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<b>set </b><font color="Maroon" title="c4">D</font> =  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="glib_005.html#K8" title="GLIB_005:func.8">.flowSeq</a> <font color="Maroon" title="c2">EL</font></span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:35_1_1_1"/>
not <font color="Maroon" title="c3">W</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">IT</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">IT</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="glib_005.html#K8" title="GLIB_005:func.8">.flowSeq</a> <font color="Maroon" title="c2">EL</font></span>)</span> &amp; ( for <font color="Olive" title="b2">k</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="glib_005.html#K8" title="GLIB_005:func.8">.flowSeq</a> <font color="Maroon" title="c2">EL</font></span>)</span> holds <br/><font color="Olive" title="b1">IT</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> ) )</span><br/><a NAME="E2:35_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">W</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="glib_001.html#T137" title="GLIB_001:th.137">GLIB_001:137</a></i>;<br/><a NAME="E3:35_1_1_1"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> </span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="glib_001.html#D17" title="GLIB_001:def.17">GLIB_001:def 17</a></i>;<br/><b>then consider </b><font color="Maroon" title="c5">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:35_1_1_1"/><i><font color="Green" title="E30">A1</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><b>consider </b><font color="Maroon" title="c6">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:35_1_1_1"/><i><font color="Green" title="E31">A2</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> </span>)</span> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> )
 <b>by </b><i><a class="txt" href="#E5:35_1_1_1"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E8:35_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="glib_005.html#K8" title="GLIB_005:func.8">.flowSeq</a> <font color="Maroon" title="c2">EL</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:35"><i><font color="Green" title="E29">A</font></i></a>, <a class="txt" href="#E7:35_1_1_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="glib_005.html#D15" target="_self" title="GLIB_005:def.15">Def18</a></i>;<br/><a NAME="E9:35_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="glib_005.html#K8" title="GLIB_005:func.8">.flowSeq</a> <font color="Maroon" title="c2">EL</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="glib_005.html#K8" title="GLIB_005:func.8">.flowSeq</a> <font color="Maroon" title="c2">EL</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c7">D</font> =  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="glib_005.html#K8" title="GLIB_005:func.8">.flowSeq</a> <font color="Maroon" title="c2">EL</font></span>)</span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <b>-&gt; </b>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> = $1;<br/><b>consider </b><font color="Maroon" title="c8">IT</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c7">D</font><b> such that </b><br/><a NAME="E12:35_1_1_1"/><i><font color="Green" title="E32">A3</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c7">D</font> holds  <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">IT</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font>)
 <b>from </b><i><a class="ref" href="graph_5.html#S2" title="GRAPH_5:sch.2">GRAPH_5:sch 2</a>();<br/></i><b>reconsider </b><font color="Maroon" title="c9">IT</font> = <font color="Maroon" title="c8">IT</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/><b>take </b><font color="Maroon" title="c10">IT</font> = <font color="Maroon" title="c9">IT</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">IT</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="glib_005.html#K8" title="GLIB_005:func.8">.flowSeq</a> <font color="Maroon" title="c2">EL</font></span>)</span> &amp; ( for <font color="Olive" title="b1">k</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="glib_005.html#K8" title="GLIB_005:func.8">.flowSeq</a> <font color="Maroon" title="c2">EL</font></span>)</span> holds <br/><font color="Maroon" title="c10">IT</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> ) )</span><br/><b>thus </b><a NAME="E14:35_1_1_1"/>
<font color="Maroon" title="c10">IT</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="glib_005.html#K8" title="GLIB_005:func.8">.flowSeq</a> <font color="Maroon" title="c2">EL</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">k</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="glib_005.html#K8" title="GLIB_005:func.8">.flowSeq</a> <font color="Maroon" title="c2">EL</font></span>)</span> holds <br/><font color="Maroon" title="c10">IT</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font></span><br/><b>let </b><font color="Maroon" title="c11">k</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"> ( <font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="glib_005.html#K8" title="GLIB_005:func.8">.flowSeq</a> <font color="Maroon" title="c2">EL</font></span>)</span> implies <font color="Maroon" title="c10">IT</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font> )</span><br/><b>assume </b><a NAME="E15:35_1_1_1"/>
<font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="glib_005.html#K8" title="GLIB_005:func.8">.flowSeq</a> <font color="Maroon" title="c2">EL</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">IT</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font></span><br/><b>hence </b><a NAME="E16:35_1_1_1"/>
<font color="Maroon" title="c10">IT</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font>
 <b>by </b><i><a class="txt" href="#E12:35_1_1_1"><i><font color="Green" title="E32">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:35_1_1"/>
( ( not <font color="Maroon" title="c3">W</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> implies  ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="glib_005.html#K8" title="GLIB_005:func.8">.flowSeq</a> <font color="Maroon" title="c2">EL</font></span>)</span> &amp; ( for <font color="Olive" title="b2">k</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="glib_005.html#K8" title="GLIB_005:func.8">.flowSeq</a> <font color="Maroon" title="c2">EL</font></span>)</span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> ) ) ) &amp; ( <font color="Maroon" title="c3">W</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> implies  ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 ; <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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