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<b>let </b><font color="Maroon" title="c1">T</font> be    <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</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">f</font> being  <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b2">R</font> being  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b1">f</font> is <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a> &amp; <font color="Olive" title="b2">R</font> is <a href="pscomp_1.html#V3" title="PSCOMP_1:attr.3">open</a> holds <br/><span class="p1">(<span class="default"><a href="pscomp_1.html#K1" title="PSCOMP_1:func.1">"</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b2">R</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a></span><br/><b>let </b><font color="Maroon" title="c2">f</font> be   <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"> for <font color="Olive" title="b1">R</font> being  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Maroon" title="c2">f</font> is <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a> &amp; <font color="Olive" title="b1">R</font> is <a href="pscomp_1.html#V3" title="PSCOMP_1:attr.3">open</a> holds <br/><span class="p1">(<span class="default"><a href="pscomp_1.html#K1" title="PSCOMP_1:func.1">"</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">R</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a></span><br/><b>let </b><font color="Maroon" title="c3">R</font> be   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> is <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a> &amp; <font color="Maroon" title="c3">R</font> is <a href="pscomp_1.html#V3" title="PSCOMP_1:attr.3">open</a> implies <span class="p1">(<span class="default"><a href="pscomp_1.html#K1" title="PSCOMP_1:func.1">"</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">R</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> )</span><br/>





<b>assume </b><a NAME="E1:87"/><i><font color="Green" title="E54">A1</font></i>: 
<font color="Maroon" title="c2">f</font> is <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c3">R</font> is <a href="pscomp_1.html#V3" title="PSCOMP_1:attr.3">open</a> or <span class="p1">(<span class="default"><a href="pscomp_1.html#K1" title="PSCOMP_1:func.1">"</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">R</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> )</span><br/>

<b>assume </b><a NAME="E2:87"/><i><font color="Green" title="E55">A2</font></i>: 
<font color="Maroon" title="c3">R</font> is <a href="pscomp_1.html#V3" title="PSCOMP_1:attr.3">open</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="pscomp_1.html#K1" title="PSCOMP_1:func.1">"</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">R</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a></span><br/>

<b>let </b><font color="Maroon" title="c4">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K1" title="PSCOMP_1:func.1">"</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">R</font> or <font color="Maroon" title="c4">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )</span><br/>

<b>assume </b><a NAME="E3:87"/>
<font color="Maroon" title="c4">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K1" title="PSCOMP_1:func.1">"</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a></span><br/>

<b>then consider </b><font color="Maroon" title="c5">eR</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:87"/><i><font color="Green" title="E56">A3</font></i>: 
<font color="Maroon" title="c5">eR</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <b>and </b><br/><a NAME="E6:87"/><i><font color="Green" title="E57">A4</font></i>: 
<font color="Maroon" title="c5">eR</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R</font>
 <b>and </b><br/><a NAME="E7:87"/><i><font color="Green" title="E58">A5</font></i>: 
<font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K1" title="PSCOMP_1:func.1">"</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">eR</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T115" title="FUNCT_2:th.115">FUNCT_2:115</a></i>;<br/>
<a NAME="E8:87"/><i><font color="Green" title="E59">A6</font></i>: 
<font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c5">eR</font>
 
<b>by </b><i><a class="txt" href="#E5:87"><i><font color="Green" title="E56">A3</font></i></a>, <a class="txt" href="#E7:87"><i><font color="Green" title="E58">A5</font></i></a>, <a class="ref" href="pscomp_1.html#D2" target="_self" title="PSCOMP_1:def.2">Def2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">eR</font> = <font color="Maroon" title="c5">eR</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="txt" href="#E5:87"><i><font color="Green" title="E56">A3</font></i></a></i>;<br/>
<a NAME="E10:87"/>
<font color="Maroon" title="c6">eR</font> is <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="#E2:87"><i><font color="Green" title="E55">A2</font></i></a>, <a class="txt" href="#E6:87"><i><font color="Green" title="E57">A4</font></i></a>, <a class="ref" href="pscomp_1.html#D5" target="_self" title="PSCOMP_1:def.5">Def5</a></i>;<br/>
<b>hence </b><a NAME="E11:87"/>
<font color="Maroon" title="c4">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="#E1:87"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="#E8:87"><i><font color="Green" title="E59">A6</font></i></a>, <a class="ref" href="pscomp_1.html#T54" target="_self" title="PSCOMP_1:th.54">Th54</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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