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<span class="kw">let </span><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</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">P</font>, <font color="Olive" title="b2">Q</font> being   <a href="ltlaxio3.html#NM1" title="LTLAXIO3:NM.1">PNPair</a>  st <font color="Maroon" title="c3">A</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c4">B</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> <span class="p1">(<span class="default"><font color="Olive" title="b1">P</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> &amp; <font color="Olive" title="b2">Q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ltlaxio4.html#K9" title="LTLAXIO4:func.9">compn</a> <font color="Olive" title="b1">P</font> holds <br/> <a href="ltlaxio2.html#K1" title="LTLAXIO2:func.1">untn</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</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> <span class="p1">(<span class="default"><font color="Olive" title="b2">Q</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">P</font>, <font color="Maroon" title="c6">Q</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"> ( <font color="Maroon" title="c3">A</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c4">B</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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> &amp; <font color="Maroon" title="c6">Q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ltlaxio4.html#K9" title="LTLAXIO4:func.9">compn</a> <font color="Maroon" title="c5">P</font> implies  <a href="ltlaxio2.html#K1" title="LTLAXIO2:func.1">untn</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:47"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
<font color="Maroon" title="c3">A</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c4">B</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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:47"/><span class="lab"><font color="Green" title="E25">A2</font></span>: 
<font color="Maroon" title="c6">Q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ltlaxio4.html#K9" title="LTLAXIO4:func.9">compn</a> <font color="Maroon" title="c5">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">  <a href="ltlaxio2.html#K1" title="LTLAXIO2:func.1">untn</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c7">Q1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio3.html#K1" title="LTLAXIO3:func.1">**</a></span>)</span>,<span class="p2">(<span class="default"><a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio3.html#K1" title="LTLAXIO3:func.1">**</a></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span><span class="kw"> such that </span><br/><a NAME="E4:47"/>
<font color="Maroon" title="c6">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">Q1</font>
 <span class="kw">and </span><br/><a NAME="E5:47"/><span class="lab"><font color="Green" title="E26">A3</font></span>: 
<font color="Maroon" title="c7">Q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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">P</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47"><span class="lab"><font color="Green" title="E25">A2</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E7:47"/>
<font color="Maroon" title="c7">Q1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">x</font>
 <span class="kw">and </span><br/><a NAME="E8:47"/><span class="lab"><font color="Green" title="E27">A4</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="ltlaxio4.html#K5" title="LTLAXIO4:func.5">comp</a> <font color="Olive" title="b1">R</font></span>)</span> where <font color="Olive" title="b1">R</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio3.html#K1" title="LTLAXIO3:func.1">**</a></span>)</span>,<span class="p2">(<span class="default"><a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio3.html#K1" title="LTLAXIO3:func.1">**</a></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> : <font color="Olive" title="b1">R</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">P</font> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a>, <a class="txt" href="#E5:47"><span class="lab"><font color="Green" title="E26">A3</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">R</font> being   <a href="ltlaxio3.html#NM1" title="LTLAXIO3:NM.1">PNPair</a><span class="kw"> such that </span><br/><a NAME="E10:47"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ltlaxio4.html#K5" title="LTLAXIO4:func.5">comp</a> <font color="Maroon" title="c9">R</font>
 <span class="kw">and </span><br/><a NAME="E11:47"/><span class="lab"><font color="Green" title="E28">A5</font></span>: 
<font color="Maroon" title="c9">R</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">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:47"><span class="lab"><font color="Green" title="E27">A4</font></span></a></span>;<br/>
<a NAME="E12:47"/>
 ex <font color="Olive" title="b1">R1</font> being   <a href="ltlaxio3.html#NM1" title="LTLAXIO3:NM.1">PNPair</a> st <br/>( <font color="Olive" title="b1">R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">R</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">R1</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ltlaxio4.html#K7" title="LTLAXIO4:func.7">untn</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span></span>)</span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">R1</font> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ltlaxio4.html#K7" title="LTLAXIO4:func.7">untn</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:47"><span class="lab"><font color="Green" title="E28">A5</font></span></a></span>;<br/>
<a NAME="E13:47"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A6</font></span>: 
 <a href="ltlaxio2.html#K1" title="LTLAXIO2:func.1">untn</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</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> <span class="p1">(<span class="default"><font color="Maroon" title="c9">R</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E24">A1</font></span></a></span>;<br/>
<a NAME="E14:47"/>
<font color="Maroon" title="c6">Q</font> <a href="ltlaxio4.html#R1" title="LTLAXIO4:pred.1">is_completion_of</a> <font color="Maroon" title="c9">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:47"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E2:47"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="ltlaxio4.html#T19" target="_self" onmouseover="rs('ltlaxio4/T19')" onmouseout="rh()">Th19</a></span>;<br/>
<a NAME="E15:47"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">R</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>
 
;<br/>
<span class="kw">hence </span><a NAME="E16:47"/>
 <a href="ltlaxio2.html#K1" title="LTLAXIO2:func.1">untn</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:47"><span class="lab"><font color="Green" title="E29">A6</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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