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

<span class="kw">let </span><font color="Maroon" title="c1">p</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>, <font color="Olive" title="b2">T1</font> being  <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a>  st <font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">T</font> holds <br/><font color="Olive" title="b2">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">T</font> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <font color="Maroon" title="c1">p</font>,<font color="Olive" title="b2">T1</font></span>)</span> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c1">p</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">T</font>, <font color="Maroon" title="c3">T1</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"> ( <font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T</font> implies <font color="Maroon" title="c3">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">T</font> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">T1</font></span>)</span> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c1">p</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:80"/><span class="lab"><font color="Green" title="E33">A1</font></span>: 
<font color="Maroon" title="c1">p</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="c3">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">T</font> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">T1</font></span>)</span> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c1">p</font></span><br/>

<a NAME="E2:80"/><span class="lab"><font color="Green" title="E34">A2</font></span>: 
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T</font> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">T1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:80"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="trees_1.html#D12" target="_self" title="TREES_1:def.12">Def12</a></span>;<br/>
<span class="kw">thus </span><a NAME="E3:80"/>
<font color="Maroon" title="c3">T1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">T</font> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">T1</font></span>)</span> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c1">p</font>
  <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">T</font> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">T1</font></span>)</span> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c1">p</font> <a href="trees_1.html#NR1" title="TREES_1:NR.1">is_a_prefix_of</a> <font color="Maroon" title="c3">T1</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T1</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">T</font> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">T1</font></span>)</span> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c1">p</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:80_1"/><span class="lab"><font color="Green" title="E35">A3</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T1</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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">T</font> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">T1</font></span>)</span> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c1">p</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">q</font> = <font color="Maroon" title="c4">x</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="#E1:80_1"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="ref" href="trees_1.html#T44" target="_self" title="TREES_1:th.44">Th44</a></span>;<br/>
<a NAME="E3:80_1"/><span class="lab"><font color="Green" title="E36">A4</font></span>: 
<font color="Maroon" title="c1">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T</font> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">T1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:80"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="txt" href="#E1:80_1"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="ref" href="trees_1.html#D12" target="_self" title="TREES_1:def.12">Def12</a></span>;<br/>
<span class="kw">thus </span><a NAME="E4:80_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">T</font> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">T1</font></span>)</span> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:80"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="txt" href="#E3:80_1"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="ref" href="trees_1.html#D9" target="_self" title="TREES_1:def.9">Def9</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">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">T</font> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">T1</font></span>)</span> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c1">p</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E4:80"/><span class="lab"><font color="Green" title="E35">A5</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">T</font> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">T1</font></span>)</span> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c1">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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T1</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">q</font> = <font color="Maroon" title="c4">x</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="#E4:80"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="ref" href="trees_1.html#T44" target="_self" title="TREES_1:th.44">Th44</a></span>;<br/>
<a NAME="E6:80"/><span class="lab"><font color="Green" title="E36">A6</font></span>: 
<font color="Maroon" title="c1">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T</font> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">T1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:80"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="txt" href="#E4:80"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="ref" href="trees_1.html#D9" target="_self" title="TREES_1:def.9">Def9</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E37">A7</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="80_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:80_2"/>
<font color="Maroon" title="c1">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T</font>
 <span class="kw">and </span><br/><a NAME="E2:80_2"/><span class="lab"><font color="Green" title="E38">A8</font></span>: 
not <font color="Maroon" title="c1">p</font> <a href="trees_1.html#NR3" title="TREES_1:NR.3">is_a_proper_prefix_of</a> <font color="Maroon" title="c1">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">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="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T1</font></span><br/><a NAME="E3:80_2"/><span class="lab"><font color="Green" title="E39">A9</font></span>: 
<font color="Maroon" title="c1">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c1">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_1.html#T8" target="_self" title="TREES_1:th.8">Th8</a></span>;<br/><span class="lab"><font color="Green" title="E40">A10</font></span>: <font color="Maroon" title="c1">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">q</font> = 
<font color="Maroon" title="c1">p</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:80_2"><span class="lab"><font color="Green" title="E38">A8</font></span></a>, <a class="txt" href="#E3:80_2"><span class="lab"><font color="Green" title="E39">A9</font></span></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></span>
<br/>.= 
<font color="Maroon" title="c1">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</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="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></span>
;<br/><a NAME="E5:80_2"/><span class="lab"><font color="Green" title="E41">A11</font></span>: 
<font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:80_2"><span class="lab"><font color="Green" title="E40">A10</font></span></a>, <a class="ref" href="finseq_1.html#T46" title="FINSEQ_1:th.46">FINSEQ_1:46</a></span>;<br/><span class="kw">thus </span><a NAME="E6:80_2"/>
<font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:80_2"><span class="lab"><font color="Green" title="E41">A11</font></span></a>, <a class="ref" href="trees_1.html#T47" target="_self" title="TREES_1:th.47">Th47</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="E8:80"/><span class="lab"><font color="Green" title="E38">A12</font></span>: 
(  ex <font color="Olive" title="b1">r</font> being   <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>  st <br/>( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T1</font> &amp; <font color="Maroon" title="c1">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b1">r</font> ) implies <font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T46" title="FINSEQ_1:th.46">FINSEQ_1:46</a></span>;<br/>
<span class="kw">thus </span><a NAME="E9:80"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:80"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="txt" href="#E6:80"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="txt" href="#E7:80"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="txt" href="#E8:80"><span class="lab"><font color="Green" title="E38">A12</font></span></a>, <a class="ref" href="trees_1.html#D12" target="_self" title="TREES_1:def.12">Def12</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>
