<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">P</font>, <font color="Maroon" title="c2">Q</font>, <font color="Maroon" title="c3">R</font> be   <a href="finseq_1.html#V4" title="FINSEQ_1:attr.4">FinSequence-membered</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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"><font color="Maroon" title="c1">P</font> <a href="polnot_1.html#K2" title="POLNOT_1:func.2">^</a> <font color="Maroon" title="c2">Q</font></span>)</span> <a href="polnot_1.html#K2" title="POLNOT_1:func.2">^</a> <font color="Maroon" title="c3">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">P</font> <a href="polnot_1.html#K2" title="POLNOT_1:func.2">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Q</font> <a href="polnot_1.html#K2" title="POLNOT_1:func.2">^</a> <font color="Maroon" title="c3">R</font></span>)</span></span><br/>





<a NAME="E1:8"/>
 for <font color="Olive" title="b1">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="polnot_1.html#K2" title="POLNOT_1:func.2">^</a> <font color="Maroon" title="c2">Q</font></span>)</span> <a href="polnot_1.html#K2" title="POLNOT_1:func.2">^</a> <font color="Maroon" title="c3">R</font> iff <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> <a href="polnot_1.html#K2" title="POLNOT_1:func.2">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Q</font> <a href="polnot_1.html#K2" title="POLNOT_1:func.2">^</a> <font color="Maroon" title="c3">R</font></span>)</span> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="8_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="polnot_1.html#K2" title="POLNOT_1:func.2">^</a> <font color="Maroon" title="c2">Q</font></span>)</span> <a href="polnot_1.html#K2" title="POLNOT_1:func.2">^</a> <font color="Maroon" title="c3">R</font> iff <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> <a href="polnot_1.html#K2" title="POLNOT_1:func.2">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Q</font> <a href="polnot_1.html#K2" title="POLNOT_1:func.2">^</a> <font color="Maroon" title="c3">R</font></span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E1:8_1"/>
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="polnot_1.html#K2" title="POLNOT_1:func.2">^</a> <font color="Maroon" title="c2">Q</font></span>)</span> <a href="polnot_1.html#K2" title="POLNOT_1:func.2">^</a> <font color="Maroon" title="c3">R</font> implies <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> <a href="polnot_1.html#K2" title="POLNOT_1:func.2">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Q</font> <a href="polnot_1.html#K2" title="POLNOT_1:func.2">^</a> <font color="Maroon" title="c3">R</font></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="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> <a href="polnot_1.html#K2" title="POLNOT_1:func.2">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Q</font> <a href="polnot_1.html#K2" title="POLNOT_1:func.2">^</a> <font color="Maroon" title="c3">R</font></span>)</span> implies <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="polnot_1.html#K2" title="POLNOT_1:func.2">^</a> <font color="Maroon" title="c2">Q</font></span>)</span> <a href="polnot_1.html#K2" title="POLNOT_1:func.2">^</a> <font color="Maroon" title="c3">R</font> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="8_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:8_1_1"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="polnot_1.html#K2" title="POLNOT_1:func.2">^</a> <font color="Maroon" title="c2">Q</font></span>)</span> <a href="polnot_1.html#K2" title="POLNOT_1:func.2">^</a> <font color="Maroon" title="c3">R</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> <a href="polnot_1.html#K2" title="POLNOT_1:func.2">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Q</font> <a href="polnot_1.html#K2" title="POLNOT_1:func.2">^</a> <font color="Maroon" title="c3">R</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">u</font>, <font color="Maroon" title="c6">r</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E3:8_1_1"/><span class="lab"><font color="Green" title="E4">A2</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">u</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c6">r</font>
 <span class="kw">and </span><br/><a NAME="E4:8_1_1"/><span class="lab"><font color="Green" title="E5">A3</font></span>: 
<font color="Maroon" title="c5">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">P</font> <a href="polnot_1.html#K2" title="POLNOT_1:func.2">^</a> <font color="Maroon" title="c2">Q</font>
 <span class="kw">and </span><br/><a NAME="E5:8_1_1"/><span class="lab"><font color="Green" title="E6">A4</font></span>: 
<font color="Maroon" title="c6">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">R</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polnot_1.html#D2" target="_self" onmouseover="rs('polnot_1/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">p</font>, <font color="Maroon" title="c8">q</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E7:8_1_1"/><span class="lab"><font color="Green" title="E7">A5</font></span>: 
<font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c8">q</font>
 <span class="kw">and </span><br/><a NAME="E8:8_1_1"/><span class="lab"><font color="Green" title="E8">A6</font></span>: 
<font color="Maroon" title="c7">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 <span class="kw">and </span><br/><a NAME="E9:8_1_1"/><span class="lab"><font color="Green" title="E9">A7</font></span>: 
<font color="Maroon" title="c8">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8_1_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="ref" href="polnot_1.html#D2" target="_self" onmouseover="rs('polnot_1/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<a NAME="E10:8_1_1"/><span class="lab"><font color="Green" title="E10">A8</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c6">r</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:8_1_1"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="txt" href="#E7:8_1_1"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="ref" href="finseq_1.html#T32" onmouseover="rs('finseq_1/T32')" onmouseout="rh()">FINSEQ_1:32</a></span>;<br/>
<a NAME="E11:8_1_1"/>
<font color="Maroon" title="c8">q</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c6">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Q</font> <a href="polnot_1.html#K2" title="POLNOT_1:func.2">^</a> <font color="Maroon" title="c3">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1_1"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="txt" href="#E9:8_1_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="ref" href="polnot_1.html#D2" target="_self" onmouseover="rs('polnot_1/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:8_1_1"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> <a href="polnot_1.html#K2" title="POLNOT_1:func.2">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Q</font> <a href="polnot_1.html#K2" title="POLNOT_1:func.2">^</a> <font color="Maroon" title="c3">R</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:8_1_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E10:8_1_1"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="ref" href="polnot_1.html#D2" target="_self" onmouseover="rs('polnot_1/D2')" onmouseout="rh()">Def2</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="E2:8_1"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> <a href="polnot_1.html#K2" title="POLNOT_1:func.2">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Q</font> <a href="polnot_1.html#K2" title="POLNOT_1:func.2">^</a> <font color="Maroon" title="c3">R</font></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="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="polnot_1.html#K2" title="POLNOT_1:func.2">^</a> <font color="Maroon" title="c2">Q</font></span>)</span> <a href="polnot_1.html#K2" title="POLNOT_1:func.2">^</a> <font color="Maroon" title="c3">R</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">p</font>, <font color="Maroon" title="c6">t</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E4:8_1"/><span class="lab"><font color="Green" title="E4">A9</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c6">t</font>
 <span class="kw">and </span><br/><a NAME="E5:8_1"/><span class="lab"><font color="Green" title="E5">A10</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 <span class="kw">and </span><br/><a NAME="E6:8_1"/><span class="lab"><font color="Green" title="E6">A11</font></span>: 
<font color="Maroon" title="c6">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Q</font> <a href="polnot_1.html#K2" title="POLNOT_1:func.2">^</a> <font color="Maroon" title="c3">R</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polnot_1.html#D2" target="_self" onmouseover="rs('polnot_1/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">q</font>, <font color="Maroon" title="c8">r</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E8:8_1"/><span class="lab"><font color="Green" title="E7">A12</font></span>: 
<font color="Maroon" title="c6">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c8">r</font>
 <span class="kw">and </span><br/><a NAME="E9:8_1"/><span class="lab"><font color="Green" title="E8">A13</font></span>: 
<font color="Maroon" title="c7">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Q</font>
 <span class="kw">and </span><br/><a NAME="E10:8_1"/><span class="lab"><font color="Green" title="E9">A14</font></span>: 
<font color="Maroon" title="c8">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:8_1"><span class="lab"><font color="Green" title="E6">A11</font></span></a>, <a class="ref" href="polnot_1.html#D2" target="_self" onmouseover="rs('polnot_1/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<a NAME="E11:8_1"/><span class="lab"><font color="Green" title="E10">A15</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c7">q</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c8">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8_1"><span class="lab"><font color="Green" title="E4">A9</font></span></a>, <a class="txt" href="#E8:8_1"><span class="lab"><font color="Green" title="E7">A12</font></span></a>, <a class="ref" href="finseq_1.html#T32" onmouseover="rs('finseq_1/T32')" onmouseout="rh()">FINSEQ_1:32</a></span>;<br/>
<a NAME="E12:8_1"/>
<font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c7">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">P</font> <a href="polnot_1.html#K2" title="POLNOT_1:func.2">^</a> <font color="Maroon" title="c2">Q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1"><span class="lab"><font color="Green" title="E5">A10</font></span></a>, <a class="txt" href="#E9:8_1"><span class="lab"><font color="Green" title="E8">A13</font></span></a>, <a class="ref" href="polnot_1.html#D2" target="_self" onmouseover="rs('polnot_1/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:8_1"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="polnot_1.html#K2" title="POLNOT_1:func.2">^</a> <font color="Maroon" title="c2">Q</font></span>)</span> <a href="polnot_1.html#K2" title="POLNOT_1:func.2">^</a> <font color="Maroon" title="c3">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:8_1"><span class="lab"><font color="Green" title="E9">A14</font></span></a>, <a class="txt" href="#E11:8_1"><span class="lab"><font color="Green" title="E10">A15</font></span></a>, <a class="ref" href="polnot_1.html#D2" target="_self" onmouseover="rs('polnot_1/D2')" onmouseout="rh()">Def2</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">hence </span><a NAME="E2:8"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="polnot_1.html#K2" title="POLNOT_1:func.2">^</a> <font color="Maroon" title="c2">Q</font></span>)</span> <a href="polnot_1.html#K2" title="POLNOT_1:func.2">^</a> <font color="Maroon" title="c3">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">P</font> <a href="polnot_1.html#K2" title="POLNOT_1:func.2">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Q</font> <a href="polnot_1.html#K2" title="POLNOT_1:func.2">^</a> <font color="Maroon" title="c3">R</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T2" onmouseover="rs('tarski/T2')" onmouseout="rh()">TARSKI:2</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>
