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<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">X</font> be  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 <font color="Maroon" title="c1">T</font>;<br/><b>let </b><font color="Maroon" title="c3">Y</font> be  <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/><b>let </b><font color="Maroon" title="c4">f</font> be  <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a> <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <font color="Maroon" title="c1">T</font>;<br/>







<b>assume </b><a NAME="E1:102"/><i><font color="Green" title="E62">A1</font></i>: 
<font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">Y1</font> = <font color="Maroon" title="c3">Y</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/>
<i><font color="Green" title="E63">A2</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">||</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c3">Y</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">Y</font>
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">Y</font>
<b>by </b><i><a class="ref" href="rfunct_2.html#T5" title="RFUNCT_2:th.5">RFUNCT_2:5</a></i>
;<br/>
<i><font color="Green" title="E64">A3</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">||</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">X</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">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/>.= 
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">X</font>
<b>by </b><i><a class="ref" href="rfunct_2.html#T5" title="RFUNCT_2:th.5">RFUNCT_2:5</a></i>
;<br/>
<a NAME="E5:102"/><i><font color="Green" title="E65">A4</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">||</a> <font color="Maroon" title="c5">Y1</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c5">Y1</font></span>)</span> is <a href="seq_4.html#V2" title="SEQ_4:attr.2">bounded_below</a>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#D11" target="_self" title="PSCOMP_1:def.11">Def11</a></i>;<br/>
<a NAME="E6:102"/>
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">Y</font>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E1:102"><i><font color="Green" title="E62">A1</font></i></a>, <a class="ref" href="relat_1.html#T156" title="RELAT_1:th.156">RELAT_1:156</a></i>;<br/>
<b>hence </b><a NAME="E7:102"/>
 <a href="pscomp_1.html#K10" title="PSCOMP_1:func.10">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">||</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K10" title="PSCOMP_1:func.10">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">||</a> <font color="Maroon" title="c2">X</font></span>)</span>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E3:102"><i><font color="Green" title="E63">A2</font></i></a>, <a class="txt" href="pscomp_1.html#E4:102"><i><font color="Green" title="E64">A3</font></i></a>, <a class="txt" href="pscomp_1.html#E5:102"><i><font color="Green" title="E65">A4</font></i></a>, <a class="ref" href="seq_4.html#T64" title="SEQ_4:th.64">SEQ_4:64</a></i>;<br/>


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