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<span class="kw">let </span><font color="Maroon" title="c5">f1</font>, <font color="Maroon" title="c6">f2</font> be    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font></span>)</span>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">v</font> being   <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font> holds <br/> ( ( <font color="Olive" title="b1">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c3">ll</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">r</font> implies <font color="Maroon" title="c5">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  ) &amp; ( not <font color="Olive" title="b1">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c3">ll</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">r</font> implies <font color="Maroon" title="c5">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  ) ) ) &amp; ( for <font color="Olive" title="b1">v</font> being   <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font> holds <br/> ( ( <font color="Olive" title="b1">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c3">ll</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">r</font> implies <font color="Maroon" title="c6">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  ) &amp; ( not <font color="Olive" title="b1">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c3">ll</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">r</font> implies <font color="Maroon" title="c6">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  ) ) ) implies <font color="Maroon" title="c5">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:15_1_2"/><span class="lab"><font color="Green" title="E6">A3</font></span>: 
for <font color="Olive" title="b1">v</font> being   <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font> holds <br/> ( ( <font color="Olive" title="b1">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c3">ll</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">r</font> implies <font color="Maroon" title="c5">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  ) &amp; ( not <font color="Olive" title="b1">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c3">ll</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">r</font> implies <font color="Maroon" title="c5">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  ) )
 <span class="kw">and </span><br/><a NAME="E2:15_1_2"/><span class="lab"><font color="Green" title="E7">A4</font></span>: 
for <font color="Olive" title="b1">v</font> being   <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font> holds <br/> ( ( <font color="Olive" title="b1">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c3">ll</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">r</font> implies <font color="Maroon" title="c6">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  ) &amp; ( not <font color="Olive" title="b1">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c3">ll</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">r</font> implies <font color="Maroon" title="c6">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f2</font></span><br/>

<a NAME="E3:15_1_2"/>
for <font color="Olive" title="b1">v</font> being   <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font> holds  <font color="Maroon" title="c5">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">v</font> be    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:15_1_2_1_1"/>
( <font color="Maroon" title="c7">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c3">ll</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">r</font> or  not <font color="Maroon" title="c7">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c3">ll</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">r</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:15_1_2_1_1_1"/><span class="lab"><font color="Green" title="E8">A5</font></span>: 
<font color="Maroon" title="c7">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c3">ll</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font></span><br/><div class="add"><a NAME="E2:15_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_2"><span class="lab"><font color="Green" title="E6">A3</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:15_1_2_1_1_1"/>
<font color="Maroon" title="c5">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15_1_2"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="txt" href="#E1:15_1_2_1_1_1"><span class="lab"><font color="Green" title="E8">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="15_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:15_1_2_1_1_2"/><span class="lab"><font color="Green" title="E8">A6</font></span>: 
not <font color="Maroon" title="c7">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c3">ll</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font></span><br/><div class="add"><a NAME="E2:15_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_2"><span class="lab"><font color="Green" title="E6">A3</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:15_1_2_1_1_2"/>
<font color="Maroon" title="c5">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15_1_2"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="txt" href="#E1:15_1_2_1_1_2"><span class="lab"><font color="Green" title="E8">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">hence </span><a NAME="E4:15_1_2"/>
<font color="Maroon" title="c5">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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