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<span class="kw">let </span><font color="Maroon" title="c3">P</font>, <font color="Maroon" title="c4">Q</font>, <font color="Maroon" title="c5">R</font> be   <a href="ltlaxio3.html#NM1" title="LTLAXIO3:NM.1">PNPair</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">T</font> being    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c3">P</font>  st <font color="Maroon" title="c5">R</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b1">T</font> &amp; <font color="Maroon" title="c4">Q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ltlaxio4.html#K8" title="LTLAXIO4:func.8">untn</a> <font color="Maroon" title="c5">R</font> holds <br/> <a href="ltlaxio4.html#K5" title="LTLAXIO4:func.5">comp</a> <font color="Maroon" title="c4">Q</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="ltlaxio4.html#K11" title="LTLAXIO4:func.11">rngr</a> <font color="Olive" title="b1">T</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">T</font> be    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c3">P</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">R</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">T</font> &amp; <font color="Maroon" title="c4">Q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ltlaxio4.html#K8" title="LTLAXIO4:func.8">untn</a> <font color="Maroon" title="c5">R</font> implies  <a href="ltlaxio4.html#K5" title="LTLAXIO4:func.5">comp</a> <font color="Maroon" title="c4">Q</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="ltlaxio4.html#K11" title="LTLAXIO4:func.11">rngr</a> <font color="Maroon" title="c6">T</font> )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:62"/><span class="lab"><font color="Green" title="E31">A1</font></span>: 
<font color="Maroon" title="c5">R</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">T</font>
 <span class="kw">and </span><br/><a NAME="E2:62"/><span class="lab"><font color="Green" title="E32">A2</font></span>: 
<font color="Maroon" title="c4">Q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ltlaxio4.html#K8" title="LTLAXIO4:func.8">untn</a> <font color="Maroon" title="c5">R</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="ltlaxio4.html#K5" title="LTLAXIO4:func.5">comp</a> <font color="Maroon" title="c4">Q</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="ltlaxio4.html#K11" title="LTLAXIO4:func.11">rngr</a> <font color="Maroon" title="c6">T</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c7">S</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ltlaxio4.html#K5" title="LTLAXIO4:func.5">comp</a> <font color="Maroon" title="c4">Q</font> or <font color="Maroon" title="c7">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ltlaxio4.html#K11" title="LTLAXIO4:func.11">rngr</a> <font color="Maroon" title="c6">T</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:62"/><span class="lab"><font color="Green" title="E33">A3</font></span>: 
<font color="Maroon" title="c7">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ltlaxio4.html#K5" title="LTLAXIO4:func.5">comp</a> <font color="Maroon" title="c4">Q</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ltlaxio4.html#K11" title="LTLAXIO4:func.11">rngr</a> <font color="Maroon" title="c6">T</font></span><br/>

<a NAME="E4:62"/>
 <a href="ltlaxio4.html#K5" title="LTLAXIO4:func.5">comp</a> <font color="Maroon" title="c4">Q</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="ltlaxio4.html#K6" title="LTLAXIO4:func.6">comp</a> <span class="p1">(<span class="default"><a href="ltlaxio4.html#K8" title="LTLAXIO4:func.8">untn</a> <font color="Maroon" title="c5">R</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio4.html#T14" target="_self" onmouseover="rs('ltlaxio4/T14')" onmouseout="rh()">Th14</a>, <a class="txt" href="#E2:62"><span class="lab"><font color="Green" title="E32">A2</font></span></a></span>;<br/>
<a NAME="E5:62"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A4</font></span>: 
<font color="Maroon" title="c7">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ltlaxio4.html#K9" title="LTLAXIO4:func.9">compn</a> <font color="Maroon" title="c5">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62"><span class="lab"><font color="Green" title="E33">A3</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">t</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E7:62"/><span class="lab"><font color="Green" title="E35">A5</font></span>: 
<font color="Maroon" title="c8">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c6">T</font>
 <span class="kw">and </span><br/><a NAME="E8:62"/><span class="lab"><font color="Green" title="E36">A6</font></span>: 
<font color="Maroon" title="c6">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">R</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a>, <a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E31">A1</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">t</font> = <font color="Maroon" title="c8">t</font> as    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c6">T</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:62"><span class="lab"><font color="Green" title="E35">A5</font></span></a></span>;<br/>
<a NAME="E10:62"/>
 <a href="trees_9.html#K2" title="TREES_9:func.2">succ</a> (<font color="Maroon" title="c6">T</font>,<font color="Maroon" title="c9">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="ltlaxio4.html#M2" title="LTLAXIO4:mode.2">Enumeration</a> of  <a href="ltlaxio4.html#K9" title="LTLAXIO4:func.9">compn</a> <font color="Maroon" title="c5">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio4.html#D11" target="_self" onmouseover="rs('ltlaxio4/D11')" onmouseout="rh()">Def11</a>, <a class="txt" href="#E8:62"><span class="lab"><font color="Green" title="E36">A6</font></span></a></span>;<br/>
<a NAME="E11:62"/><span class="kw">then </span>
<font color="Maroon" title="c7">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><a href="trees_9.html#K2" title="TREES_9:func.2">succ</a> (<font color="Maroon" title="c6">T</font>,<font color="Maroon" title="c9">t</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin3.html#D1" onmouseover="rs('rlaffin3/D1')" onmouseout="rh()">RLAFFIN3:def 1</a>, <a class="txt" href="#E5:62"><span class="lab"><font color="Green" title="E34">A4</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">t1</font> being    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c6">T</font><span class="kw"> such that </span><br/><a NAME="E13:62"/><span class="lab"><font color="Green" title="E37">A7</font></span>: 
<font color="Maroon" title="c7">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c10">t1</font>
 <span class="kw">and </span><br/><a NAME="E14:62"/><span class="lab"><font color="Green" title="E38">A8</font></span>: 
<font color="Maroon" title="c10">t1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c9">t</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_9.html#T42" onmouseover="rs('trees_9/T42')" onmouseout="rh()">TREES_9:42</a></span>;<br/>
<a NAME="E15:62"/>
 ex <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Maroon" title="c10">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">t</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Olive" title="b1">n</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Maroon" title="c9">t</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Olive" title="b1">n</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c6">T</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:62"><span class="lab"><font color="Green" title="E38">A8</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:62"/>
<font color="Maroon" title="c7">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ltlaxio4.html#K11" title="LTLAXIO4:func.11">rngr</a> <font color="Maroon" title="c6">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:62"><span class="lab"><font color="Green" title="E37">A7</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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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