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<span class="kw">let </span><font color="Maroon" title="c1">r</font>, <font color="Maroon" title="c2">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <br/>  for <font color="Olive" title="b2">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="c1">r</font>,<font color="Maroon" title="c2">s</font>)</span>)</span><br/>  for <font color="Olive" title="b3">C</font> being    <a href="rcomp_3.html#M1" title="RCOMP_3:mode.1">IntervalCover</a> of <font color="Olive" title="b2">F</font><br/>  for <font color="Olive" title="b4">G</font> being    <a href="rcomp_3.html#M2" title="RCOMP_3:mode.2">IntervalCoverPts</a> of <font color="Olive" title="b3">C</font>  st <font color="Olive" title="b2">F</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">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="c1">r</font>,<font color="Maroon" title="c2">s</font>)</span>)</span> &amp; <font color="Olive" title="b2">F</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  &amp; <font color="Olive" title="b2">F</font> is  <a href="rcomp_3.html#V1" title="RCOMP_3:attr.1">connected</a>  &amp; <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">G</font> holds <br/><span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">n</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c1">r</font>,<font color="Maroon" title="c2">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#NM13" title="STRUCT_0:NM.13">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="c1">r</font>,<font color="Maroon" title="c2">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="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">G</font> holds <br/><span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">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="c1">r</font>,<font color="Maroon" title="c2">s</font>)</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c4">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="c4">F</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">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="c1">r</font>,<font color="Maroon" title="c2">s</font>)</span>)</span> &amp; <font color="Maroon" title="c4">F</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  &amp; <font color="Maroon" title="c4">F</font> is  <a href="rcomp_3.html#V1" title="RCOMP_3:attr.1">connected</a>  &amp; <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">G</font> holds <br/><span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">C</font> be    <a href="rcomp_3.html#M1" title="RCOMP_3:mode.1">IntervalCover</a> of <font color="Maroon" title="c4">F</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c5">C</font>  st <font color="Maroon" title="c4">F</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">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="c1">r</font>,<font color="Maroon" title="c2">s</font>)</span>)</span> &amp; <font color="Maroon" title="c4">F</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  &amp; <font color="Maroon" title="c4">F</font> is  <a href="rcomp_3.html#V1" title="RCOMP_3:attr.1">connected</a>  &amp; <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">G</font> holds <br/><span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">G</font> be    <a href="rcomp_3.html#M2" title="RCOMP_3:mode.2">IntervalCoverPts</a> of <font color="Maroon" title="c5">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">F</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">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="c1">r</font>,<font color="Maroon" title="c2">s</font>)</span>)</span> &amp; <font color="Maroon" title="c4">F</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  &amp; <font color="Maroon" title="c4">F</font> is  <a href="rcomp_3.html#V1" title="RCOMP_3:attr.1">connected</a>  &amp; <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">G</font> implies <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> )</span><br/>











<span class="kw">set </span><font color="Maroon" title="c7">L</font> =  <a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> (<font color="Maroon" title="c1">r</font>,<font color="Maroon" title="c2">s</font>);<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:87"/><span class="lab"><font color="Green" title="E63">A1</font></span>: 
( <font color="Maroon" title="c4">F</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">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="c1">r</font>,<font color="Maroon" title="c2">s</font>)</span>)</span> &amp; <font color="Maroon" title="c4">F</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  )
 <span class="kw">and </span><br/><a NAME="E2:87"/><span class="lab"><font color="Green" title="E64">A2</font></span>: 
<font color="Maroon" title="c4">F</font> is  <a href="rcomp_3.html#V1" title="RCOMP_3:attr.1">connected</a> 
 <span class="kw">and </span><br/><a NAME="E3:87"/><span class="lab"><font color="Green" title="E65">A3</font></span>: 
<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font>
 <span class="kw">and </span><br/><a NAME="E4:87"/><span class="lab"><font color="Green" title="E66">A4</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">and </span><br/><a NAME="E5:87"/><span class="lab"><font color="Green" title="E67">A5</font></span>: 
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font></span><br/>

<a NAME="E6:87"/><span class="lab"><font color="Green" title="E68">A6</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">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="c5">C</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="txt" href="#E2:87"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="txt" href="#E3:87"><span class="lab"><font color="Green" title="E65">A3</font></span></a>, <a class="ref" href="rcomp_3.html#D3" target="_self" title="RCOMP_3:def.3">Def3</a></span>;<br/>
<a NAME="E7:87"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A7</font></span>: 
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:87"><span class="lab"><font color="Green" title="E67">A5</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E8:87"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A8</font></span>: 
<font color="Maroon" title="c5">C</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:87"><span class="lab"><font color="Green" title="E66">A4</font></span></a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</a></span>;<br/>
<a NAME="E9:87"/>
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:87"><span class="lab"><font color="Green" title="E66">A4</font></span></a>, <a class="txt" href="#E7:87"><span class="lab"><font color="Green" title="E69">A7</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E10:87"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A9</font></span>: 
<font color="Maroon" title="c5">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E11:87"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="txt" href="#E2:87"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="txt" href="#E3:87"><span class="lab"><font color="Green" title="E65">A3</font></span></a>, <a class="ref" href="rcomp_3.html#D2" target="_self" title="RCOMP_3:def.2">Def2</a></span>;<br/>
<a NAME="E12:87"/><span class="kw">then </span>
<font color="Maroon" title="c5">C</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:87"><span class="lab"><font color="Green" title="E70">A8</font></span></a>, <a class="txt" href="#E10:87"><span class="lab"><font color="Green" title="E71">A9</font></span></a></span>;<br/>
<a NAME="E13:87"/><span class="kw">then </span>
<font color="Maroon" title="c5">C</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">n</font> is   <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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="c1">r</font>,<font color="Maroon" title="c2">s</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:87"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="ref" href="rcomp_3.html#D1" target="_self" title="RCOMP_3:def.1">Def1</a></span>;<br/>
<a NAME="E14:87"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A10</font></span>: 
<font color="Maroon" title="c5">C</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">n</font> is  <a href="xxreal_2.html#V6" title="XXREAL_2:attr.6">interval</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_3.html#T43" target="_self" title="RCOMP_3:th.43">Th56</a></span>;<br/>
<a NAME="E15:87"/><span class="lab"><font color="Green" title="E73">A11</font></span>: 
 not <font color="Maroon" title="c5">C</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">n</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="txt" href="#E2:87"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="txt" href="#E3:87"><span class="lab"><font color="Green" title="E65">A3</font></span></a>, <a class="txt" href="#E4:87"><span class="lab"><font color="Green" title="E66">A4</font></span></a>, <a class="txt" href="#E7:87"><span class="lab"><font color="Green" title="E69">A7</font></span></a>, <a class="ref" href="rcomp_3.html#D2" target="_self" title="RCOMP_3:def.2">Def2</a></span>;<br/>
<a NAME="E16:87"/><span class="lab"><font color="Green" title="E74">A12</font></span>: 
<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:87"><span class="lab"><font color="Green" title="E67">A5</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E17:87"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/>
<a NAME="E18:87"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A13</font></span>: 
<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:87"><span class="lab"><font color="Green" title="E74">A12</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E19:87"/><span class="lab"><font color="Green" title="E76">A14</font></span>: 
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:87"><span class="lab"><font color="Green" title="E66">A4</font></span></a>, <a class="txt" href="#E5:87"><span class="lab"><font color="Green" title="E67">A5</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E20:87"/><span class="lab"><font color="Green" title="E77">A15</font></span>: 
<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:87_1"/>
( <font color="Maroon" title="c1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> or <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">s</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:87"><span class="lab"><font color="Green" title="E65">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_1"><span class="kw">suppose </span></a><a NAME="E1:87_1_1"/>
<font color="Maroon" title="c1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font></span><br/><div class="add"><a NAME="E2:87_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">r</font>,<font color="Maroon" title="c1">r</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="txt" href="#E2:87"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="ref" href="rcomp_3.html#T50" target="_self" title="RCOMP_3:th.50">Th64</a></span>;<br/><a NAME="E3:87_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A16</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>;<br/><a NAME="E4:87_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c1">r</font>,<font color="Maroon" title="c2">s</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="txt" href="#E2:87"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="txt" href="#E3:87"><span class="lab"><font color="Green" title="E65">A3</font></span></a>, <a class="ref" href="rcomp_3.html#T61" target="_self" title="RCOMP_3:th.61">Th75</a></span>;<br/><a NAME="E5:87_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A17</font></span>: 
( <font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> &amp; <font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>;<br/><a NAME="E6:87_1_1"/>
( <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c5">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">r</font>,<font color="Maroon" title="c2">s</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="txt" href="#E2:87"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="txt" href="#E3:87"><span class="lab"><font color="Green" title="E65">A3</font></span></a>, <a class="txt" href="#E4:87"><span class="lab"><font color="Green" title="E66">A4</font></span></a>, <a class="txt" href="#E7:87"><span class="lab"><font color="Green" title="E69">A7</font></span></a>, <a class="txt" href="#E3:87_1_1"><span class="lab"><font color="Green" title="E78">A16</font></span></a>, <a class="ref" href="rcomp_3.html#T52" target="_self" title="RCOMP_3:th.52">Th66</a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E7:87_1_1"/>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:87_1_1"><span class="lab"><font color="Green" title="E79">A17</font></span></a>, <a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_2"><span class="kw">suppose </span></a><a NAME="E1:87_1_2"/>
<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font></span><br/><div class="add"><a NAME="E2:87_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">G</font> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="txt" href="#E2:87"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="ref" href="rcomp_3.html#T65" target="_self" title="RCOMP_3:th.65">Th79</a></span>;<br/><a NAME="E3:87_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A18</font></span>: 
<font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:87"><span class="lab"><font color="Green" title="E76">A14</font></span></a>, <a class="txt" href="#E18:87"><span class="lab"><font color="Green" title="E75">A13</font></span></a>, <a class="txt" href="#E20:87"><span class="lab"><font color="Green" title="E77">A15</font></span></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a></span>;<br/><a NAME="E4:87_1_2"/><span class="lab"><font color="Green" title="E79">A19</font></span>: 
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="txt" href="#E2:87"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="txt" href="#E3:87"><span class="lab"><font color="Green" title="E65">A3</font></span></a>, <a class="ref" href="rcomp_3.html#T60" target="_self" title="RCOMP_3:th.60">Th74</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:87_1_2_1"/>
( ( <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) or ( <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">G</font> ) or ( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) or ( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">G</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:87"><span class="lab"><font color="Green" title="E66">A4</font></span></a>, <a class="txt" href="#E16:87"><span class="lab"><font color="Green" title="E74">A12</font></span></a>, <a class="txt" href="#E4:87_1_2"><span class="lab"><font color="Green" title="E79">A19</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_2_1_1"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:87_1_2_1_1"/><span class="lab"><font color="Green" title="E80">A20</font></span>: 
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">and </span><br/><a NAME="E2:87_1_2_1_1"/><span class="lab"><font color="Green" title="E81">A21</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font></span><br/><div class="add"><a NAME="E3:87_1_2_1_1"/>
<font color="Maroon" title="c6">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c1">r</font>,<font color="Maroon" title="c2">s</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="txt" href="#E2:87"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="txt" href="#E3:87"><span class="lab"><font color="Green" title="E65">A3</font></span></a>, <a class="txt" href="#E6:87"><span class="lab"><font color="Green" title="E68">A6</font></span></a>, <a class="txt" href="#E2:87_1_2_1_1"><span class="lab"><font color="Green" title="E81">A21</font></span></a>, <a class="ref" href="rcomp_3.html#T61" target="_self" title="RCOMP_3:th.61">Th75</a></span>;<br/><a NAME="E4:87_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A22</font></span>: 
( <font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> &amp; <font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>;<br/><a NAME="E5:87_1_2_1_1"/>
<font color="Maroon" title="c5">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">r</font>,<font color="Maroon" title="c2">s</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="txt" href="#E2:87"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="txt" href="#E3:87"><span class="lab"><font color="Green" title="E65">A3</font></span></a>, <a class="txt" href="#E6:87"><span class="lab"><font color="Green" title="E68">A6</font></span></a>, <a class="txt" href="#E2:87_1_2_1_1"><span class="lab"><font color="Green" title="E81">A21</font></span></a>, <a class="ref" href="rcomp_3.html#T52" target="_self" title="RCOMP_3:th.52">Th66</a></span>;<br/><span class="kw">hence </span><a NAME="E6:87_1_2_1_1"/>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87_1_2_1_1"><span class="lab"><font color="Green" title="E80">A20</font></span></a>, <a class="txt" href="#E4:87_1_2_1_1"><span class="lab"><font color="Green" title="E82">A22</font></span></a>, <a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_2_1_2"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:87_1_2_1_2"/><span class="lab"><font color="Green" title="E80">A23</font></span>: 
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">and </span><br/><a NAME="E2:87_1_2_1_2"/><span class="lab"><font color="Green" title="E81">A24</font></span>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font></span><br/><div class="add"><a NAME="E3:87_1_2_1_2"/>
<font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">C</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">C</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="txt" href="#E2:87"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="txt" href="#E3:87"><span class="lab"><font color="Green" title="E65">A3</font></span></a>, <a class="txt" href="#E2:87_1_2_1_2"><span class="lab"><font color="Green" title="E81">A24</font></span></a>, <a class="ref" href="rcomp_3.html#D3" target="_self" title="RCOMP_3:def.3">Def3</a></span>;<br/><a NAME="E4:87_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A25</font></span>: 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">C</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T4" title="XXREAL_1:th.4">XXREAL_1:4</a></span>;<br/><a NAME="E5:87_1_2_1_2"/>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:87"><span class="lab"><font color="Green" title="E68">A6</font></span></a>, <a class="txt" href="#E2:87_1_2_1_2"><span class="lab"><font color="Green" title="E81">A24</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E6:87_1_2_1_2"/><span class="kw">then </span>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">C</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">C</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="txt" href="#E2:87"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="txt" href="#E3:87"><span class="lab"><font color="Green" title="E65">A3</font></span></a>, <a class="ref" href="rcomp_3.html#D2" target="_self" title="RCOMP_3:def.2">Def2</a></span>;<br/><a NAME="E7:87_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A26</font></span>: 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">C</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87_1_2_1_2"><span class="lab"><font color="Green" title="E80">A23</font></span></a>, <a class="txt" href="#E4:87_1_2_1_2"><span class="lab"><font color="Green" title="E82">A25</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E8:87_1_2_1_2"/><span class="lab"><font color="Green" title="E84">A27</font></span>: 
( <font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> &amp; <font color="Maroon" title="c1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="txt" href="#E2:87"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="txt" href="#E3:87"><span class="lab"><font color="Green" title="E65">A3</font></span></a>, <a class="ref" href="rcomp_3.html#D3" target="_self" title="RCOMP_3:def.3">Def3</a>, <a class="ref" href="rcomp_3.html#T57" target="_self" title="RCOMP_3:th.57">Th71</a></span>;<br/><a NAME="E9:87_1_2_1_2"/>
<font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">C</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="txt" href="#E2:87"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="txt" href="#E3:87"><span class="lab"><font color="Green" title="E65">A3</font></span></a>, <a class="txt" href="#E1:87_1_2_1_2"><span class="lab"><font color="Green" title="E80">A23</font></span></a>, <a class="txt" href="#E2:87_1_2_1_2"><span class="lab"><font color="Green" title="E81">A24</font></span></a>, <a class="ref" href="rcomp_3.html#T62" target="_self" title="RCOMP_3:th.62">Th76</a></span>;<br/><a NAME="E10:87_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:87"><span class="lab"><font color="Green" title="E70">A8</font></span></a>, <a class="txt" href="#E14:87"><span class="lab"><font color="Green" title="E72">A10</font></span></a>, <a class="txt" href="#E15:87"><span class="lab"><font color="Green" title="E73">A11</font></span></a>, <a class="txt" href="#E7:87_1_2_1_2"><span class="lab"><font color="Green" title="E83">A26</font></span></a>, <a class="ref" href="rcomp_3.html#T30" target="_self" title="RCOMP_3:th.30">Th42</a></span>;<br/><span class="kw">hence </span><a NAME="E11:87_1_2_1_2"/>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:87"><span class="lab"><font color="Green" title="E70">A8</font></span></a>, <a class="txt" href="#E14:87"><span class="lab"><font color="Green" title="E72">A10</font></span></a>, <a class="txt" href="#E1:87_1_2_1_2"><span class="lab"><font color="Green" title="E80">A23</font></span></a>, <a class="txt" href="#E8:87_1_2_1_2"><span class="lab"><font color="Green" title="E84">A27</font></span></a>, <a class="ref" href="xxreal_2.html#D12" title="XXREAL_2:def.12">XXREAL_2:def 12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_2_1_3"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:87_1_2_1_3"/><span class="lab"><font color="Green" title="E80">A28</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">and </span><br/><a NAME="E2:87_1_2_1_3"/><span class="lab"><font color="Green" title="E81">A29</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font></span><br/><div class="add"><a NAME="E3:87_1_2_1_3"/>
1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87_1_2_1_3"><span class="lab"><font color="Green" title="E80">A28</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E4:87_1_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A30</font></span>: 
( <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 &amp; <font color="Maroon" title="c3">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T3" title="INT_1:th.3">INT_1:3</a>, <a class="ref" href="int_1.html#T7" title="INT_1:th.7">INT_1:7</a></span>;<br/><a NAME="E5:87_1_2_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">C</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">C</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="txt" href="#E2:87"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="txt" href="#E3:87"><span class="lab"><font color="Green" title="E65">A3</font></span></a>, <a class="txt" href="#E20:87"><span class="lab"><font color="Green" title="E77">A15</font></span></a>, <a class="txt" href="#E2:87_1_2_1_3"><span class="lab"><font color="Green" title="E81">A29</font></span></a>, <a class="ref" href="rcomp_3.html#D3" target="_self" title="RCOMP_3:def.3">Def3</a></span>;<br/><a NAME="E6:87_1_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A31</font></span>: 
<font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">C</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T4" title="XXREAL_1:th.4">XXREAL_1:4</a></span>;<br/><a NAME="E7:87_1_2_1_3"/>
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">C</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">C</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="txt" href="#E2:87"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="txt" href="#E3:87"><span class="lab"><font color="Green" title="E65">A3</font></span></a>, <a class="txt" href="#E6:87"><span class="lab"><font color="Green" title="E68">A6</font></span></a>, <a class="txt" href="#E2:87_1_2_1_3"><span class="lab"><font color="Green" title="E81">A29</font></span></a>, <a class="txt" href="#E4:87_1_2_1_3"><span class="lab"><font color="Green" title="E82">A30</font></span></a>, <a class="ref" href="rcomp_3.html#D2" target="_self" title="RCOMP_3:def.2">Def2</a></span>;<br/><a NAME="E8:87_1_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A32</font></span>: 
<font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">C</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:87_1_2_1_3"><span class="lab"><font color="Green" title="E83">A31</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E9:87_1_2_1_3"/>
<font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c6">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="txt" href="#E2:87"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="txt" href="#E3:87"><span class="lab"><font color="Green" title="E65">A3</font></span></a>, <a class="ref" href="rcomp_3.html#D3" target="_self" title="RCOMP_3:def.3">Def3</a></span>;<br/><a NAME="E10:87_1_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A33</font></span>: 
<font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="txt" href="#E2:87"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="txt" href="#E3:87"><span class="lab"><font color="Green" title="E65">A3</font></span></a>, <a class="txt" href="#E6:87"><span class="lab"><font color="Green" title="E68">A6</font></span></a>, <a class="txt" href="#E8:87"><span class="lab"><font color="Green" title="E70">A8</font></span></a>, <a class="txt" href="#E2:87_1_2_1_3"><span class="lab"><font color="Green" title="E81">A29</font></span></a>, <a class="ref" href="rcomp_3.html#T59" target="_self" title="RCOMP_3:th.59">Th73</a></span>;<br/><a NAME="E11:87_1_2_1_3"/>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">C</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="txt" href="#E2:87"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="txt" href="#E3:87"><span class="lab"><font color="Green" title="E65">A3</font></span></a>, <a class="txt" href="#E6:87"><span class="lab"><font color="Green" title="E68">A6</font></span></a>, <a class="txt" href="#E1:87_1_2_1_3"><span class="lab"><font color="Green" title="E80">A28</font></span></a>, <a class="txt" href="#E2:87_1_2_1_3"><span class="lab"><font color="Green" title="E81">A29</font></span></a>, <a class="ref" href="rcomp_3.html#T63" target="_self" title="RCOMP_3:th.63">Th77</a></span>;<br/><a NAME="E12:87_1_2_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:87"><span class="lab"><font color="Green" title="E70">A8</font></span></a>, <a class="txt" href="#E14:87"><span class="lab"><font color="Green" title="E72">A10</font></span></a>, <a class="txt" href="#E15:87"><span class="lab"><font color="Green" title="E73">A11</font></span></a>, <a class="txt" href="#E8:87_1_2_1_3"><span class="lab"><font color="Green" title="E84">A32</font></span></a>, <a class="ref" href="rcomp_3.html#T30" target="_self" title="RCOMP_3:th.30">Th42</a></span>;<br/><span class="kw">hence </span><a NAME="E13:87_1_2_1_3"/>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:87"><span class="lab"><font color="Green" title="E70">A8</font></span></a>, <a class="txt" href="#E14:87"><span class="lab"><font color="Green" title="E72">A10</font></span></a>, <a class="txt" href="#E10:87_1_2_1_3"><span class="lab"><font color="Green" title="E85">A33</font></span></a>, <a class="ref" href="xxreal_2.html#D12" title="XXREAL_2:def.12">XXREAL_2:def 12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_2_1_4"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:87_1_2_1_4"/><span class="lab"><font color="Green" title="E80">A34</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">and </span><br/><a NAME="E2:87_1_2_1_4"/><span class="lab"><font color="Green" title="E81">A35</font></span>: 
<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font></span><br/><div class="add"><a NAME="E3:87_1_2_1_4"/><span class="lab"><font color="Green" title="E82">A36</font></span>: 
<font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">C</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="txt" href="#E2:87"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="txt" href="#E3:87"><span class="lab"><font color="Green" title="E65">A3</font></span></a>, <a class="txt" href="#E4:87"><span class="lab"><font color="Green" title="E66">A4</font></span></a>, <a class="txt" href="#E2:87_1_2_1_4"><span class="lab"><font color="Green" title="E81">A35</font></span></a>, <a class="ref" href="rcomp_3.html#T62" target="_self" title="RCOMP_3:th.62">Th76</a></span>;<br/><a NAME="E4:87_1_2_1_4"/>
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:87"><span class="lab"><font color="Green" title="E67">A5</font></span></a>, <a class="txt" href="#E6:87"><span class="lab"><font color="Green" title="E68">A6</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E5:87_1_2_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A37</font></span>: 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">C</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="txt" href="#E2:87"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="txt" href="#E3:87"><span class="lab"><font color="Green" title="E65">A3</font></span></a>, <a class="txt" href="#E1:87_1_2_1_4"><span class="lab"><font color="Green" title="E80">A34</font></span></a>, <a class="ref" href="rcomp_3.html#T63" target="_self" title="RCOMP_3:th.63">Th77</a></span>;<br/><a NAME="E6:87_1_2_1_4"/><span class="kw">then </span>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">C</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:87_1_2"><span class="lab"><font color="Green" title="E78">A18</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E7:87_1_2_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A38</font></span>: 
<font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:87"><span class="lab"><font color="Green" title="E70">A8</font></span></a>, <a class="txt" href="#E14:87"><span class="lab"><font color="Green" title="E72">A10</font></span></a>, <a class="txt" href="#E15:87"><span class="lab"><font color="Green" title="E73">A11</font></span></a>, <a class="txt" href="#E3:87_1_2_1_4"><span class="lab"><font color="Green" title="E82">A36</font></span></a>, <a class="ref" href="rcomp_3.html#T30" target="_self" title="RCOMP_3:th.30">Th42</a></span>;<br/><a NAME="E8:87_1_2_1_4"/>
<font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">C</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:87_1_2"><span class="lab"><font color="Green" title="E78">A18</font></span></a>, <a class="txt" href="#E3:87_1_2_1_4"><span class="lab"><font color="Green" title="E82">A36</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E9:87_1_2_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:87"><span class="lab"><font color="Green" title="E70">A8</font></span></a>, <a class="txt" href="#E14:87"><span class="lab"><font color="Green" title="E72">A10</font></span></a>, <a class="txt" href="#E15:87"><span class="lab"><font color="Green" title="E73">A11</font></span></a>, <a class="txt" href="#E5:87_1_2_1_4"><span class="lab"><font color="Green" title="E83">A37</font></span></a>, <a class="ref" href="rcomp_3.html#T30" target="_self" title="RCOMP_3:th.30">Th42</a></span>;<br/><span class="kw">hence </span><a NAME="E10:87_1_2_1_4"/>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:87"><span class="lab"><font color="Green" title="E70">A8</font></span></a>, <a class="txt" href="#E14:87"><span class="lab"><font color="Green" title="E72">A10</font></span></a>, <a class="txt" href="#E7:87_1_2_1_4"><span class="lab"><font color="Green" title="E84">A38</font></span></a>, <a class="ref" href="xxreal_2.html#D12" title="XXREAL_2:def.12">XXREAL_2:def 12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
