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<span class="kw">set </span><font color="Maroon" title="c4">D</font> =  <a href="relset_1.html#K2" title="RELSET_1:func.2">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"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">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"/><span class="kw">then </span>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T137" title="GLIB_001:th.137">GLIB_001:137</a></span>;<br/><a NAME="E3:35_1_1_1"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D17" title="GLIB_001:def.17">GLIB_001:def 17</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:35_1_1_1"/><span class="lab"><font color="Green" title="E29">A2</font></span>: 
<font color="Maroon" title="c5">y</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> <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c6">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:35_1_1_1"/><span class="lab"><font color="Green" title="E30">A3</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 <span class="kw">and </span><br/><a NAME="E8:35_1_1_1"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1_1"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E9:35_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#K1" title="RELSET_1:func.1">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E7:35_1_1_1"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="ref" href="glib_005.html#D15" target="_self" title="GLIB_005:def.15">Def15</a></span>;<br/><a NAME="E10:35_1_1_1"/><span class="kw">then </span>
<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="funct_1.html#K1" title="FUNCT_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#K2" title="RELSET_1:func.2">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">D</font> =  <a href="relset_1.html#K2" title="RELSET_1:func.2">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/><span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <span class="kw">-&gt; </span>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> = $1;<br/><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E13:35_1_1_1"/><span class="lab"><font color="Green" title="E31">A4</font></span>: 
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>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="graph_5.html#S2" title="GRAPH_5:sch.2">GRAPH_5:sch 2</a>();<br/></span><span class="kw">reconsider </span><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/><span class="kw">take </span><font color="Maroon" title="c10">IT</font> = <font color="Maroon" title="c9">IT</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">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/><span class="kw">thus </span><a NAME="E15:35_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#K2" title="RELSET_1:func.2">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K2" title="RELSET_1:func.2">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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K2" title="RELSET_1:func.2">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/><span class="kw">assume </span><a NAME="E16:35_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#K2" title="RELSET_1:func.2">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">hence </span><a NAME="E17:35_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:35_1_1_1"><span class="lab"><font color="Green" title="E31">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><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#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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