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<span class="kw">let </span><font color="Maroon" title="c1">S</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</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="c1">S</font> is  <a href="substut1.html#V5" title="SUBSTUT1:attr.5">Sub_negative</a>  implies  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K27" title="SUBSTUT1:func.27">Sub_the_argument_of</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:68"/>
<font color="Maroon" title="c1">S</font> is  <a href="substut1.html#V5" title="SUBSTUT1:attr.5">Sub_negative</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K27" title="SUBSTUT1:func.27">Sub_the_argument_of</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c2">S9</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a> <span class="kw"> such that </span><br/><a NAME="E3:68"/><span class="lab"><font color="Green" title="E38">A1</font></span>: 
<font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="substut1.html#K20" title="SUBSTUT1:func.20">Sub_not</a> <font color="Maroon" title="c2">S9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="substut1.html#D26" target="_self" title="SUBSTUT1:def.26">Def26</a></span>;<br/>
<a NAME="E4:68"/><span class="lab"><font color="Green" title="E39">A2</font></span>: 
 <a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S9</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang1.html#D18" title="QC_LANG1:def.18">QC_LANG1:def 18</a></span>;<br/>
<a NAME="E5:68"/>
<font color="Maroon" title="c1">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S9</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:68"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>;<br/>
<a NAME="E6:68"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A3</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">S9</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:68"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="ref" href="qc_lang1.html#T45" title="QC_LANG1:th.45">QC_LANG1:45</a></span>;<br/>
<a NAME="E7:68"/>
<span class="p1">(<span class="default"><a href="substut1.html#K27" title="SUBSTUT1:func.27">Sub_the_argument_of</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">S9</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:68"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="ref" href="substut1.html#D30" target="_self" title="SUBSTUT1:def.30">Def30</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:68"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K27" title="SUBSTUT1:func.27">Sub_the_argument_of</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:68"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="ref" href="qc_lang2.html#T2" title="QC_LANG2:th.2">QC_LANG2:2</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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