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<b>let </b><font color="Maroon" title="c4">IT1</font>, <font color="Maroon" title="c5">IT2</font> be    <a href="glib_005.html#M3" title="GLIB_005:mode.3">FF:ELabelingSeq</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">IT1</font> <a href="glib_005.html#K12" title="GLIB_005:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c4">IT1</font> <a href="glib_005.html#K12" title="GLIB_005:func.12">.</a> <span class="p1">(<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> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="glib_005.html#K11" title="GLIB_005:func.11">FF:Step</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">IT1</font> <a href="glib_005.html#K12" title="GLIB_005:func.12">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font> ) &amp; <font color="Maroon" title="c5">IT2</font> <a href="glib_005.html#K12" title="GLIB_005:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c5">IT2</font> <a href="glib_005.html#K12" title="GLIB_005:func.12">.</a> <span class="p1">(<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> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="glib_005.html#K11" title="GLIB_005:func.11">FF:Step</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">IT2</font> <a href="glib_005.html#K12" title="GLIB_005:func.12">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font> ) implies <font color="Maroon" title="c4">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">IT2</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:40_1_2"/><i><font color="Green" title="E33">A15</font></i>: 
( <font color="Maroon" title="c4">IT1</font> <a href="glib_005.html#K12" title="GLIB_005:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c4">IT1</font> <a href="glib_005.html#K12" title="GLIB_005:func.12">.</a> <span class="p1">(<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> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="glib_005.html#K11" title="GLIB_005:func.11">FF:Step</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">IT1</font> <a href="glib_005.html#K12" title="GLIB_005:func.12">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font> ) )
 <b>and </b><br/><a NAME="E2:40_1_2"/><i><font color="Green" title="E34">A16</font></i>: 
( <font color="Maroon" title="c5">IT2</font> <a href="glib_005.html#K12" title="GLIB_005:func.12">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c5">IT2</font> <a href="glib_005.html#K12" title="GLIB_005:func.12">.</a> <span class="p1">(<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> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="glib_005.html#K11" title="GLIB_005:func.11">FF:Step</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">IT2</font> <a href="glib_005.html#K12" title="GLIB_005:func.12">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">IT2</font></span><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b><font color="Maroon" title="c4">IT1</font> <a href="glib_005.html#K12" title="GLIB_005:func.12">.</a> $1 <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c5">IT2</font> <a href="glib_005.html#K12" title="GLIB_005:func.12">.</a> $1;<br/>
<a NAME="E3:40_1_2"/><i><font color="Green" title="E35">A17</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="txt" href="#E1:40_1_2"><i><font color="Green" title="E33">A15</font></i></a>, <a class="txt" href="#E2:40_1_2"><i><font color="Green" title="E34">A16</font></i></a></i>;<br/>
<div><i><font color="Green" title="E36">A18</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">n</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/><b>assume </b><a NAME="E1:40_1_2_1"/><i><font color="Green" title="E37">A19</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><a NAME="E2:40_1_2_1"/><i><font color="Green" title="E38">A21</font></i>: 
<font color="Maroon" title="c5">IT2</font> <a href="glib_005.html#K12" title="GLIB_005:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="glib_005.html#K11" title="GLIB_005:func.11">FF:Step</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">IT2</font> <a href="glib_005.html#K12" title="GLIB_005:func.12">.</a> <font color="Maroon" title="c6">n</font></span>)</span>,<font color="Maroon" title="c2">source</font>,<font color="Maroon" title="c3">sink</font>
 <b>by </b><i><a class="txt" href="#E2:40_1_2"><i><font color="Green" title="E34">A16</font></i></a></i>;<br/><b>thus </b><a NAME="E3:40_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <b>by </b><i><a class="txt" href="#E1:40_1_2_1"><i><font color="Green" title="E37">A19</font></i></a>, <a class="txt" href="#E1:40_1_2"><i><font color="Green" title="E33">A15</font></i></a>, <a class="txt" href="#E2:40_1_2_1"><i><font color="Green" title="E38">A21</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>
<a NAME="E5:40_1_2"/>
for <font color="Olive" title="b1">n</font> being   <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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E3:40_1_2"><i><font color="Green" title="E35">A17</font></i></a>, <a class="txt" href="#E4:40_1_2"><i><font color="Green" title="E36">A18</font></i></a>);<br/></i>
<a NAME="E6:40_1_2"/><b>then </b>
for <font color="Olive" title="b1">n</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon" title="c4">IT1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">IT2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>
 
;<br/>
<b>hence </b><a NAME="E7:40_1_2"/>
<font color="Maroon" title="c4">IT1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c5">IT2</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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