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<span class="kw">let </span><font color="Maroon" title="c1">PS</font> be   <a href="parsp_1.html#NM1" title="PARSP_1:NM.1">ParSp</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">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">p</font>, <font color="Olive" title="b5">q</font>, <font color="Olive" title="b6">r</font>, <font color="Olive" title="b7">s</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">PS</font>  st  not <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="parsp_1.html#R1" title="PARSP_1:pred.1">'||'</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="parsp_1.html#R1" title="PARSP_1:pred.1">'||'</a> <font color="Olive" title="b4">p</font>,<font color="Olive" title="b5">q</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font> <a href="parsp_1.html#R1" title="PARSP_1:pred.1">'||'</a> <font color="Olive" title="b4">p</font>,<font color="Olive" title="b6">r</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font> <a href="parsp_1.html#R1" title="PARSP_1:pred.1">'||'</a> <font color="Olive" title="b4">p</font>,<font color="Olive" title="b7">s</font> &amp; <font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> <a href="parsp_1.html#R1" title="PARSP_1:pred.1">'||'</a> <font color="Olive" title="b5">q</font>,<font color="Olive" title="b6">r</font> &amp; <font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> <a href="parsp_1.html#R1" title="PARSP_1:pred.1">'||'</a> <font color="Olive" title="b5">q</font>,<font color="Olive" title="b7">s</font> holds <br/><font color="Olive" title="b6">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">s</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</font>, <font color="Maroon" title="c5">p</font>, <font color="Maroon" title="c6">q</font>, <font color="Maroon" title="c7">r</font>, <font color="Maroon" title="c8">s</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">PS</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"> (  not <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="parsp_1.html#R1" title="PARSP_1:pred.1">'||'</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="parsp_1.html#R1" title="PARSP_1:pred.1">'||'</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="parsp_1.html#R1" title="PARSP_1:pred.1">'||'</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c7">r</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="parsp_1.html#R1" title="PARSP_1:pred.1">'||'</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c8">s</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="parsp_1.html#R1" title="PARSP_1:pred.1">'||'</a> <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">r</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="parsp_1.html#R1" title="PARSP_1:pred.1">'||'</a> <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c8">s</font> implies <font color="Maroon" title="c7">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> )</span><br/>















<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:55"/><span class="lab"><font color="Green" title="E42">A1</font></span>: 
 not <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="parsp_1.html#R1" title="PARSP_1:pred.1">'||'</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>
 <span class="kw">and </span><br/><a NAME="E2:55"/><span class="lab"><font color="Green" title="E43">A2</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="parsp_1.html#R1" title="PARSP_1:pred.1">'||'</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>
 <span class="kw">and </span><br/><a NAME="E3:55"/><span class="lab"><font color="Green" title="E44">A3</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="parsp_1.html#R1" title="PARSP_1:pred.1">'||'</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c7">r</font>
 <span class="kw">and </span><br/><a NAME="E4:55"/><span class="lab"><font color="Green" title="E45">A4</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="parsp_1.html#R1" title="PARSP_1:pred.1">'||'</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c8">s</font>
 <span class="kw">and </span><br/><a NAME="E5:55"/><span class="lab"><font color="Green" title="E46">A5</font></span>: 
<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="parsp_1.html#R1" title="PARSP_1:pred.1">'||'</a> <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">r</font>
 <span class="kw">and </span><br/><a NAME="E6:55"/><span class="lab"><font color="Green" title="E47">A6</font></span>: 
<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="parsp_1.html#R1" title="PARSP_1:pred.1">'||'</a> <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c8">s</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="c7">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font></span><br/>

<div><span class="lab"><font color="Green" title="E48">A7</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="55_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"> ( <font color="Maroon" title="c5">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">q</font> implies <font color="Maroon" title="c7">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> )</span></a><div class="add"><a NAME="E1:55_1"/>
<font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="ref" href="parsp_1.html#T18" target="_self" onmouseover="rs('parsp_1/T18')" onmouseout="rh()">Th18</a></span>;<br/><a NAME="E2:55_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A8</font></span>: 
<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">r</font> <a href="parsp_1.html#R1" title="PARSP_1:pred.1">'||'</a> <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c8">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:55"><span class="lab"><font color="Green" title="E46">A5</font></span></a>, <a class="txt" href="#E6:55"><span class="lab"><font color="Green" title="E47">A6</font></span></a>, <a class="ref" href="parsp_1.html#D12" target="_self" onmouseover="rs('parsp_1/D12')" onmouseout="rh()">Def11</a></span>;<br/><a NAME="E3:55_1"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="ref" href="parsp_1.html#D12" target="_self" onmouseover="rs('parsp_1/D12')" onmouseout="rh()">Def11</a></span>;<br/><a NAME="E4:55_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A9</font></span>: 
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c7">r</font> <a href="parsp_1.html#R1" title="PARSP_1:pred.1">'||'</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c8">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:55"><span class="lab"><font color="Green" title="E44">A3</font></span></a>, <a class="txt" href="#E4:55"><span class="lab"><font color="Green" title="E45">A4</font></span></a>, <a class="ref" href="parsp_1.html#D12" target="_self" onmouseover="rs('parsp_1/D12')" onmouseout="rh()">Def11</a></span>;<br/><span class="kw">assume </span><a NAME="E5:55_1"/>
<font color="Maroon" title="c5">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">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" title="c7">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font></span><br/><a NAME="E6:55_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font> <a href="parsp_1.html#R1" title="PARSP_1:pred.1">'||'</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c7">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="txt" href="#E2:55"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E3:55"><span class="lab"><font color="Green" title="E44">A3</font></span></a>, <a class="txt" href="#E5:55"><span class="lab"><font color="Green" title="E46">A5</font></span></a>, <a class="ref" href="parsp_1.html#T31" target="_self" onmouseover="rs('parsp_1/T31')" onmouseout="rh()">Th31</a></span>;<br/><span class="kw">hence </span><a NAME="E7:55_1"/>
<font color="Maroon" title="c7">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:55_1"><span class="lab"><font color="Green" title="E50">A9</font></span></a>, <a class="txt" href="#E2:55_1"><span class="lab"><font color="Green" title="E49">A8</font></span></a>, <a class="ref" href="parsp_1.html#T33" target="_self" onmouseover="rs('parsp_1/T33')" onmouseout="rh()">Th33</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="E8:55"/>
( <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</font> implies ( <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> &amp; <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="txt" href="#E3:55"><span class="lab"><font color="Green" title="E44">A3</font></span></a>, <a class="txt" href="#E4:55"><span class="lab"><font color="Green" title="E45">A4</font></span></a>, <a class="txt" href="#E5:55"><span class="lab"><font color="Green" title="E46">A5</font></span></a>, <a class="txt" href="#E6:55"><span class="lab"><font color="Green" title="E47">A6</font></span></a>, <a class="ref" href="parsp_1.html#T32" target="_self" onmouseover="rs('parsp_1/T32')" onmouseout="rh()">Th32</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:55"/>
<font color="Maroon" title="c7">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:55"><span class="lab"><font color="Green" title="E48">A7</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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