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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">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">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="b3">T</font> being    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Olive" title="b2">P</font><br/>  for <font color="Olive" title="b4">t</font> being    <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of <font color="Olive" title="b3">T</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"><span class="p2">(<span class="default"><font color="Olive" title="b3">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b4">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> holds <br/> for <font color="Olive" title="b5">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   holds <br/>(  not <font color="Olive" title="b5">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">i</font> or <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="Olive" title="b3">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b4">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Olive" title="b5">j</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> or  ex <font color="Olive" title="b6">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b6">k</font> &amp; <font color="Olive" title="b6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b5">j</font> &amp; <font color="Maroon" title="c3">A</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="Olive" title="b3">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b4">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Olive" title="b6">k</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">i</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">  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><br/>  for <font color="Olive" title="b3">t</font> being    <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of <font color="Olive" title="b2">T</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"><span class="p2">(<span class="default"><font color="Olive" title="b2">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c5">i</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> holds <br/> for <font color="Olive" title="b4">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   holds <br/>(  not <font color="Olive" title="b4">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">i</font> or <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="Olive" title="b2">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Olive" title="b4">j</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> or  ex <font color="Olive" title="b5">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b5">k</font> &amp; <font color="Olive" title="b5">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">j</font> &amp; <font color="Maroon" title="c3">A</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="Olive" title="b2">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Olive" title="b5">k</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> ) )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c6">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="c7">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="c7">P</font><br/>  for <font color="Olive" title="b2">t</font> being    <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of <font color="Olive" title="b1">T</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"><span class="p2">(<span class="default"><font color="Olive" title="b1">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c5">i</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> holds <br/> for <font color="Olive" title="b3">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   holds <br/>(  not <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">i</font> or <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="Olive" title="b1">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Olive" title="b3">j</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> or  ex <font color="Olive" title="b4">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">k</font> &amp; <font color="Olive" title="b4">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">j</font> &amp; <font color="Maroon" title="c3">A</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="Olive" title="b1">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Olive" title="b4">k</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c8">T</font> be    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c7">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">  for <font color="Olive" title="b1">t</font> being    <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of <font color="Maroon" title="c8">T</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"><span class="p2">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c5">i</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> holds <br/> for <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   holds <br/>(  not <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">i</font> or <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="c8">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Olive" title="b2">j</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> or  ex <font color="Olive" title="b3">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">k</font> &amp; <font color="Olive" title="b3">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> &amp; <font color="Maroon" title="c3">A</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="c8">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Olive" title="b3">k</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c9">t</font> be    <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of <font color="Maroon" title="c8">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="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="c8">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c5">i</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> implies  for <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   holds <br/>(  not <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">i</font> or <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="c8">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Olive" title="b1">j</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> or  ex <font color="Olive" title="b2">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">k</font> &amp; <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">j</font> &amp; <font color="Maroon" title="c3">A</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="c8">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Olive" title="b2">k</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> ) ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:66"/><span class="lab"><font color="Green" title="E34">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"><span class="p2">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c5">i</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</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">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   holds <br/>(  not <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">i</font> or <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="c8">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Olive" title="b1">j</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> or  ex <font color="Olive" title="b2">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">k</font> &amp; <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">j</font> &amp; <font color="Maroon" title="c3">A</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="c8">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Olive" title="b2">k</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> ) )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c10">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><a NAME="E3:66"/><span class="lab"><font color="Green" title="E35">A2</font></span>: 
<font color="Maroon" title="c10">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">i</font>
 <span class="kw">and </span><br/><a NAME="E4:66"/><span class="lab"><font color="Green" title="E36">A3</font></span>: 
 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"><span class="p2">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c10">j</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:66"/><span class="lab"><font color="Green" title="E37">A4</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">j</font> holds <br/> not <font color="Maroon" title="c3">A</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="c8">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</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"> contradiction</span><br/>

<a NAME="E6:66"/><span class="lab"><font color="Green" title="E38">A5</font></span>: 
<font color="Maroon" title="c10">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:66"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_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="E1:66_1"/>
( <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c10">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:66"><span class="lab"><font color="Green" title="E38">A5</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1"><span class="kw">suppose </span></a><a NAME="E1:66_1_1"/>
<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <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"> contradiction</span><br/><div class="add"><a NAME="E2:66_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c10">j</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> <span class="p1">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c5">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio4.html#D13" target="_self" onmouseover="rs('ltlaxio4/D13')" onmouseout="rh()">Def13</a></span>;<br/><a NAME="E3:66_1_1"/><span class="kw">then </span>
 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="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <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; <span class="p1">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <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="c8">T</font> )
 ;<br/><a NAME="E4:66_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c10">j</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ltlaxio4.html#K9" title="LTLAXIO4:func.9">compn</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c5">i</font></span>)</span></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></span>;<br/><span class="kw">hence </span><a NAME="E5:66_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio4.html#T24" target="_self" onmouseover="rs('ltlaxio4/T24')" onmouseout="rh()">Th24</a>, <a class="txt" href="#E1:66"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="txt" href="#E4:66"><span class="lab"><font color="Green" title="E36">A3</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_2"><span class="kw">suppose </span></a><a NAME="E1:66_1_2"/><span class="lab"><font color="Green" title="E39">A6</font></span>: 
<font color="Maroon" title="c10">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <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"> contradiction</span><br/><div class="add"><a NAME="E2:66_1_2"/>
<font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">j1</font> = <font color="Maroon" title="c10">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a>, <a class="txt" href="#E1:66_1_2"><span class="lab"><font color="Green" title="E39">A6</font></span></a>, <a class="ref" href="nat_1.html#T21" onmouseover="rs('nat_1/T21')" onmouseout="rh()">NAT_1:21</a></span>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span>( <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> $1 &amp; $1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">j</font> implies <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="c8">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> $1</span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> );<br/><a NAME="E4:66_1_2"/>
<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><a NAME="E5:66_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c10">j</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> <span class="p1">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c11">j1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio4.html#D13" target="_self" onmouseover="rs('ltlaxio4/D13')" onmouseout="rh()">Def13</a></span>;<br/><a NAME="E6:66_1_2"/><span class="kw">then </span>
 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="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c11">j1</font></span>)</span> <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; <span class="p1">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c11">j1</font></span>)</span> <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="c8">T</font> )
 ;<br/><a NAME="E7:66_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A7</font></span>: 
<font color="Maroon" title="c8">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c10">j</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ltlaxio4.html#K9" title="LTLAXIO4:func.9">compn</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c11">j1</font></span>)</span></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></span>;<br/><div><span class="lab"><font color="Green" title="E41">A8</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="66_1_2_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">k</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">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">k</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="c12">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/><span class="kw">set </span><font color="Maroon" title="c13">k1</font> = <font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/><span class="kw">assume </span><a NAME="E1:66_1_2_1"/><span class="lab"><font color="Green" title="E42">A9</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">k</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="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/><span class="kw">thus </span><a NAME="E2:66_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
  <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="66_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:66_1_2_1_1"/><span class="lab"><font color="Green" title="E43">A10</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">and </span><br/><a NAME="E2:66_1_2_1_1"/><span class="lab"><font color="Green" title="E44">A11</font></span>: 
<font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">j</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"><span class="p2">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span></span><br/>

<a NAME="E3:66_1_2_1_1"/><span class="lab"><font color="Green" title="E45">A12</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a>, <a class="txt" href="#E1:66_1_2_1_1"><span class="lab"><font color="Green" title="E43">A10</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_2_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="E1:66_1_2_1_1_1"/>
( <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">k</font> or <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">k</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:66_1_2_1_1"><span class="lab"><font color="Green" title="E45">A12</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_2_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:66_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E46">A13</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">k</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"><span class="p2">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c14">Pk1</font> = <font color="Maroon" title="c8">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>;<br/><a NAME="E2:66_1_2_1_1_1_1"/>
<font color="Maroon" title="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c12">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio4.html#D13" target="_self" onmouseover="rs('ltlaxio4/D13')" onmouseout="rh()">Def13</a></span>;<br/><a NAME="E3:66_1_2_1_1_1_1"/><span class="kw">then </span>
 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="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <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; <span class="p1">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <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="c8">T</font> )
 ;<br/><a NAME="E4:66_1_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ltlaxio4.html#K9" title="LTLAXIO4:func.9">compn</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c12">k</font></span>)</span></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></span>;<br/><a NAME="E5:66_1_2_1_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c3">A</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="c8">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> or <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="c8">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio4.html#T24" target="_self" onmouseover="rs('ltlaxio4/T24')" onmouseout="rh()">Th24</a>, <a class="txt" href="#E2:66_1_2_1_1"><span class="lab"><font color="Green" title="E44">A11</font></span></a>, <a class="ref" href="nat_1.html#T16" onmouseover="rs('nat_1/T16')" onmouseout="rh()">NAT_1:16</a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a>, <a class="txt" href="#E1:66_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E46">A13</font></span></a>, <a class="txt" href="#E1:66_1_2_1"><span class="lab"><font color="Green" title="E42">A9</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:66_1_2_1_1_1_1"/>
<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="c8">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:66"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="txt" href="#E1:66_1_2_1_1"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="txt" href="#E2:66_1_2_1_1"><span class="lab"><font color="Green" title="E44">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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_2_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:66_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E46">A14</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">k</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"><span class="p2">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c14">Pk1</font> = <font color="Maroon" title="c8">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>;<br/><a NAME="E2:66_1_2_1_1_1_2"/>
<font color="Maroon" title="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c5">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E46">A14</font></span></a>, <a class="ref" href="ltlaxio4.html#D13" target="_self" onmouseover="rs('ltlaxio4/D13')" onmouseout="rh()">Def13</a></span>;<br/><a NAME="E3:66_1_2_1_1_1_2"/><span class="kw">then </span>
 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="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <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; <span class="p1">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <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="c8">T</font> )
 ;<br/><a NAME="E4:66_1_2_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ltlaxio4.html#K9" title="LTLAXIO4:func.9">compn</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c5">i</font></span>)</span></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></span>;<br/><a NAME="E5:66_1_2_1_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c3">A</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="c8">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> or <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="c8">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio4.html#T24" target="_self" onmouseover="rs('ltlaxio4/T24')" onmouseout="rh()">Th24</a>, <a class="txt" href="#E1:66"><span class="lab"><font color="Green" title="E34">A1</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:66_1_2_1_1_1_2"/>
<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="c8">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:66"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="txt" href="#E1:66_1_2_1_1"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="txt" href="#E2:66_1_2_1_1"><span class="lab"><font color="Green" title="E44">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><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><a NAME="E9:66_1_2"/><span class="lab"><font color="Green" title="E42">A15</font></span>: 
( <font color="Maroon" title="c10">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">j</font> &amp; <font color="Maroon" title="c10">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T30" onmouseover="rs('xreal_1/T30')" onmouseout="rh()">XREAL_1:30</a>, <a class="txt" href="#E1:66_1_2"><span class="lab"><font color="Green" title="E39">A6</font></span></a>, <a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><a NAME="E10:66_1_2"/><span class="lab"><font color="Green" title="E43">A16</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 ;<br/><a NAME="E11:66_1_2"/>
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E10:66_1_2"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E8:66_1_2"><span class="lab"><font color="Green" title="E41">A8</font></span></a>);<br/></span><a NAME="E12:66_1_2"/><span class="kw">then </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="c8">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c11">j1</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:66_1_2"><span class="lab"><font color="Green" title="E42">A15</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E13:66_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio4.html#T24" target="_self" onmouseover="rs('ltlaxio4/T24')" onmouseout="rh()">Th24</a>, <a class="txt" href="#E7:66_1_2"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="txt" href="#E4:66"><span class="lab"><font color="Green" title="E36">A3</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
