<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">m</font>, <font color="Maroon" title="c2">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"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font>, <font color="Olive" title="b3">h</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>   st <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font> <a href="graphsp.html#R2" title="GRAPHSP:pred.2">equal_at</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">n</font> &amp; <font color="Olive" title="b2">g</font>,<font color="Olive" title="b3">h</font> <a href="graphsp.html#R2" title="GRAPHSP:pred.2">equal_at</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">n</font> holds <br/><font color="Olive" title="b1">f</font>,<font color="Olive" title="b3">h</font> <a href="graphsp.html#R2" title="GRAPHSP:pred.2">equal_at</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">n</font></span><br/><b>let </b><font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c4">g</font>, <font color="Maroon" title="c5">h</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font> <a href="graphsp.html#R2" title="GRAPHSP:pred.2">equal_at</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c5">h</font> <a href="graphsp.html#R2" title="GRAPHSP:pred.2">equal_at</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">n</font> implies <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">h</font> <a href="graphsp.html#R2" title="GRAPHSP:pred.2">equal_at</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">n</font> )</span><br/>









<b>assume </b><a NAME="E1:76"/><i><font color="Green" title="E64">A1</font></i>: 
( <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font> <a href="graphsp.html#R2" title="GRAPHSP:pred.2">equal_at</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c5">h</font> <a href="graphsp.html#R2" title="GRAPHSP:pred.2">equal_at</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">n</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">h</font> <a href="graphsp.html#R2" title="GRAPHSP:pred.2">equal_at</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">n</font></span><br/>

<a NAME="E2:76"/><b>then </b><i><font color="Green" title="E65">A2</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> &amp; ( for <font color="Olive" title="b1">k</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>   st <font color="Olive" title="b1">k</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="c3">f</font> &amp; <font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> holds <br/><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">k</font> ) )
 
<b>by </b><i><a class="ref" href="graphsp.html#D16" target="_self" title="GRAPHSP:def.16">Def16</a></i>;<br/>
<a NAME="E3:76"/><i><font color="Green" title="E66">A3</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">h</font> &amp; ( for <font color="Olive" title="b1">k</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>   st <font color="Olive" title="b1">k</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="c4">g</font> &amp; <font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> holds <br/><font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">k</font> ) )
 
<b>by </b><i><a class="txt" href="#E1:76"><i><font color="Green" title="E64">A1</font></i></a>, <a class="ref" href="graphsp.html#D16" target="_self" title="GRAPHSP:def.16">Def16</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">k</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" title="c6">k</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="c3">f</font> &amp; <font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> &amp; <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> implies <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font> )</span><br/><b>assume </b><a NAME="E1:76_1"/><i><font color="Green" title="E67">A4</font></i>: 
( <font color="Maroon" title="c6">k</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="c3">f</font> &amp; <font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> &amp; <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span><br/><b>hence </b><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font> = 
<font color="Maroon" title="c4">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font>
<b>by </b><i><a class="txt" href="#E1:76"><i><font color="Green" title="E64">A1</font></i></a>, <a class="ref" href="graphsp.html#D16" target="_self" title="GRAPHSP:def.16">Def16</a></i>
<br/>.= 
<font color="Maroon" title="c5">h</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font>
<b>by </b><i><a class="txt" href="#E1:76"><i><font color="Green" title="E64">A1</font></i></a>, <a class="txt" href="#E2:76"><i><font color="Green" title="E65">A2</font></i></a>, <a class="txt" href="#E1:76_1"><i><font color="Green" title="E67">A4</font></i></a>, <a class="ref" href="graphsp.html#D16" target="_self" title="GRAPHSP:def.16">Def16</a></i>
;<br/> <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="E5:76"/>
<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">h</font> <a href="graphsp.html#R2" title="GRAPHSP:pred.2">equal_at</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="#E2:76"><i><font color="Green" title="E65">A2</font></i></a>, <a class="txt" href="#E3:76"><i><font color="Green" title="E66">A3</font></i></a>, <a class="ref" href="graphsp.html#D16" target="_self" title="GRAPHSP:def.16">Def16</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>
