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<span class="kw">consider </span><font color="Maroon" title="c4">Q</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E2:71_1_1"/><span class="lab"><font color="Green" title="E37">A1</font></span>: 
<font color="Maroon" title="c4">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ltlaxio4.html#K5" title="LTLAXIO4:func.5">comp</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" onmouseover="rs('xboole_0/D1')" onmouseout="rh()">XBOOLE_0:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">Pg</font> being   <a href="ltlaxio3.html#NV3" title="LTLAXIO3:NV.3">consistent</a>  <a href="ltlaxio3.html#NM1" title="LTLAXIO3:NM.1">PNPair</a><span class="kw"> such that </span><br/><a NAME="E4:71_1_1"/>
<font color="Maroon" title="c5">Pg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Q</font>
 <span class="kw">and </span><br/><a NAME="E5:71_1_1"/><span class="lab"><font color="Green" title="E38">A2</font></span>: 
<font color="Maroon" title="c5">Pg</font> <a href="ltlaxio4.html#R1" title="LTLAXIO4:pred.1">is_completion_of</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71_1_1"><span class="lab"><font color="Green" title="E37">A1</font></span></a></span>;<br/>
<a NAME="E6:71_1_1"/><span class="lab"><font color="Green" title="E39">A3</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Pg</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="#E5:71_1_1"><span class="lab"><font color="Green" title="E38">A2</font></span></a></span>;<br/>
<a NAME="E7:71_1_1"/><span class="lab"><font color="Green" title="E40">A4</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Pg</font> <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="#E5:71_1_1"><span class="lab"><font color="Green" title="E38">A2</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">Pg</font> = <font color="Maroon" title="c5">Pg</font> as   <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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:71_1_1"><span class="lab"><font color="Green" title="E38">A2</font></span></a>, <a class="ref" href="ltlaxio4.html#T13" target="_self" onmouseover="rs('ltlaxio4/T13')" onmouseout="rh()">Th13</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">T</font> =  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">t</font> =  the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <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> ,   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <a href="ltlaxio1.html#K10" title="LTLAXIO1:func.10">props</a>] <span class="kw">means </span><font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is    <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>  : ( <font color="Olive" title="b1">p</font> is  <a href="hilbert2.html#V3" title="HILBERT2:attr.3">simple</a>  &amp; <font color="Olive" title="b1">p</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"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c1">P</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> ) </span> } </span> ;<br/>
<div><span class="lab"><font color="Green" title="E41">A5</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="71_1_1_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">x</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>   ex <font color="Olive" title="b2">y</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <a href="ltlaxio1.html#K10" title="LTLAXIO1:func.10">props</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <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> ; <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">  ex <font color="Olive" title="b1">y</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <a href="ltlaxio1.html#K10" title="LTLAXIO1:func.10">props</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><span class="kw">set </span><font color="Maroon" title="c10">y</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is    <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>  : ( <font color="Olive" title="b1">p</font> is  <a href="hilbert2.html#V3" title="HILBERT2:attr.3">simple</a>  &amp; <font color="Olive" title="b1">p</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"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> ) </span> } </span> ;<br/><a NAME="E1:71_1_1_1"/>
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is    <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>  : ( <font color="Olive" title="b1">p</font> is  <a href="hilbert2.html#V3" title="HILBERT2:attr.3">simple</a>  &amp; <font color="Olive" title="b1">p</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"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> ) </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ltlaxio1.html#K10" title="LTLAXIO1:func.10">props</a> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="71_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c11">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is    <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>  : ( <font color="Olive" title="b1">p</font> is  <a href="hilbert2.html#V3" title="HILBERT2:attr.3">simple</a>  &amp; <font color="Olive" title="b1">p</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"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> ) </span> } </span>  or <font color="Maroon" title="c11">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ltlaxio1.html#K10" title="LTLAXIO1:func.10">props</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:71_1_1_1_1"/>
<font color="Maroon" title="c11">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is    <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>  : ( <font color="Olive" title="b1">p</font> is  <a href="hilbert2.html#V3" title="HILBERT2:attr.3">simple</a>  &amp; <font color="Olive" title="b1">p</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"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> ) </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"> <font color="Maroon" title="c11">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ltlaxio1.html#K10" title="LTLAXIO1:func.10">props</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">p</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E3:71_1_1_1_1"/><span class="lab"><font color="Green" title="E42">A6</font></span>: 
<font color="Maroon" title="c12">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z</font>
 <span class="kw">and </span><br/><a NAME="E4:71_1_1_1_1"/><span class="lab"><font color="Green" title="E43">A7</font></span>: 
<font color="Maroon" title="c12">p</font> is  <a href="hilbert2.html#V3" title="HILBERT2:attr.3">simple</a> 
 <span class="kw">and </span><br/><a NAME="E5:71_1_1_1_1"/>
<font color="Maroon" title="c12">p</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"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>
 ;<br/>
<a NAME="E6:71_1_1_1_1"/>
 ex <font color="Olive" title="b1">n</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="c12">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilbert2.html#K1" title="HILBERT2:func.1">prop</a> <font color="Olive" title="b1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:71_1_1_1_1"><span class="lab"><font color="Green" title="E43">A7</font></span></a>, <a class="ref" href="hilbert2.html#D8" onmouseover="rs('hilbert2/D8')" onmouseout="rh()">HILBERT2:def 8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:71_1_1_1_1"/>
<font color="Maroon" title="c11">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ltlaxio1.html#K10" title="LTLAXIO1:func.10">props</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#D10" onmouseover="rs('ltlaxio1/D10')" onmouseout="rh()">LTLAXIO1:def 10</a>, <a class="txt" href="#E3:71_1_1_1_1"><span class="lab"><font color="Green" title="E42">A6</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">y</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is    <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>  : ( <font color="Olive" title="b1">p</font> is  <a href="hilbert2.html#V3" title="HILBERT2:attr.3">simple</a>  &amp; <font color="Olive" title="b1">p</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"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> ) </span> } </span>  as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <a href="ltlaxio1.html#K10" title="LTLAXIO1:func.10">props</a> ;<br/><a NAME="E3:71_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c11">y</font>]
 ;<br/><span class="kw">hence </span><a NAME="E4:71_1_1_1"/>
 ex <font color="Olive" title="b1">y</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <a href="ltlaxio1.html#K10" title="LTLAXIO1:func.10">props</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</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"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c9">M</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <a href="ltlaxio1.html#K10" title="LTLAXIO1:func.10">props</a></span>)</span><span class="kw"> such that </span><br/><a NAME="E11:71_1_1"/><span class="lab"><font color="Green" title="E42">A8</font></span>: 
 for <font color="Olive" title="b1">n</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  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c9">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" onmouseover="rs('funct_2/S3')" onmouseout="rh()">FUNCT_2:sch 3</a>(<a class="txt" href="#E9:71_1_1"><span class="lab"><font color="Green" title="E41">A5</font></span></a>);<br/></span>
<span class="kw">set </span><font color="Maroon" title="c10">s</font> =  <a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <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> ] <span class="kw">means </span> for <font color="Olive" title="b1">i</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="c1">P</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <span class="p1">(<span class="default"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p2">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> holds <br/>( <span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Maroon" title="c1">P</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 iff <font color="Maroon" title="c1">P</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"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</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#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> );<br/>
<div><span class="lab"><font color="Green" title="E43">A9</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="71_1_1_2"><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">i</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> <br/>  for <font color="Olive" title="b2">p</font> being    <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>   st <font color="Olive" title="b2">p</font> is  <a href="hilbert2.html#V3" title="HILBERT2:attr.3">simple</a>  holds <br/>( ( <span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Olive" title="b2">p</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"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</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#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> ) &amp; ( <font color="Olive" title="b2">p</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"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</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#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> implies <span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">i</font> be    <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> ; <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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a>   st <font color="Olive" title="b1">p</font> is  <a href="hilbert2.html#V3" title="HILBERT2:attr.3">simple</a>  holds <br/>( ( <span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">i</font>,<font color="Olive" title="b1">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Olive" title="b1">p</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"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c11">i</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> ) &amp; ( <font color="Olive" title="b1">p</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"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c11">i</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> implies <span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">i</font>,<font color="Olive" title="b1">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c12">p</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"> ( <font color="Maroon" title="c12">p</font> is  <a href="hilbert2.html#V3" title="HILBERT2:attr.3">simple</a>  implies ( ( <span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Maroon" title="c12">p</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"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c11">i</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> ) &amp; ( <font color="Maroon" title="c12">p</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"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c11">i</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> implies <span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) ) )</span><br/><span class="kw">assume </span><a NAME="E1:71_1_1_2"/><span class="lab"><font color="Green" title="E44">A10</font></span>: 
<font color="Maroon" title="c12">p</font> is  <a href="hilbert2.html#V3" title="HILBERT2:attr.3">simple</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"> ( ( <span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Maroon" title="c12">p</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"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c11">i</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> ) &amp; ( <font color="Maroon" title="c12">p</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"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c11">i</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> implies <span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )</span><br/><a NAME="E2:71_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A11</font></span>: 
 ex <font color="Olive" title="b1">n</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="c12">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilbert2.html#K1" title="HILBERT2:func.1">prop</a> <font color="Olive" title="b1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hilbert2.html#D8" onmouseover="rs('hilbert2/D8')" onmouseout="rh()">HILBERT2:def 8</a></span>;<br/><a NAME="E3:71_1_1_2"/><span class="lab"><font color="Green" title="E46">A12</font></span>: 
<font color="Maroon" title="c9">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is    <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>  : ( <font color="Olive" title="b1">q</font> is  <a href="hilbert2.html#V3" title="HILBERT2:attr.3">simple</a>  &amp; <font color="Olive" title="b1">q</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"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c11">i</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:71_1_1"><span class="lab"><font color="Green" title="E42">A8</font></span></a></span>;<br/><div><span class="kw">hereby </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"> ( <font color="Maroon" title="c12">p</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"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c11">i</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> implies <span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:71_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.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"> <font color="Maroon" title="c12">p</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"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c11">i</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span></span><br/><a NAME="E2:71_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71_1_1_2"><span class="lab"><font color="Green" title="E45">A11</font></span></a>, <a class="ref" href="ltlaxio1.html#D11" onmouseover="rs('ltlaxio1/D11')" onmouseout="rh()">LTLAXIO1:def 11</a></span>;<br/><a NAME="E3:71_1_1_2_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">q</font> being    <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>  st <br/>( <font color="Maroon" title="c12">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> &amp; <font color="Olive" title="b1">q</font> is  <a href="hilbert2.html#V3" title="HILBERT2:attr.3">simple</a>  &amp; <font color="Olive" title="b1">q</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"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c11">i</font></span>)</span></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="#E3:71_1_1_2"><span class="lab"><font color="Green" title="E46">A12</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:71_1_1_2_1"/>
<font color="Maroon" title="c12">p</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"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c11">i</font></span>)</span></span>)</span> <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"> verum</span><br/></div>
<span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E5:71_1_1_2"/>
<font color="Maroon" title="c12">p</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"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c11">i</font></span>)</span></span>)</span> <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"> <span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E6:71_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:71_1_1_2"><span class="lab"><font color="Green" title="E46">A12</font></span></a>, <a class="txt" href="#E1:71_1_1_2"><span class="lab"><font color="Green" title="E44">A10</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:71_1_1_2"/>
<span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">p</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#D11" onmouseover="rs('ltlaxio1/D11')" onmouseout="rh()">LTLAXIO1:def 11</a>, <a class="txt" href="#E2:71_1_1_2"><span class="lab"><font color="Green" title="E45">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="lab"><font color="Green" title="E44">A13</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="71_1_1_3"><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">n</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  <font color="Maroon">S<sub>2</sub></font>[ <a href="hilbert2.html#K1" title="HILBERT2:func.1">prop</a> <font color="Olive" title="b1">n</font>]</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">n</font> be    <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> ; <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>2</sub></font>[ <a href="hilbert2.html#K1" title="HILBERT2:func.1">prop</a> <font color="Maroon" title="c11">n</font>]</span><br/><a NAME="E1:71_1_1_3"/>
 <a href="hilbert2.html#K1" title="HILBERT2:func.1">prop</a> <font color="Maroon" title="c11">n</font> is  <a href="hilbert2.html#V3" title="HILBERT2:attr.3">simple</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hilbert2.html#D8" onmouseover="rs('hilbert2/D8')" onmouseout="rh()">HILBERT2:def 8</a></span>;<br/><span class="kw">hence </span><a NAME="E2:71_1_1_3"/>
<font color="Maroon">S<sub>2</sub></font>[ <a href="hilbert2.html#K1" title="HILBERT2:func.1">prop</a> <font color="Maroon" title="c11">n</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:71_1_1"><span class="lab"><font color="Green" title="E43">A9</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="lab"><font color="Green" title="E45">A14</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="71_1_1_4"><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">r</font>, <font color="Olive" title="b2">q</font> being    <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>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">r</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">q</font>] holds <br/>( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">r</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Olive" title="b2">q</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">r</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Olive" title="b2">q</font>] )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">r</font>, <font color="Maroon" title="c12">q</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"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">r</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">q</font>] implies ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">r</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c12">q</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">r</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c12">q</font>] ) )</span><br/><span class="kw">set </span><font color="Maroon" title="c13">ruq</font> = <font color="Maroon" title="c11">r</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c12">q</font>;<br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:71_1_1_4"/><span class="lab"><font color="Green" title="E46">A15</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">r</font>]
 <span class="kw">and </span><br/><a NAME="E2:71_1_1_4"/><span class="lab"><font color="Green" title="E47">A16</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">q</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">r</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c12">q</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">r</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c12">q</font>] )</span><br/><span class="kw">thus </span><a NAME="E3:71_1_1_4"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">r</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c12">q</font>]
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">r</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c12">q</font>]</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="71_1_1_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">i</font> be    <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> ; <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="c11">r</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c12">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <span class="p1">(<span class="default"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p2">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c14">i</font></span>)</span></span>)</span> implies ( <span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c12">q</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 iff <font color="Maroon" title="c11">r</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c12">q</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"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c14">i</font></span>)</span></span>)</span> <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>3</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span> not <span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c12">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">Q</font> =  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p1">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c14">i</font></span>)</span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>4</sub></font>[   <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">means </span> ex <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 <br/>( <font color="Maroon" title="c14">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="c1">P</font> &amp;  not <span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">k</font>,<font color="Maroon" title="c11">r</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 );<br/>
<span class="kw">assume </span><a NAME="E1:71_1_1_4_1"/><span class="lab"><font color="Green" title="E48">A17</font></span>: 
<font color="Maroon" title="c11">r</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c12">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <span class="p1">(<span class="default"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p2">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c14">i</font></span>)</span></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"> ( <span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c12">q</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 iff <font color="Maroon" title="c11">r</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c12">q</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"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c14">i</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> )</span><br/>

<div><span class="kw">hereby </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"> ( <font color="Maroon" title="c11">r</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c12">q</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"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c14">i</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> implies <span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c12">q</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:71_1_1_4_1_1"/><span class="lab"><font color="Green" title="E49">A18</font></span>: 
<span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c12">q</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.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"> <font color="Maroon" title="c11">r</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c12">q</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"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c14">i</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span></span><br/><span class="kw">assume </span><a NAME="E2:71_1_1_4_1_1"/>
 not <font color="Maroon" title="c11">r</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c12">q</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"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c14">i</font></span>)</span></span>)</span> <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"> contradiction</span><br/><a NAME="E3:71_1_1_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A19</font></span>: 
<font color="Maroon" title="c11">r</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c12">q</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"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c14">i</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="#E1:71_1_1_4_1"><span class="lab"><font color="Green" title="E48">A17</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c16">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><br/><a NAME="E5:71_1_1_4_1_1"/><span class="lab"><font color="Green" title="E51">A20</font></span>: 
( <font color="Maroon" title="c16">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c14">i</font> &amp; <span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c16">j</font>,<font color="Maroon" title="c12">q</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; (  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="c14">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="c16">j</font> holds <br/><span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">k</font>,<font color="Maroon" title="c11">r</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71_1_1_4_1_1"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="ref" href="ltlaxio4.html#T5" target="_self" onmouseover="rs('ltlaxio4/T5')" onmouseout="rh()">Th5</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_1_4_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="E1:71_1_1_4_1_1_1"/>
( <font color="Maroon" title="c12">q</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"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c16">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="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 <br/>( <font color="Maroon" title="c14">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="c16">j</font> &amp; <font color="Maroon" title="c11">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</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> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio4.html#T30" target="_self" onmouseover="rs('ltlaxio4/T30')" onmouseout="rh()">Th30</a>, <a class="txt" href="#E3:71_1_1_4_1_1"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="txt" href="#E5:71_1_1_4_1_1"><span class="lab"><font color="Green" title="E51">A20</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_1_4_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:71_1_1_4_1_1_1_1"/>
<font color="Maroon" title="c12">q</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"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c16">j</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/><div class="add"><a NAME="E2:71_1_1_4_1_1_1_1"/><span class="kw">then </span>
(  not <font color="Maroon" title="c12">q</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"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c16">j</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> &amp; <font color="Maroon" title="c12">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <span class="p1">(<span class="default"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p2">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c16">j</font></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio3.html#T30" onmouseover="rs('ltlaxio3/T30')" onmouseout="rh()">LTLAXIO3:30</a>, <a class="ref" href="xboole_0.html#T3" onmouseover="rs('xboole_0/T3')" onmouseout="rh()">XBOOLE_0:3</a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:71_1_1_4_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71_1_1_4"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E5:71_1_1_4_1_1"><span class="lab"><font color="Green" title="E51">A20</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="71_1_1_4_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:71_1_1_4_1_1_1_2"/>
 ex <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 <br/>( <font color="Maroon" title="c14">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="c16">j</font> &amp; <font color="Maroon" title="c11">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</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/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">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> <span class="kw"> such that </span><br/><a NAME="E3:71_1_1_4_1_1_1_2"/><span class="lab"><font color="Green" title="E52">A21</font></span>: 
( <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">k</font> &amp; <font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">j</font> )
 <span class="kw">and </span><br/><a NAME="E4:71_1_1_4_1_1_1_2"/><span class="lab"><font color="Green" title="E53">A22</font></span>: 
<font color="Maroon" title="c11">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c17">k</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span>
 ;<br/><a NAME="E5:71_1_1_4_1_1_1_2"/>
(  not <font color="Maroon" title="c11">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c17">k</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> &amp; <font color="Maroon" title="c11">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <span class="p1">(<span class="default"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p2">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c17">k</font></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio3.html#T30" onmouseover="rs('ltlaxio3/T30')" onmouseout="rh()">LTLAXIO3:30</a>, <a class="txt" href="#E4:71_1_1_4_1_1_1_2"><span class="lab"><font color="Green" title="E53">A22</font></span></a>, <a class="ref" href="xboole_0.html#T3" onmouseover="rs('xboole_0/T3')" onmouseout="rh()">XBOOLE_0:3</a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E6:71_1_1_4_1_1_1_2"/><span class="kw">then </span>
 not <span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c17">k</font>,<font color="Maroon" title="c11">r</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71_1_1_4"><span class="lab"><font color="Green" title="E46">A15</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:71_1_1_4_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:71_1_1_4_1_1_1_2"><span class="lab"><font color="Green" title="E52">A21</font></span></a>, <a class="txt" href="#E5:71_1_1_4_1_1"><span class="lab"><font color="Green" title="E51">A20</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>

<span class="kw">assume </span><a NAME="E3:71_1_1_4_1"/>
<font color="Maroon" title="c11">r</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c12">q</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"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c14">i</font></span>)</span></span>)</span> <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"> <span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c12">q</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">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><br/><a NAME="E5:71_1_1_4_1"/><span class="lab"><font color="Green" title="E49">A23</font></span>: 
<font color="Maroon" title="c16">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c14">i</font>
 <span class="kw">and </span><br/><a NAME="E6:71_1_1_4_1"/><span class="lab"><font color="Green" title="E50">A24</font></span>: 
<font color="Maroon" title="c12">q</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"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c16">j</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:71_1_1_4_1"/><span class="lab"><font color="Green" title="E51">A25</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="c14">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="c16">j</font> holds <br/><font color="Maroon" title="c11">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</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#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio4.html#D14" target="_self" onmouseover="rs('ltlaxio4/D14')" onmouseout="rh()">Def14</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E52">A26</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="71_1_1_4_1_2"><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="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="c14">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="c16">j</font> holds <br/><span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">k</font>,<font color="Maroon" title="c11">r</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c17">k</font> be    <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> ; <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="c14">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">k</font> &amp; <font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">j</font> implies <span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c17">k</font>,<font color="Maroon" title="c11">r</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><a NAME="E1:71_1_1_4_1_2"/>
( <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">k</font> &amp; <font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">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"> <span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c17">k</font>,<font color="Maroon" title="c11">r</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E2:71_1_1_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A27</font></span>: 
<font color="Maroon" title="c11">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c17">k</font></span>)</span></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="#E7:71_1_1_4_1"><span class="lab"><font color="Green" title="E51">A25</font></span></a></span>;<br/><a NAME="E3:71_1_1_4_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <span class="p1">(<span class="default"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p2">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c17">k</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E4:71_1_1_4_1_2"/>
<span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c17">k</font>,<font color="Maroon" title="c11">r</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71_1_1_4"><span class="lab"><font color="Green" title="E46">A15</font></span></a>, <a class="txt" href="#E2:71_1_1_4_1_2"><span class="lab"><font color="Green" title="E53">A27</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>
<a NAME="E9:71_1_1_4_1"/>
<font color="Maroon" title="c12">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <span class="p1">(<span class="default"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p2">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c16">j</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:71_1_1_4_1"><span class="lab"><font color="Green" title="E50">A24</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E10:71_1_1_4_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c16">j</font>,<font color="Maroon" title="c12">q</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:71_1_1_4_1"><span class="lab"><font color="Green" title="E50">A24</font></span></a>, <a class="txt" href="#E2:71_1_1_4"><span class="lab"><font color="Green" title="E47">A16</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:71_1_1_4_1"/>
<span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c12">q</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:71_1_1_4_1"><span class="lab"><font color="Green" title="E52">A26</font></span></a>, <a class="ref" href="ltlaxio4.html#T5" target="_self" onmouseover="rs('ltlaxio4/T5')" onmouseout="rh()">Th5</a>, <a class="txt" href="#E5:71_1_1_4_1"><span class="lab"><font color="Green" title="E49">A23</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><span class="kw">thus </span><a NAME="E4:71_1_1_4"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">r</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c12">q</font>]
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="71_1_1_4_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">i</font> be    <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> ; <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="c11">r</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c12">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <span class="p1">(<span class="default"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p2">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c14">i</font></span>)</span></span>)</span> implies ( <span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c12">q</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 iff <font color="Maroon" title="c11">r</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c12">q</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"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c14">i</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c15">Q</font> =  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p1">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c14">i</font></span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E1:71_1_1_4_2"/><span class="lab"><font color="Green" title="E48">A28</font></span>: 
<font color="Maroon" title="c11">r</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c12">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <span class="p1">(<span class="default"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p2">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c14">i</font></span>)</span></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"> ( <span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c12">q</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 iff <font color="Maroon" title="c11">r</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c12">q</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"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c14">i</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> )</span><br/>

<a NAME="E2:71_1_1_4_2"/><span class="lab"><font color="Green" title="E49">A29</font></span>: 
 <a href="ltlaxio3.html#K5" title="LTLAXIO3:func.5">tau</a> <span class="p1">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <span class="p2">(<span class="default"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c14">i</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <span class="p1">(<span class="default"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p2">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c14">i</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio3.html#D11" onmouseover="rs('ltlaxio3/D11')" onmouseout="rh()">LTLAXIO3:def 11</a></span>;<br/>
<a NAME="E3:71_1_1_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A30</font></span>: 
<font color="Maroon" title="c11">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <span class="p1">(<span class="default"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p2">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c14">i</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71_1_1_4_2"><span class="lab"><font color="Green" title="E48">A28</font></span></a>, <a class="ref" href="ltlaxio3.html#T19" onmouseover="rs('ltlaxio3/T19')" onmouseout="rh()">LTLAXIO3:19</a></span>;<br/>
<a NAME="E4:71_1_1_4_2"/><span class="lab"><font color="Green" title="E51">A31</font></span>: 
<font color="Maroon" title="c12">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <span class="p1">(<span class="default"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p2">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c14">i</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71_1_1_4_2"><span class="lab"><font color="Green" title="E48">A28</font></span></a>, <a class="ref" href="ltlaxio3.html#T19" onmouseover="rs('ltlaxio3/T19')" onmouseout="rh()">LTLAXIO3:19</a>, <a class="txt" href="#E2:71_1_1_4_2"><span class="lab"><font color="Green" title="E49">A29</font></span></a></span>;<br/>
<a NAME="E5:71_1_1_4_2"/><span class="lab"><font color="Green" title="E52">A32</font></span>: 
( <span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c11">r</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c11">r</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboolean.html#D3" onmouseover="rs('xboolean/D3')" onmouseout="rh()">XBOOLEAN:def 3</a></span>;<br/>
<div><span class="kw">hereby </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"> ( <font color="Maroon" title="c11">r</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c12">q</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"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c14">i</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> implies <span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c12">q</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:71_1_1_4_2_1"/>
<span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c12">q</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.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"> <font color="Maroon" title="c11">r</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c12">q</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"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c14">i</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span></span><br/><a NAME="E2:71_1_1_4_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c11">r</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="xboolean.html#K6" title="XBOOLEAN:func.6">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c12">q</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#D11" onmouseover="rs('ltlaxio1/D11')" onmouseout="rh()">LTLAXIO1:def 11</a></span>;<br/><a NAME="E3:71_1_1_4_2_1"/><span class="kw">then </span>
(  not <font color="Maroon" title="c11">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c14">i</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> or <font color="Maroon" title="c12">q</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"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c14">i</font></span>)</span></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="#E5:71_1_1_4_2"><span class="lab"><font color="Green" title="E52">A32</font></span></a>, <a class="txt" href="#E1:71_1_1_4"><span class="lab"><font color="Green" title="E46">A15</font></span></a>, <a class="txt" href="#E2:71_1_1_4"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E3:71_1_1_4_2"><span class="lab"><font color="Green" title="E50">A30</font></span></a>, <a class="txt" href="#E4:71_1_1_4_2"><span class="lab"><font color="Green" title="E51">A31</font></span></a></span>;<br/><a NAME="E4:71_1_1_4_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c11">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c14">i</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> or <font color="Maroon" title="c12">q</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"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c14">i</font></span>)</span></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="#E3:71_1_1_4_2"><span class="lab"><font color="Green" title="E50">A30</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E5:71_1_1_4_2_1"/>
<font color="Maroon" title="c11">r</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c12">q</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"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c14">i</font></span>)</span></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="ltlaxio3.html#T33" onmouseover="rs('ltlaxio3/T33')" onmouseout="rh()">LTLAXIO3:33</a>, <a class="txt" href="#E1:71_1_1_4_2"><span class="lab"><font color="Green" title="E48">A28</font></span></a>, <a class="txt" href="#E3:71_1_1_4_2"><span class="lab"><font color="Green" title="E50">A30</font></span></a>, <a class="txt" href="#E4:71_1_1_4_2"><span class="lab"><font color="Green" title="E51">A31</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>

<span class="kw">assume </span><a NAME="E7:71_1_1_4_2"/>
<font color="Maroon" title="c11">r</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c12">q</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"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c14">i</font></span>)</span></span>)</span> <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"> <span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c12">q</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/>

<a NAME="E8:71_1_1_4_2"/><span class="kw">then </span>
( <font color="Maroon" title="c11">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c14">i</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> or <font color="Maroon" title="c12">q</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"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c14">i</font></span>)</span></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="ltlaxio3.html#T33" onmouseover="rs('ltlaxio3/T33')" onmouseout="rh()">LTLAXIO3:33</a>, <a class="txt" href="#E1:71_1_1_4_2"><span class="lab"><font color="Green" title="E48">A28</font></span></a>, <a class="txt" href="#E3:71_1_1_4_2"><span class="lab"><font color="Green" title="E50">A30</font></span></a>, <a class="txt" href="#E4:71_1_1_4_2"><span class="lab"><font color="Green" title="E51">A31</font></span></a></span>;<br/>
<a NAME="E9:71_1_1_4_2"/><span class="kw">then </span>
(  not <font color="Maroon" title="c11">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c14">i</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> or <font color="Maroon" title="c12">q</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"> the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c14">i</font></span>)</span></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="xboole_0.html#T3" onmouseover="rs('xboole_0/T3')" onmouseout="rh()">XBOOLE_0:3</a>, <a class="ref" href="ltlaxio3.html#T30" onmouseover="rs('ltlaxio3/T30')" onmouseout="rh()">LTLAXIO3:30</a></span>;<br/>
<a NAME="E10:71_1_1_4_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c11">r</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="xboolean.html#K6" title="XBOOLEAN:func.6">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c12">q</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71_1_1_4"><span class="lab"><font color="Green" title="E46">A15</font></span></a>, <a class="txt" href="#E2:71_1_1_4"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E3:71_1_1_4_2"><span class="lab"><font color="Green" title="E50">A30</font></span></a>, <a class="txt" href="#E4:71_1_1_4_2"><span class="lab"><font color="Green" title="E51">A31</font></span></a>, <a class="txt" href="#E5:71_1_1_4_2"><span class="lab"><font color="Green" title="E52">A32</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:71_1_1_4_2"/>
<span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c12">q</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio1.html#D11" onmouseover="rs('ltlaxio1/D11')" onmouseout="rh()">LTLAXIO1:def 11</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>
<a NAME="E15:71_1_1"/><span class="lab"><font color="Green" title="E46">A33</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="ltlaxio1.html#NK2" title="LTLAXIO1:NK.2">TFALSUM</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio3.html#T32" onmouseover="rs('ltlaxio3/T32')" onmouseout="rh()">LTLAXIO3:32</a>, <a class="ref" href="ltlaxio1.html#D11" onmouseover="rs('ltlaxio1/D11')" onmouseout="rh()">LTLAXIO1:def 11</a></span>;<br/>
<a NAME="E16:71_1_1"/><span class="lab"><font color="Green" title="E47">A34</font></span>: 
 for <font color="Olive" title="b1">A</font> being    <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>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">A</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="hilbert2.html#S2" onmouseover="rs('hilbert2/S2')" onmouseout="rh()">HILBERT2:sch 2</a>(<a class="txt" href="#E15:71_1_1"><span class="lab"><font color="Green" title="E46">A33</font></span></a>, <a class="txt" href="#E13:71_1_1"><span class="lab"><font color="Green" title="E44">A13</font></span></a>, <a class="txt" href="#E14:71_1_1"><span class="lab"><font color="Green" title="E45">A14</font></span></a>);<br/></span>
<span class="lab"><font color="Green" title="E48">A35</font></span>:  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p1">(<span class="default"> the   <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a>   <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of  the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> = 
 the    <a href="ltlaxio4.html#M3" title="LTLAXIO4:mode.3">pnptree</a> of <font color="Maroon" title="c6">Pg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>
<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/>.= 
<font color="Maroon" title="c6">Pg</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/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="71_1_1_6"><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">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">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"> ( <font color="Maroon" title="c11">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">set </span><font color="Maroon" title="c12">nA</font> = <span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i</font>;<br/><span class="kw">set </span><font color="Maroon" title="c13">A</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i</font>;<br/><span class="kw">assume </span><a NAME="E1:71_1_1_6"/><span class="lab"><font color="Green" title="E49">A36</font></span>: 
<font color="Maroon" title="c11">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span></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"> <span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E2:71_1_1_6"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <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="ltlaxio2.html#D4" onmouseover="rs('ltlaxio2/D4')" onmouseout="rh()">LTLAXIO2:def 4</a></span>;<br/><a NAME="E3:71_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A37</font></span>: 
<font color="Maroon" title="c11">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <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="#E1:71_1_1_6"><span class="lab"><font color="Green" title="E49">A36</font></span></a>, <a class="ref" href="finseq_3.html#T29" onmouseover="rs('finseq_3/T29')" onmouseout="rh()">FINSEQ_3:29</a></span>;<br/><a NAME="E4:71_1_1_6"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</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="c3">P</font> <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="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/><a NAME="E5:71_1_1_6"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i</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="c3">P</font> <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="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a>, <a class="txt" href="#E3:71_1_1_6"><span class="lab"><font color="Green" title="E50">A37</font></span></a></span>;<br/><a NAME="E6:71_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A38</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i</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">Pg</font> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> &amp;  not <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i</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">Pg</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="#E7:71_1_1"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="ref" href="ltlaxio3.html#T30" onmouseover="rs('ltlaxio3/T30')" onmouseout="rh()">LTLAXIO3:30</a>, <a class="ref" href="xboole_0.html#T3" onmouseover="rs('xboole_0/T3')" onmouseout="rh()">XBOOLE_0:3</a></span>;<br/><a NAME="E7:71_1_1_6"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Pg</font> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</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="c6">Pg</font>
 <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></span>;<br/><a NAME="E8:71_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A39</font></span>: 
 not <span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:71_1_1_6"><span class="lab"><font color="Green" title="E51">A38</font></span></a>, <a class="txt" href="#E17:71_1_1"><span class="lab"><font color="Green" title="E48">A35</font></span></a>, <a class="txt" href="#E16:71_1_1"><span class="lab"><font color="Green" title="E47">A34</font></span></a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> = 
<span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p2">(<span class="default"><a href="ltlaxio1.html#K1" title="LTLAXIO1:func.1">'not'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i</font></span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio2.html#T8" onmouseover="rs('ltlaxio2/T8')" onmouseout="rh()">LTLAXIO2:8</a>, <a class="txt" href="#E3:71_1_1_6"><span class="lab"><font color="Green" title="E50">A37</font></span></a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:71_1_1_6"><span class="lab"><font color="Green" title="E52">A39</font></span></a>, <a class="ref" href="xboolean.html#D3" onmouseover="rs('xboolean/D3')" onmouseout="rh()">XBOOLEAN:def 3</a>, <a class="ref" href="ltlaxio1.html#T5" onmouseover="rs('ltlaxio1/T5')" onmouseout="rh()">LTLAXIO1:5</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>
<a NAME="E19:71_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A40</font></span>: 
<span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon">H<sub>2</sub></font>( <a href="ltlaxio2.html#K4" title="LTLAXIO2:func.4">nega</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K9" title="LTLAXIO3:func.9">`2</a></span>)</span>)</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio4.html#T6" target="_self" onmouseover="rs('ltlaxio4/T6')" onmouseout="rh()">Th6</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="71_1_1_7"><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">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">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"> ( <font color="Maroon" title="c11">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> implies <span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">set </span><font color="Maroon" title="c12">A</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i</font>;<br/><span class="kw">assume </span><a NAME="E1:71_1_1_7"/>
<font color="Maroon" title="c11">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</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"> <span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E2:71_1_1_7"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i</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="c3">P</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="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>;<br/><a NAME="E3:71_1_1_7"/><span class="kw">then </span>
(  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Pg</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="c6">Pg</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i</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">Pg</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="#E6:71_1_1"><span class="lab"><font color="Green" title="E39">A3</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:71_1_1_7"/>
<span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:71_1_1"><span class="lab"><font color="Green" title="E48">A35</font></span></a>, <a class="txt" href="#E16:71_1_1"><span class="lab"><font color="Green" title="E47">A34</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>
<a NAME="E21:71_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a> )</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio4.html#T6" target="_self" onmouseover="rs('ltlaxio4/T6')" onmouseout="rh()">Th6</a></span>;<br/>
<a NAME="E22:71_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="ltlaxio1.html#K11" title="LTLAXIO1:func.11">SAT</a> <font color="Maroon" title="c9">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:71_1_1"><span class="lab"><font color="Green" title="E49">A40</font></span></a>, <a class="ref" href="ltlaxio1.html#T7" onmouseover="rs('ltlaxio1/T7')" onmouseout="rh()">LTLAXIO1:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E23:71_1_1"/>
<font color="Maroon" title="c3">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a>  is  <a href="ltlaxio4.html#V1" title="LTLAXIO4:attr.1">satisfiable</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"> verum</span><br/>


</div>
