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<b>let </b><font color="Maroon" title="c1">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> &amp; <font color="Olive" title="b1">X</font> is <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a> holds <br/>(  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">inf</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="pscomp_1.html#K10" title="PSCOMP_1:func.10">||</a> <font color="Olive" title="b1">X</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c1">p</font> <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#K9" title="PSCOMP_1:func.9">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="pscomp_1.html#K10" title="PSCOMP_1:func.10">||</a> <font color="Olive" title="b1">X</font></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c2">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c2">X</font> is <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a> implies (  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">inf</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="pscomp_1.html#K10" title="PSCOMP_1:func.10">||</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c1">p</font> <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#K9" title="PSCOMP_1:func.9">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="pscomp_1.html#K10" title="PSCOMP_1:func.10">||</a> <font color="Maroon" title="c2">X</font></span>)</span> ) )</span><br/>



<b>set </b><font color="Maroon" title="c3">T</font> =  <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2;<br/>
<b>assume </b><a NAME="E1:11"/><i><font color="Green" title="E12">A1</font></i>: 
( <font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c2">X</font> is <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">inf</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="pscomp_1.html#K10" title="PSCOMP_1:func.10">||</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c1">p</font> <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#K9" title="PSCOMP_1:func.9">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="pscomp_1.html#K10" title="PSCOMP_1:func.10">||</a> <font color="Maroon" title="c2">X</font></span>)</span> )</span><br/>

<b>then reconsider </b><font color="Maroon" title="c4">X</font> = <font color="Maroon" title="c2">X</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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>reconsider </b><font color="Maroon" title="c5">p'</font> = <font color="Maroon" title="c1">p</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">X</font></span>)</span> <b>by </b><i><a class="txt" href="jordan1c.html#E1:11"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E4:11"/><i><font color="Green" title="E13">A2</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">X</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<i><font color="Green" title="E14">A3</font></i>: <span class="p1">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="pscomp_1.html#K10" title="PSCOMP_1:func.10">||</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">X</font> = 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">X</font>
<b>by </b><i><a class="ref" href="pscomp_1.html#D26" title="PSCOMP_1:def.26">PSCOMP_1:def 26</a></i>
<br/>.= 
<a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">X</font>
<b>by </b><i><a class="ref" href="relat_1.html#T162" title="RELAT_1:th.162">RELAT_1:162</a></i>
;<br/>
<a NAME="E6:11"/><i><font color="Green" title="E15">A4</font></i>: 
<a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">X</font> is <a href="seq_4.html#V3" title="SEQ_4:attr.3">bounded</a>
 
<b>by </b><i><a class="txt" href="jordan1c.html#E1:11"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="jct_misc.html#T23" title="JCT_MISC:th.23">JCT_MISC:23</a></i>;<br/>
<a NAME="E7:11"/><b>then </b><i><font color="Green" title="E16">A5</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="pscomp_1.html#K10" title="PSCOMP_1:func.10">||</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">X</font> is <a href="seq_4.html#V1" title="SEQ_4:attr.1">bounded_above</a>
 
<b>by </b><i><a class="txt" href="jordan1c.html#E5:11"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="seq_4.html#D3" title="SEQ_4:def.3">SEQ_4:def 3</a></i>;<br/>
<a NAME="E8:11"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="pscomp_1.html#K10" title="PSCOMP_1:func.10">||</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">X</font> is <a href="seq_4.html#V2" title="SEQ_4:attr.2">bounded_below</a>
 
<b>by </b><i><a class="txt" href="jordan1c.html#E5:11"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="jordan1c.html#E6:11"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="seq_4.html#D3" title="SEQ_4:def.3">SEQ_4:def 3</a></i>;<br/>
<a NAME="E9:11"/><b>then </b><i><font color="Green" title="E17">A6</font></i>: 
<a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="pscomp_1.html#K10" title="PSCOMP_1:func.10">||</a> <font color="Maroon" title="c4">X</font> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a>
 
<b>by </b><i><a class="txt" href="jordan1c.html#E4:11"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="pscomp_1.html#D11" title="PSCOMP_1:def.11">PSCOMP_1:def 11</a></i>;<br/>
<a NAME="E10:11"/>
<a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="pscomp_1.html#K10" title="PSCOMP_1:func.10">||</a> <font color="Maroon" title="c4">X</font> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a>
 
<b>by </b><i><a class="txt" href="jordan1c.html#E4:11"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="jordan1c.html#E7:11"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="pscomp_1.html#D12" title="PSCOMP_1:def.12">PSCOMP_1:def 12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">f</font> = <a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="pscomp_1.html#K10" title="PSCOMP_1:func.10">||</a> <font color="Maroon" title="c4">X</font> as  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">X</font></span>)</span> <b>by </b><i><a class="txt" href="jordan1c.html#E9:11"><i><font color="Green" title="E17">A6</font></i></a>, <a class="ref" href="seq_2.html#D5" title="SEQ_2:def.5">SEQ_2:def 5</a></i>;<br/>
<a NAME="E12:11"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="pscomp_1.html#K10" title="PSCOMP_1:func.10">||</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 
<b>by </b><i><a class="txt" href="jordan1c.html#E1:11"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="pscomp_1.html#T70" title="PSCOMP_1:th.70">PSCOMP_1:70</a></i>;<br/>
<a NAME="E13:11"/><b>then </b>
(  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">inf</a> <font color="Maroon" title="c6">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c1">p</font> <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#K9" title="PSCOMP_1:func.9">sup</a> <font color="Maroon" title="c6">f</font> )
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T47" title="PSCOMP_1:th.47">PSCOMP_1:47</a>, <a class="ref" href="pscomp_1.html#T50" title="PSCOMP_1:th.50">PSCOMP_1:50</a></i>;<br/>
<b>hence </b><a NAME="E14:11"/>
(  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">inf</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="pscomp_1.html#K10" title="PSCOMP_1:func.10">||</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c1">p</font> <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#K9" title="PSCOMP_1:func.9">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="pscomp_1.html#K10" title="PSCOMP_1:func.10">||</a> <font color="Maroon" title="c2">X</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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