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



<span class="kw">assume </span><a NAME="E1:10"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</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">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">W</font> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c2">p</font></span>)</span></span><br/>

<div><span class="lab"><font color="Green" title="E7">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">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"> ( ( <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> implies <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c2">p</font></span>)</span> ) &amp; ( <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c2">p</font></span>)</span> implies <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> ) )</span><br/><span class="kw">thus </span><a NAME="E1:10_1"/>
( <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> implies <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c2">p</font></span>)</span> )
  <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">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c2">p</font></span>)</span> implies <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:10_1_1"/><span class="lab"><font color="Green" title="E8">A3</font></span>: 
<font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</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">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c2">p</font></span>)</span></span><br/>

<div><span class="lab"><font color="Green" title="E9">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:10_1_1_1"/><span class="lab"><font color="Green" title="E10">A5</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="trees_1.html#NR3" title="TREES_1:NR.3">is_a_proper_prefix_of</a> <font color="Maroon" title="c3">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  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="c1">W</font> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Maroon" title="c3">q</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b1">r</font> )</span><br/><a NAME="E2:10_1_1_1"/><span class="lab"><font color="Green" title="E11">A6</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c3">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10_1_1_1"><span class="lab"><font color="Green" title="E10">A5</font></span></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c4">r</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E4:10_1_1_1"/><span class="lab"><font color="Green" title="E12">A7</font></span>: 
<font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c4">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10_1_1_1"><span class="lab"><font color="Green" title="E11">A6</font></span></a>, <a class="ref" href="trees_1.html#T8" title="TREES_1:th.8">TREES_1:8</a></span>;<br/><a NAME="E5:10_1_1_1"/><span class="lab"><font color="Green" title="E13">A8</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:10_1_1_1"><span class="lab"><font color="Green" title="E12">A7</font></span></a>, <a class="ref" href="finseq_1.html#T43" title="FINSEQ_1:th.43">FINSEQ_1:43</a></span>;<br/><a NAME="E6:10_1_1_1"/><span class="lab"><font color="Green" title="E14">A9</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:10_1_1_1"><span class="lab"><font color="Green" title="E13">A8</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c5">r</font> = <font color="Maroon" title="c4">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="#E6:10_1_1_1"><span class="lab"><font color="Green" title="E14">A9</font></span></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c6">r</font> = <font color="Maroon" title="c5">r</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="c6">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Maroon" title="c3">q</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c6">r</font> )</span><br/><span class="kw">thus </span><a NAME="E8:10_1_1_1"/>
( <font color="Maroon" title="c6">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Maroon" title="c3">q</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c6">r</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E1:10_1_1"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:10_1_1_1"><span class="lab"><font color="Green" title="E12">A7</font></span></a>, <a class="ref" href="trees_1.html#D9" title="TREES_1:def.9">TREES_1:def 9</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="E3:10_1_1"/>
<font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c2">p</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E1:10_1_1"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E2:10_1_1"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="ref" href="trees_1.html#D12" title="TREES_1:def.12">TREES_1:def 12</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">assume </span><span class="kw">that </span><br/><a NAME="E2:10_1"/><span class="lab"><font color="Green" title="E8">A10</font></span>: 
<font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c2">p</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:10_1"/><span class="lab"><font color="Green" title="E9">A11</font></span>: 
not <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E4:10_1"/><span class="lab"><font color="Green" title="E10">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="c1">W</font> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Maroon" title="c3">q</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b1">r</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E2:10_1"><span class="lab"><font color="Green" title="E8">A10</font></span></a>, <a class="txt" href="#E3:10_1"><span class="lab"><font color="Green" title="E9">A11</font></span></a>, <a class="ref" href="trees_1.html#D12" title="TREES_1:def.12">TREES_1:def 12</a></span>;<br/><span class="kw">thus </span><a NAME="E5:10_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E3:10_1"><span class="lab"><font color="Green" title="E9">A11</font></span></a>, <a class="txt" href="#E4:10_1"><span class="lab"><font color="Green" title="E10">A12</font></span></a>, <a class="ref" href="trees_1.html#D9" title="TREES_1:def.9">TREES_1:def 9</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="E3:10"/>
<font color="Maroon" title="c1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">W</font> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c2">p</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="ref" href="trees_2.html#T7" target="_self" title="TREES_2:th.7">Th7</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>
