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<b>let </b><font color="Maroon" title="c1">C</font> be  <a href="abcmiz_1.html#V3" title="ABCMIZ_1:attr.3">initialized</a> <a href="abcmiz_1.html#NM5" title="ABCMIZ_1:NM.5">ConstructorSignature</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font> being   <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon" title="c1">C</font>, <a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon" title="c1">C</font><br/> for <font color="Olive" title="b2">f</font> being  <a href="abcmiz_1.html#NM13" title="ABCMIZ_1:NM.13">valuation</a> of <font color="Maroon" title="c1">C</font> holds  <span class="p1">(<span class="default"><a href="abcmiz_1.html#K37" title="ABCMIZ_1:func.37">Non</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="abcmiz_1.html#K60" title="ABCMIZ_1:func.60">at</a> <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_1.html#K37" title="ABCMIZ_1:func.37">Non</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="abcmiz_1.html#K60" title="ABCMIZ_1:func.60">at</a> <font color="Olive" title="b2">f</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">a</font> be    <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon" title="c1">C</font>, <a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="abcmiz_1.html#NM13" title="ABCMIZ_1:NM.13">valuation</a> of <font color="Maroon" title="c1">C</font> holds  <span class="p1">(<span class="default"><a href="abcmiz_1.html#K37" title="ABCMIZ_1:func.37">Non</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="abcmiz_1.html#K60" title="ABCMIZ_1:func.60">at</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_1.html#K37" title="ABCMIZ_1:func.37">Non</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="abcmiz_1.html#K60" title="ABCMIZ_1:func.60">at</a> <font color="Olive" title="b1">f</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="abcmiz_1.html#NM13" title="ABCMIZ_1:NM.13">valuation</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K37" title="ABCMIZ_1:func.37">Non</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="abcmiz_1.html#K60" title="ABCMIZ_1:func.60">at</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_1.html#K37" title="ABCMIZ_1:func.37">Non</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="abcmiz_1.html#K60" title="ABCMIZ_1:func.60">at</a> <font color="Maroon" title="c3">f</font></span>)</span></span><br/>





<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="260_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:260_1"/>
( <font color="Maroon" title="c2">a</font> is <a href="abcmiz_1.html#V6" title="ABCMIZ_1:attr.6">positive</a> or  not <font color="Maroon" title="c2">a</font> is <a href="abcmiz_1.html#V6" title="ABCMIZ_1:attr.6">positive</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="260_1_1"><b>suppose </b></a><a NAME="E1:260_1_1"/>
<font color="Maroon" title="c2">a</font> is <a href="abcmiz_1.html#V6" title="ABCMIZ_1:attr.6">positive</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K37" title="ABCMIZ_1:func.37">Non</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="abcmiz_1.html#K60" title="ABCMIZ_1:func.60">at</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_1.html#K37" title="ABCMIZ_1:func.37">Non</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="abcmiz_1.html#K60" title="ABCMIZ_1:func.60">at</a> <font color="Maroon" title="c3">f</font></span>)</span></span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c4">b</font> = <font color="Maroon" title="c2">a</font> as  <a href="abcmiz_1.html#V6" title="ABCMIZ_1:attr.6">positive</a>  <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon" title="c1">C</font>, <a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon" title="c1">C</font> ;<br/><b>reconsider </b><font color="Maroon" title="c5">af</font> = <font color="Maroon" title="c4">b</font> <a href="abcmiz_1.html#K61" title="ABCMIZ_1:func.61">at</a> <font color="Maroon" title="c3">f</font> as  <a href="abcmiz_1.html#V6" title="ABCMIZ_1:attr.6">positive</a>  <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon" title="c1">C</font>, <a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon" title="c1">C</font> ;<br/><b>thus </b><span class="p1">(<span class="default"><a href="abcmiz_1.html#K37" title="ABCMIZ_1:func.37">Non</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="abcmiz_1.html#K60" title="ABCMIZ_1:func.60">at</a> <font color="Maroon" title="c3">f</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="abcmiz_1.html#K30" title="ABCMIZ_1:func.30">term</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <font color="Maroon" title="c3">f</font>
<b>by </b><i><a class="ref" href="abcmiz_1.html#T59" target="_self" title="ABCMIZ_1:th.59">Th44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="abcmiz_1.html#K30" title="ABCMIZ_1:func.30">term</a> <font color="Maroon" title="c5">af</font>
<b>by </b><i><a class="ref" href="abcmiz_1.html#D60" target="_self" title="ABCMIZ_1:def.60">SUBST</a></i>
<br/>.= 
 <a href="abcmiz_1.html#K37" title="ABCMIZ_1:func.37">Non</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="abcmiz_1.html#K60" title="ABCMIZ_1:func.60">at</a> <font color="Maroon" title="c3">f</font></span>)</span>
<b>by </b><i><a class="ref" href="abcmiz_1.html#T59" target="_self" title="ABCMIZ_1:th.59">Th44</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="260_1_2"><b>suppose </b></a><a NAME="E1:260_1_2"/>
not <font color="Maroon" title="c2">a</font> is <a href="abcmiz_1.html#V6" title="ABCMIZ_1:attr.6">positive</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K37" title="ABCMIZ_1:func.37">Non</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="abcmiz_1.html#K60" title="ABCMIZ_1:func.60">at</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_1.html#K37" title="ABCMIZ_1:func.37">Non</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="abcmiz_1.html#K60" title="ABCMIZ_1:func.60">at</a> <font color="Maroon" title="c3">f</font></span>)</span></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c4">b</font> being    <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon" title="c1">C</font>, <a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E3:260_1_2"/><i><font color="Green" title="E123">A0</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="abcmiz_1.html#K30" title="ABCMIZ_1:func.30">term</a> <font color="Maroon" title="c4">b</font> &amp;  <a href="abcmiz_1.html#K37" title="ABCMIZ_1:func.37">Non</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> )
 <b>by </b><i><a class="ref" href="abcmiz_1.html#T60" target="_self" title="ABCMIZ_1:th.60">Th45g</a></i>;<br/><a NAME="E4:260_1_2"/><i><font color="Green" title="E124">A1</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="abcmiz_1.html#K60" title="ABCMIZ_1:func.60">at</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="abcmiz_1.html#K30" title="ABCMIZ_1:func.30">term</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="abcmiz_1.html#K60" title="ABCMIZ_1:func.60">at</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:260_1_2"><i><font color="Green" title="E123">A0</font></i></a>, <a class="ref" href="abcmiz_1.html#D60" target="_self" title="ABCMIZ_1:def.60">SUBST</a></i>;<br/><a NAME="E5:260_1_2"/><b>then </b>
not <font color="Maroon" title="c2">a</font> <a href="abcmiz_1.html#K60" title="ABCMIZ_1:func.60">at</a> <font color="Maroon" title="c3">f</font> is <a href="abcmiz_1.html#V6" title="ABCMIZ_1:attr.6">positive</a>
 <b>by </b><i><a class="ref" href="abcmiz_1.html#D37" target="_self" title="ABCMIZ_1:def.37">POSITIVE</a></i>;<br/><a NAME="E6:260_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">k</font> being   <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon" title="c1">C</font>, <a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon" title="c1">C</font> st <br/>( <font color="Maroon" title="c2">a</font> <a href="abcmiz_1.html#K60" title="ABCMIZ_1:func.60">at</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="abcmiz_1.html#K30" title="ABCMIZ_1:func.30">term</a> <font color="Olive" title="b1">k</font> &amp;  <a href="abcmiz_1.html#K37" title="ABCMIZ_1:func.37">Non</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="abcmiz_1.html#K60" title="ABCMIZ_1:func.60">at</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> )
 <b>by </b><i><a class="ref" href="abcmiz_1.html#T60" target="_self" title="ABCMIZ_1:th.60">Th45g</a></i>;<br/><b>hence </b><a NAME="E7:260_1_2"/>
<span class="p1">(<span class="default"><a href="abcmiz_1.html#K37" title="ABCMIZ_1:func.37">Non</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="abcmiz_1.html#K60" title="ABCMIZ_1:func.60">at</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_1.html#K37" title="ABCMIZ_1:func.37">Non</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="abcmiz_1.html#K60" title="ABCMIZ_1:func.60">at</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:260_1_2"><i><font color="Green" title="E123">A0</font></i></a>, <a class="txt" href="#E4:260_1_2"><i><font color="Green" title="E124">A1</font></i></a>, <a class="ref" href="abcmiz_1.html#T44" target="_self" title="ABCMIZ_1:th.44">ThNon'</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

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