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<b>let </b><font color="Maroon" title="c8">r</font>, <font color="Maroon" title="c9">s</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c9">s</font></span>)</span><br/> for <font color="Olive" title="b2">C</font> being   <a href="rcomp_3.html#M1" title="RCOMP_3:mode.1">IntervalCover</a> of <font color="Olive" title="b1">F</font><br/> for <font color="Olive" title="b3">G</font> being   <a href="rcomp_3.html#M2" title="RCOMP_3:mode.2">IntervalCoverPts</a> of <font color="Olive" title="b2">C</font>  st <font color="Olive" title="b1">F</font> is   <a href="struct_0.html#NM11" title="STRUCT_0:NM.11">Cover</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c9">s</font></span>)</span> &amp; <font color="Olive" title="b1">F</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> &amp; <font color="Olive" title="b1">F</font> is <a href="rcomp_3.html#V1" title="RCOMP_3:attr.1">connected</a> &amp; <font color="Maroon" title="c8">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">s</font> holds <br/> not <font color="Olive" title="b3">G</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a></span><br/><b>let </b><font color="Maroon" title="c10">F</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c9">s</font></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">C</font> being   <a href="rcomp_3.html#M1" title="RCOMP_3:mode.1">IntervalCover</a> of <font color="Maroon" title="c10">F</font><br/> for <font color="Olive" title="b2">G</font> being   <a href="rcomp_3.html#M2" title="RCOMP_3:mode.2">IntervalCoverPts</a> of <font color="Olive" title="b1">C</font>  st <font color="Maroon" title="c10">F</font> is   <a href="struct_0.html#NM11" title="STRUCT_0:NM.11">Cover</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c9">s</font></span>)</span> &amp; <font color="Maroon" title="c10">F</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> &amp; <font color="Maroon" title="c10">F</font> is <a href="rcomp_3.html#V1" title="RCOMP_3:attr.1">connected</a> &amp; <font color="Maroon" title="c8">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">s</font> holds <br/> not <font color="Olive" title="b2">G</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a></span><br/><b>let </b><font color="Maroon" title="c11">C</font> be    <a href="rcomp_3.html#M1" title="RCOMP_3:mode.1">IntervalCover</a> of <font color="Maroon" title="c10">F</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">G</font> being   <a href="rcomp_3.html#M2" title="RCOMP_3:mode.2">IntervalCoverPts</a> of <font color="Maroon" title="c11">C</font>  st <font color="Maroon" title="c10">F</font> is   <a href="struct_0.html#NM11" title="STRUCT_0:NM.11">Cover</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c9">s</font></span>)</span> &amp; <font color="Maroon" title="c10">F</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> &amp; <font color="Maroon" title="c10">F</font> is <a href="rcomp_3.html#V1" title="RCOMP_3:attr.1">connected</a> &amp; <font color="Maroon" title="c8">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">s</font> holds <br/> not <font color="Olive" title="b1">G</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a></span><br/><b>let </b><font color="Maroon" title="c12">G</font> be    <a href="rcomp_3.html#M2" title="RCOMP_3:mode.2">IntervalCoverPts</a> of <font color="Maroon" title="c11">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">F</font> is   <a href="struct_0.html#NM11" title="STRUCT_0:NM.11">Cover</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c9">s</font></span>)</span> &amp; <font color="Maroon" title="c10">F</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> &amp; <font color="Maroon" title="c10">F</font> is <a href="rcomp_3.html#V1" title="RCOMP_3:attr.1">connected</a> &amp; <font color="Maroon" title="c8">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">s</font> implies  not <font color="Maroon" title="c12">G</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )</span><br/>









<b>assume </b><a NAME="E1:35"/>
( <font color="Maroon" title="c10">F</font> is   <a href="struct_0.html#NM11" title="STRUCT_0:NM.11">Cover</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c9">s</font></span>)</span> &amp; <font color="Maroon" title="c10">F</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> &amp; <font color="Maroon" title="c10">F</font> is <a href="rcomp_3.html#V1" title="RCOMP_3:attr.1">connected</a> &amp; <font color="Maroon" title="c8">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">s</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c12">G</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a></span><br/>

<a NAME="E2:35"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">C</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="rcomp_3.html#D3" title="RCOMP_3:def.3">RCOMP_3:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:35"/>
not <font color="Maroon" title="c12">G</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</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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