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<span class="kw">let </span><font color="Maroon" title="c1">q</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">T</font> being  <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a><br/> for <font color="Olive" title="b2">r</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st <font color="Maroon" title="c1">q</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">T</font> holds <br/><font color="Maroon" title="c1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">T</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">T</font> be   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a>; <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">r</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st <font color="Maroon" title="c1">q</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T</font> holds <br/><font color="Maroon" title="c1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">r</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</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="c1">q</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T</font> implies <font color="Maroon" title="c1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:55"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
<font color="Maroon" title="c1">q</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T</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="c1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">p</font> = <font color="Maroon" title="c1">q</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c3">r</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="trees_1.html#T44" target="_self" title="TREES_1:th.44">Th44</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">s</font> = <font color="Maroon" title="c4">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">q</font></span>)</span></span>)</span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T23" title="FINSEQ_1:th.23">FINSEQ_1:23</a></span>;<br/>
<a NAME="E4:55"/><span class="lab"><font color="Green" title="E24">A2</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">r</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>;<br/>
<a NAME="E5:55"/><span class="lab"><font color="Green" title="E25">A3</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:55"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E6:55"/><span class="lab"><font color="Green" title="E26">A4</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:55"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="ref" href="finseq_1.html#T21" title="FINSEQ_1:th.21">FINSEQ_1:21</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E27">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font> &amp; <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">q</font> implies <font color="Maroon" title="c1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> )</span><br/><span class="kw">assume </span><a NAME="E1:55_1"/><span class="lab"><font color="Green" title="E28">A6</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font> &amp; <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">q</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="c1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span><br/><a NAME="E2:55_1"/><span class="lab"><font color="Green" title="E29">A7</font></span>: 
<font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55_1"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E3:55_1"/><span class="lab"><font color="Green" title="E30">A8</font></span>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">q</font></span>)</span> <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="c1">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E4:55_1"/><span class="lab"><font color="Green" title="E31">A9</font></span>: 
<font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:55_1"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="txt" href="#E3:55_1"><span class="lab"><font color="Green" title="E30">A8</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><span class="kw">thus </span><a NAME="E5:55_1"/>
<font color="Maroon" title="c1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:55_1"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="txt" href="#E4:55_1"><span class="lab"><font color="Green" title="E31">A9</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</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>
<a NAME="E8:55"/><span class="lab"><font color="Green" title="E28">A10</font></span>: 
<font color="Maroon" title="c1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:55"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="txt" href="#E7:55"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></span>;<br/>
<a NAME="E9:55"/><span class="lab"><font color="Green" title="E29">A11</font></span>: 
<font color="Maroon" title="c1">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c4">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:55"><span class="lab"><font color="Green" title="E28">A10</font></span></a>, <a class="ref" href="trees_1.html#D1" target="_self" title="TREES_1:def.1">Def1</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E30">A12</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:55_2"/><span class="lab"><font color="Green" title="E31">A13</font></span>: 
<font color="Maroon" title="c1">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p</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="c1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T</font></span><br/><a NAME="E2:55_2"/><span class="lab"><font color="Green" title="E32">A14</font></span>: 
<font color="Maroon" title="c1">q</font> <a href="trees_1.html#NR3" title="TREES_1:NR.3">is_a_proper_prefix_of</a> <font color="Maroon" title="c4">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:55"><span class="lab"><font color="Green" title="E29">A11</font></span></a>, <a class="txt" href="#E1:55_2"><span class="lab"><font color="Green" title="E31">A13</font></span></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></span>;<br/><a NAME="E3:55_2"/><span class="lab"><font color="Green" title="E33">A15</font></span>: 
<font color="Maroon" title="c1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_1.html#K1" title="TREES_1:func.1">ProperPrefixes</a> <font color="Maroon" title="c4">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:55_2"><span class="lab"><font color="Green" title="E32">A14</font></span></a>, <a class="ref" href="trees_1.html#D4" target="_self" title="TREES_1:def.4">Def4</a></span>;<br/><a NAME="E4:55_2"/><span class="lab"><font color="Green" title="E34">A16</font></span>: 
 <a href="trees_1.html#K1" title="TREES_1:func.1">ProperPrefixes</a> <font color="Maroon" title="c4">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="trees_1.html#D5" target="_self" title="TREES_1:def.5">Def5</a></span>;<br/><span class="kw">thus </span><a NAME="E5:55_2"/>
<font color="Maroon" title="c1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:55_2"><span class="lab"><font color="Green" title="E33">A15</font></span></a>, <a class="txt" href="#E4:55_2"><span class="lab"><font color="Green" title="E34">A16</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">thus </span><a NAME="E11:55"/>
<font color="Maroon" title="c1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E10:55"><span class="lab"><font color="Green" title="E30">A12</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/>


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