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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> being   <a href="ltlaxio3.html#NV3" title="LTLAXIO3:NV.3">consistent</a>   <a href="ltlaxio3.html#V3" title="LTLAXIO3:attr.3">complete</a>  <a href="ltlaxio3.html#NM1" title="LTLAXIO3:NM.1">PNPair</a><br/>  for <font color="Olive" title="b2">T</font> being    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Olive" title="b1">P</font>  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; (  for <font color="Olive" title="b3">Q</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ltlaxio4.html#K11" title="LTLAXIO4:func.11">rngr</a> <font color="Olive" title="b2">T</font>  holds  not <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="b3">Q</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> ) holds <br/> for <font color="Olive" title="b3">Q</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ltlaxio4.html#K11" title="LTLAXIO4:func.11">rngr</a> <font color="Olive" title="b2">T</font> holds <br/> ( <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="b3">Q</font> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> &amp; <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="b3">Q</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c5">aub</font> = <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>;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">P</font> be   <a href="ltlaxio3.html#NV3" title="LTLAXIO3:NV.3">consistent</a>   <a href="ltlaxio3.html#V3" title="LTLAXIO3:attr.3">complete</a>  <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="c6">P</font>  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="Maroon" title="c6">P</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> &amp; (  for <font color="Olive" title="b2">Q</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ltlaxio4.html#K11" title="LTLAXIO4:func.11">rngr</a> <font color="Olive" title="b1">T</font>  holds  not <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> ) holds <br/> for <font color="Olive" title="b2">Q</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ltlaxio4.html#K11" title="LTLAXIO4:func.11">rngr</a> <font color="Olive" title="b1">T</font> holds <br/> ( <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#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> &amp; <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="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="c7">T</font> be    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">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="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="c6">P</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> &amp; (  for <font color="Olive" title="b1">Q</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ltlaxio4.html#K11" title="LTLAXIO4:func.11">rngr</a> <font color="Maroon" title="c7">T</font>  holds  not <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">Q</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> ) implies  for <font color="Olive" title="b1">Q</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ltlaxio4.html#K11" title="LTLAXIO4:func.11">rngr</a> <font color="Maroon" title="c7">T</font> holds <br/> ( <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">Q</font> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> &amp; <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">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="E1:67"/><span class="lab"><font color="Green" title="E35">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="c6">P</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:67"/><span class="lab"><font color="Green" title="E36">A2</font></span>: 
 for <font color="Olive" title="b1">Q</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ltlaxio4.html#K11" title="LTLAXIO4:func.11">rngr</a> <font color="Maroon" title="c7">T</font>  holds  not <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">Q</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</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">  for <font color="Olive" title="b1">Q</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ltlaxio4.html#K11" title="LTLAXIO4:func.11">rngr</a> <font color="Maroon" title="c7">T</font> holds <br/> ( <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">Q</font> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> &amp; <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">Q</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ] <span class="kw">means </span> for <font color="Olive" title="b1">t</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="c7">T</font>  st <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b1">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/>( <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"><span class="p2">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b1">t</font></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> &amp; <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"><span class="p2">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b1">t</font></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> );<br/>
<div><span class="lab"><font color="Green" title="E37">A3</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="67_1"><span class="kw">now </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">  for <font color="Olive" title="b1">t</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="c7">T</font><br/>  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">t</font>] &amp; <font color="Olive" title="b1">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="b2">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="c7">T</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">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="b2">n</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>]</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">t</font> be    <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="c7">T</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">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">t</font>] &amp; <font color="Maroon" title="c8">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="c7">T</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">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>]</span><br/><span class="kw">let </span><font color="Maroon" title="c9">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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">S<sub>1</sub></font>[<font color="Maroon" title="c8">t</font>] &amp; <font color="Maroon" title="c8">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="Maroon" title="c9">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="c7">T</font> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">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="Maroon" title="c9">n</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>] )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:67_1"/><span class="lab"><font color="Green" title="E38">A4</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">t</font>]
 <span class="kw">and </span><br/><a NAME="E2:67_1"/>
<font color="Maroon" title="c8">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="Maroon" title="c9">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="c7">T</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">S<sub>1</sub></font>[<font color="Maroon" title="c8">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="Maroon" title="c9">n</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>]</span><br/><span class="kw">thus </span><a NAME="E3:67_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">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="Maroon" title="c9">n</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</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><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="67_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">u</font> be    <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="c7">T</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="c10">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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="Maroon" title="c9">n</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Maroon" title="c10">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies ( <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"><span class="p2">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c10">u</font></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> &amp; <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"><span class="p2">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c10">u</font></span>)</span> <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="E1:67_1_1"/><span class="lab"><font color="Green" title="E39">A5</font></span>: 
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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="Maroon" title="c9">n</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E2:67_1_1"/>
<font color="Maroon" title="c10">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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="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"><span class="p2">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c10">u</font></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> &amp; <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"><span class="p2">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c10">u</font></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> )</span><br/>

<a NAME="E3:67_1_1"/><span class="lab"><font color="Green" title="E40">A6</font></span>: 
<font color="Maroon" title="c7">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c10">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ltlaxio4.html#K11" title="LTLAXIO4:func.11">rngr</a> <font color="Maroon" title="c7">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_1_1"><span class="lab"><font color="Green" title="E39">A5</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:67_1_1_1"/>
( <font color="Maroon" title="c8">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or  not <font color="Maroon" title="c8">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:67_1_1_1_1"/>
<font color="Maroon" title="c8">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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="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"><span class="p2">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c10">u</font></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> &amp; <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"><span class="p2">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c10">u</font></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> )</span><br/><div class="add"><a NAME="E2:67_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c8">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P</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></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">Tu</font> = <font color="Maroon" title="c7">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c10">u</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ltlaxio4.html#K9" title="LTLAXIO4:func.9">compn</a> <font color="Maroon" title="c6">P</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio4.html#T26" target="_self" onmouseover="rs('ltlaxio4/T26')" onmouseout="rh()">Th26</a>, <a class="txt" href="#E1:67_1_1"><span class="lab"><font color="Green" title="E39">A5</font></span></a></span>;<br/><a NAME="E4:67_1_1_1_1"/>
(  <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="c11">Tu</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">Tu</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="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c11">Tu</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio4.html#T22" target="_self" onmouseover="rs('ltlaxio4/T22')" onmouseout="rh()">Th22</a>, <a class="txt" href="#E1:67"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><a NAME="E5:67_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A7</font></span>: 
<font color="Maroon" title="c4">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c11">Tu</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio4.html#T8" target="_self" onmouseover="rs('ltlaxio4/T8')" onmouseout="rh()">Th8</a></span>;<br/><a NAME="E6:67_1_1_1_1"/>
 not <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="c11">Tu</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="#E2:67"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="txt" href="#E3:67_1_1"><span class="lab"><font color="Green" title="E40">A6</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:67_1_1_1_1"/>
( <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"><span class="p2">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c10">u</font></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> &amp; <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"><span class="p2">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c10">u</font></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio4.html#T25" target="_self" onmouseover="rs('ltlaxio4/T25')" onmouseout="rh()">Th25</a>, <a class="txt" href="#E1:67"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E5:67_1_1_1_1"><span class="lab"><font color="Green" title="E41">A7</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:67_1_1_1_2"/><span class="lab"><font color="Green" title="E41">A8</font></span>: 
 not <font color="Maroon" title="c8">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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="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"><span class="p2">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c10">u</font></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> &amp; <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"><span class="p2">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c10">u</font></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> )</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c11">Tu</font> = <font color="Maroon" title="c7">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c10">u</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ltlaxio4.html#K9" title="LTLAXIO4:func.9">compn</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c8">t</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio4.html#T26" target="_self" onmouseover="rs('ltlaxio4/T26')" onmouseout="rh()">Th26</a>, <a class="txt" href="#E1:67_1_1"><span class="lab"><font color="Green" title="E39">A5</font></span></a></span>;<br/><a NAME="E3:67_1_1_1_2"/>
(  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">Tu</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="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c11">Tu</font> &amp;  <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="c11">Tu</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a>, <a class="txt" href="#E1:67_1_1_1_2"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="txt" href="#E1:67_1"><span class="lab"><font color="Green" title="E38">A4</font></span></a>, <a class="ref" href="ltlaxio4.html#T22" target="_self" onmouseover="rs('ltlaxio4/T22')" onmouseout="rh()">Th22</a></span>;<br/><a NAME="E4:67_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A9</font></span>: 
<font color="Maroon" title="c4">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c11">Tu</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio4.html#T8" target="_self" onmouseover="rs('ltlaxio4/T8')" onmouseout="rh()">Th8</a></span>;<br/><a NAME="E5:67_1_1_1_2"/><span class="lab"><font color="Green" title="E43">A10</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"><span class="p2">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c8">t</font></span>)</span> <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="#E1:67_1_1_1_2"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="txt" href="#E1:67_1"><span class="lab"><font color="Green" title="E38">A4</font></span></a></span>;<br/><a NAME="E6:67_1_1_1_2"/>
 not <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="c11">Tu</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="#E2:67"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="txt" href="#E3:67_1_1"><span class="lab"><font color="Green" title="E40">A6</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:67_1_1_1_2"/>
( <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"><span class="p2">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c10">u</font></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> &amp; <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"><span class="p2">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c10">u</font></span>)</span> <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="#E5:67_1_1_1_2"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="txt" href="#E4:67_1_1_1_2"><span class="lab"><font color="Green" title="E42">A9</font></span></a>, <a class="ref" href="ltlaxio4.html#T25" target="_self" onmouseover="rs('ltlaxio4/T25')" onmouseout="rh()">Th25</a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">let </span><font color="Maroon" title="c8">Q</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ltlaxio4.html#K11" title="LTLAXIO4:func.11">rngr</a> <font color="Maroon" title="c7">T</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="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="c8">Q</font> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> &amp; <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="c8">Q</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> )</span><br/>

<a NAME="E4:67"/>
<font color="Maroon" title="c8">Q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ltlaxio4.html#K11" title="LTLAXIO4:func.11">rngr</a> <font color="Maroon" title="c7">T</font>
 
;<br/>
<a NAME="E5:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A11</font></span>: 
 ex <font color="Olive" title="b1">t</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="c7">T</font> st <br/>( <font color="Maroon" title="c8">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b1">t</font> &amp; <font color="Olive" title="b1">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 
;<br/>
<a NAME="E6:67"/><span class="lab"><font color="Green" title="E39">A12</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 
;<br/>
<a NAME="E7:67"/>
 for <font color="Olive" title="b1">t</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="c7">T</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">t</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="trees_2.html#S1" onmouseover="rs('trees_2/S1')" onmouseout="rh()">TREES_2:sch 1</a>(<a class="txt" href="#E6:67"><span class="lab"><font color="Green" title="E39">A12</font></span></a>, <a class="txt" href="#E3:67"><span class="lab"><font color="Green" title="E37">A3</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E8:67"/>
( <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="c8">Q</font> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> &amp; <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="c8">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="#E5:67"><span class="lab"><font color="Green" title="E38">A11</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/>


</div>
