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<span class="kw">let </span><font color="Maroon" title="c1">a1</font>, <font color="Maroon" title="c2">a2</font> be    <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"> ( <font color="Maroon" title="c1">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a2</font> implies  for <font color="Olive" title="b1">A</font> being    <a href="abcmiz_0.html#L1" title="ABCMIZ_0:struct.1">AdjectiveStr</a>   st  the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">a2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp;  the <a href="abcmiz_0.html#U2" title="ABCMIZ_0:sel.2">non-op</a> of <font color="Olive" title="b1">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a2</font> &amp;  the <a href="abcmiz_0.html#U2" title="ABCMIZ_0:sel.2">non-op</a> of <font color="Olive" title="b1">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a1</font> holds <br/>(  not <font color="Olive" title="b1">A</font> is  <a href="abcmiz_0.html#V4" title="ABCMIZ_0:attr.4">void</a>  &amp; <font color="Olive" title="b1">A</font> is  <a href="abcmiz_0.html#V5" title="ABCMIZ_0:attr.5">involutive</a>  &amp; <font color="Olive" title="b1">A</font> is  <a href="abcmiz_0.html#V6" title="ABCMIZ_0:attr.6">without_fixpoints</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:16"/><span class="lab"><font color="Green" title="E7">A1</font></span>: 
<font color="Maroon" title="c1">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font> being    <a href="abcmiz_0.html#L1" title="ABCMIZ_0:struct.1">AdjectiveStr</a>   st  the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">a2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp;  the <a href="abcmiz_0.html#U2" title="ABCMIZ_0:sel.2">non-op</a> of <font color="Olive" title="b1">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a2</font> &amp;  the <a href="abcmiz_0.html#U2" title="ABCMIZ_0:sel.2">non-op</a> of <font color="Olive" title="b1">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a1</font> holds <br/>(  not <font color="Olive" title="b1">A</font> is  <a href="abcmiz_0.html#V4" title="ABCMIZ_0:attr.4">void</a>  &amp; <font color="Olive" title="b1">A</font> is  <a href="abcmiz_0.html#V5" title="ABCMIZ_0:attr.5">involutive</a>  &amp; <font color="Olive" title="b1">A</font> is  <a href="abcmiz_0.html#V6" title="ABCMIZ_0:attr.6">without_fixpoints</a>  )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">A</font> be    <a href="abcmiz_0.html#L1" title="ABCMIZ_0:struct.1">AdjectiveStr</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"> (  the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">a2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp;  the <a href="abcmiz_0.html#U2" title="ABCMIZ_0:sel.2">non-op</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a2</font> &amp;  the <a href="abcmiz_0.html#U2" title="ABCMIZ_0:sel.2">non-op</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a1</font> implies (  not <font color="Maroon" title="c3">A</font> is  <a href="abcmiz_0.html#V4" title="ABCMIZ_0:attr.4">void</a>  &amp; <font color="Maroon" title="c3">A</font> is  <a href="abcmiz_0.html#V5" title="ABCMIZ_0:attr.5">involutive</a>  &amp; <font color="Maroon" title="c3">A</font> is  <a href="abcmiz_0.html#V6" title="ABCMIZ_0:attr.6">without_fixpoints</a>  ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:16"/><span class="lab"><font color="Green" title="E8">A2</font></span>: 
 the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">a1</font>,<font color="Maroon" title="c2">a2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">and </span><br/><a NAME="E3:16"/><span class="lab"><font color="Green" title="E9">A3</font></span>: 
 the <a href="abcmiz_0.html#U2" title="ABCMIZ_0:sel.2">non-op</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a2</font>
 <span class="kw">and </span><br/><a NAME="E4:16"/><span class="lab"><font color="Green" title="E10">A4</font></span>: 
 the <a href="abcmiz_0.html#U2" title="ABCMIZ_0:sel.2">non-op</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a1</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="c3">A</font> is  <a href="abcmiz_0.html#V4" title="ABCMIZ_0:attr.4">void</a>  &amp; <font color="Maroon" title="c3">A</font> is  <a href="abcmiz_0.html#V5" title="ABCMIZ_0:attr.5">involutive</a>  &amp; <font color="Maroon" title="c3">A</font> is  <a href="abcmiz_0.html#V6" title="ABCMIZ_0:attr.6">without_fixpoints</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E5:16"/>
 not  the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Maroon" title="c3">A</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E8">A2</font></span></a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="abcmiz_0.html#D4" target="_self" onmouseover="rs('abcmiz_0/D4')" onmouseout="rh()">ABCMIZ_0:def 4</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="c3">A</font> is  <a href="abcmiz_0.html#V5" title="ABCMIZ_0:attr.5">involutive</a>  &amp; <font color="Maroon" title="c3">A</font> is  <a href="abcmiz_0.html#V6" title="ABCMIZ_0:attr.6">without_fixpoints</a>  )</span><br/>

<div><span class="kw">hereby </span> <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="abcmiz_0.html#D6" target="_self" onmouseover="rs('abcmiz_0/D6')" onmouseout="rh()">ABCMIZ_0:def 6</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="c3">A</font> is  <a href="abcmiz_0.html#V6" title="ABCMIZ_0:attr.6">without_fixpoints</a> </span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c4">a</font> be   <a href="abcmiz_0.html#NM2" title="ABCMIZ_0:NM.2">adjective</a> of <font color="Maroon" title="c3">A</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">  <a href="abcmiz_0.html#K1" title="ABCMIZ_0:func.1">non-</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K1" title="ABCMIZ_0:func.1">non-</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font></span><br/><a NAME="E1:16_1"/>
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a1</font> or <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E2:16_1"/>
 <a href="abcmiz_0.html#K1" title="ABCMIZ_0:func.1">non-</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K1" title="ABCMIZ_0:func.1">non-</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E10">A4</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">let </span><font color="Maroon" title="c4">a</font> be   <a href="abcmiz_0.html#NM2" title="ABCMIZ_0:NM.2">adjective</a> of <font color="Maroon" title="c3">A</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="abcmiz_0.html#D7" target="_self" onmouseover="rs('abcmiz_0/D7')" onmouseout="rh()">ABCMIZ_0:def 7</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  <a href="abcmiz_0.html#K1" title="ABCMIZ_0:func.1">non-</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font></span><br/>

<span class="kw">assume </span><a NAME="E7:16"/><span class="lab"><font color="Green" title="E11">A5</font></span>: 
 <a href="abcmiz_0.html#K1" title="ABCMIZ_0:func.1">non-</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</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"> contradiction</span><br/>

<a NAME="E8:16"/>
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a1</font> or <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:16"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E11">A5</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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