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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:76"/><span class="lab"><font color="Green" title="E64">A1</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:76"/><span class="lab"><font color="Green" title="E65">A2</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E3:76"/><span class="lab"><font color="Green" title="E66">A3</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:76"><span class="lab"><font color="Green" title="E64">A1</font></span></a>, <a class="ref" href="graphsp.html#D16" target="_self" title="GRAPHSP:def.16">Def16</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E67">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">assume </span><a NAME="E1:76_1"/><span class="lab"><font color="Green" title="E68">A5</font></span>: 
( <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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">hence </span><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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:76"><span class="lab"><font color="Green" title="E64">A1</font></span></a>, <a class="ref" href="graphsp.html#D16" target="_self" title="GRAPHSP:def.16">Def16</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:76"><span class="lab"><font color="Green" title="E65">A2</font></span></a>, <a class="txt" href="#E3:76"><span class="lab"><font color="Green" title="E66">A3</font></span></a>, <a class="txt" href="#E1:76_1"><span class="lab"><font color="Green" title="E68">A5</font></span></a>, <a class="ref" href="graphsp.html#D16" target="_self" title="GRAPHSP:def.16">Def16</a></span>
;<br/> <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>
<a NAME="E5:76"/>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:76"><span class="lab"><font color="Green" title="E65">A2</font></span></a>, <a class="ref" href="graphsp.html#D16" target="_self" title="GRAPHSP:def.16">Def16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:76"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:76"><span class="lab"><font color="Green" title="E66">A3</font></span></a>, <a class="txt" href="#E4:76"><span class="lab"><font color="Green" title="E67">A4</font></span></a>, <a class="ref" href="graphsp.html#D16" target="_self" title="GRAPHSP:def.16">Def16</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/>


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