">PSCOMP_1:50</a></i>;<br/>
<b>hence </b><a NAME="E14:11"/>
(  <a href="pscomp_1.html#K10">inf</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="pscomp_1.html#K12">||</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K11">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="pscomp_1.html#K12">||</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 ;<br/>


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