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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>; <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 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/> for <font color="Olive" title="b2">Y</font> being <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b3">f</font> being <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>  st <font color="Olive" title="b1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">Y</font> holds <br/> <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">inf</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">f</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Olive" title="b2">Y</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">inf</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">f</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Olive" title="b1">X</font></span>)</span></span><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>; <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">Y</font> being <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b2">f</font> being <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>  st <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Y</font> holds <br/> <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">inf</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Olive" title="b1">Y</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">inf</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c2">X</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">Y</font> be  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>; <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">f</font> being <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>  st <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> holds <br/> <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">inf</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</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#K6" title="PSCOMP_1:func.6">inf</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c2">X</font></span>)</span></span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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> implies  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</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#K6" title="PSCOMP_1:func.6">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c2">X</font></span>)</span> )</span><br/>







<b>assume </b><a NAME="E1:98"/><i><font color="Green" title="E59">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</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#K6" title="PSCOMP_1:func.6">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c2">X</font></span>)</span></span><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#V2" title="COMPTS_1:attr.2">compact</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<i><font color="Green" title="E60">A2</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="pscomp_1.html#K8" title="PSCOMP_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> 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="pscomp_1.html#K8" title="PSCOMP_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="relat_1.html#T162" title="RELAT_1:th.162">RELAT_1:162</a></i>
;<br/>
<i><font color="Green" title="E61">A3</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="pscomp_1.html#K8" title="PSCOMP_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> 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="pscomp_1.html#K8" title="PSCOMP_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="relat_1.html#T162" title="RELAT_1:th.162">RELAT_1:162</a></i>
;<br/>
<a NAME="E5:98"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</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="xxreal_2.html#V3" title="XXREAL_2:attr.3">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/>
<b>hence </b><a NAME="E6:98"/>
 <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</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#K6" title="PSCOMP_1:func.6">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <font color="Maroon" title="c2">X</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:98"><i><font color="Green" title="E59">A1</font></i></a>, <a class="txt" href="#E3:98"><i><font color="Green" title="E60">A2</font></i></a>, <a class="txt" href="#E4:98"><i><font color="Green" title="E61">A3</font></i></a>, <a class="ref" href="relat_1.html#T156" title="RELAT_1:th.156">RELAT_1:156</a>, <a class="ref" href="seq_4.html#T64" title="SEQ_4:th.64">SEQ_4:64</a></i>; <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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