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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">S</font> is <a href="substut1.html#V4" title="SUBSTUT1:attr.4">Sub_atomic</a> implies <font color="Maroon" title="c1">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a>  is <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a> )</span><br/>

<b>assume </b><a NAME="E1:47"/>
<font color="Maroon" title="c1">S</font> is <a href="substut1.html#V4" title="SUBSTUT1:attr.4">Sub_atomic</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a>  is <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a></span><br/>

<b>then consider </b><font color="Maroon" title="c2">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Maroon" title="c3">P</font> being   <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Maroon" title="c2">k</font>, <font color="Maroon" title="c4">ll</font> being    <a href="qc_lang1.html#M1" title="QC_LANG1:mode.1">QC-variable_list</a> of <font color="Maroon" title="c2">k</font>, <font color="Maroon" title="c5">e</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="substut1.html#K1" title="SUBSTUT1:func.1">vSUB</a> <b> such that </b><br/><a NAME="E3:47"/><i><font color="Green" title="E23">A1</font></i>: 
<font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="substut1.html#K17" title="SUBSTUT1:func.17">Sub_P</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">ll</font>,<font color="Maroon" title="c5">e</font>
 <b>by </b><i><a class="ref" href="substut1.html#D25" target="_self" title="SUBSTUT1:def.25">Def25</a></i>;<br/>
<a NAME="E4:47"/>
<font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c4">ll</font></span>)</span>,<font color="Maroon" title="c5">e</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<b>by </b><i><a class="txt" href="#E3:47"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="substut1.html#T9" target="_self" title="SUBSTUT1:th.9">Th9</a></i>;<br/>
<a NAME="E5:47"/><b>then </b>
<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> <font color="Maroon" title="c3">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c4">ll</font>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<b>hence </b><a NAME="E6:47"/>
<font color="Maroon" title="c1">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a>  is <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a>
 <b>by </b><i><a class="ref" href="qc_lang1.html#D17" title="QC_LANG1:def.17">QC_LANG1:def 17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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