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<span class="kw">let </span><font color="Maroon" title="c1">Al</font> be    <a href="qc_lang1.html#M1" title="QC_LANG1:mode.1">QC-alphabet</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">  for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">P</font> being   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-pred_symbol</a> of <font color="Olive" title="b1">k</font>,<font color="Maroon" title="c1">Al</font><br/>  for <font color="Olive" title="b3">k</font>, <font color="Olive" title="b4">l</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">P</font> is   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-pred_symbol</a> of <font color="Olive" title="b3">k</font>,<font color="Maroon" title="c1">Al</font> &amp; <font color="Olive" title="b2">P</font> is   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-pred_symbol</a> of <font color="Olive" title="b4">l</font>,<font color="Maroon" title="c1">Al</font> holds <br/><font color="Olive" title="b3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">l</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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">  for <font color="Olive" title="b1">P</font> being   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-pred_symbol</a> of <font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c1">Al</font><br/>  for <font color="Olive" title="b2">k</font>, <font color="Olive" title="b3">l</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">P</font> is   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-pred_symbol</a> of <font color="Olive" title="b2">k</font>,<font color="Maroon" title="c1">Al</font> &amp; <font color="Olive" title="b1">P</font> is   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-pred_symbol</a> of <font color="Olive" title="b3">l</font>,<font color="Maroon" title="c1">Al</font> holds <br/><font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">l</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">P</font> be   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-pred_symbol</a> of <font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c1">Al</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">k</font>, <font color="Olive" title="b2">l</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c3">P</font> is   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-pred_symbol</a> of <font color="Olive" title="b1">k</font>,<font color="Maroon" title="c1">Al</font> &amp; <font color="Maroon" title="c3">P</font> is   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-pred_symbol</a> of <font color="Olive" title="b2">l</font>,<font color="Maroon" title="c1">Al</font> holds <br/><font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">l</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">k</font>, <font color="Maroon" title="c5">l</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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">P</font> is   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-pred_symbol</a> of <font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">Al</font> &amp; <font color="Maroon" title="c3">P</font> is   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-pred_symbol</a> of <font color="Maroon" title="c5">l</font>,<font color="Maroon" title="c1">Al</font> implies <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:2"/><span class="lab"><font color="Green" title="E3">A1</font></span>: 
( <font color="Maroon" title="c3">P</font> is   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-pred_symbol</a> of <font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">Al</font> &amp; <font color="Maroon" title="c3">P</font> is   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-pred_symbol</a> of <font color="Maroon" title="c5">l</font>,<font color="Maroon" title="c1">Al</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"> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font></span><br/>

<a NAME="E2:2"/><span class="kw">then </span>
<font color="Maroon" title="c3">P</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">l</font> <a href="qc_lang1.html#K8" title="QC_LANG1:func.8">-ary_QC-pred_symbols</a> <font color="Maroon" title="c1">Al</font>
 
;<br/>
<a NAME="E3:2"/><span class="kw">then </span>
<font color="Maroon" title="c3">P</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">P2</font> where <font color="Olive" title="b1">P2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K6" title="QC_LANG1:func.6">QC-pred_symbols</a> <font color="Maroon" title="c1">Al</font> :  <a href="qc_lang1.html#K7" title="QC_LANG1:func.7">the_arity_of</a> <font color="Olive" title="b1">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang1.html#D9" onmouseover="rs('qc_lang1/D9')" onmouseout="rh()">QC_LANG1:def 9</a></span>;<br/>
<a NAME="E4:2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E4">A2</font></span>: 
 ex <font color="Olive" title="b1">P2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K6" title="QC_LANG1:func.6">QC-pred_symbols</a> <font color="Maroon" title="c1">Al</font> st <br/>( <font color="Olive" title="b1">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font> &amp;  <a href="qc_lang1.html#K7" title="QC_LANG1:func.7">the_arity_of</a> <font color="Olive" title="b1">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> )
 
;<br/>
<a NAME="E5:2"/>
<font color="Maroon" title="c3">P</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">k</font> <a href="qc_lang1.html#K8" title="QC_LANG1:func.8">-ary_QC-pred_symbols</a> <font color="Maroon" title="c1">Al</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2"><span class="lab"><font color="Green" title="E3">A1</font></span></a></span>;<br/>
<a NAME="E6:2"/><span class="kw">then </span>
<font color="Maroon" title="c3">P</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">P1</font> where <font color="Olive" title="b1">P1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K6" title="QC_LANG1:func.6">QC-pred_symbols</a> <font color="Maroon" title="c1">Al</font> :  <a href="qc_lang1.html#K7" title="QC_LANG1:func.7">the_arity_of</a> <font color="Olive" title="b1">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang1.html#D9" onmouseover="rs('qc_lang1/D9')" onmouseout="rh()">QC_LANG1:def 9</a></span>;<br/>
<a NAME="E7:2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">P1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K6" title="QC_LANG1:func.6">QC-pred_symbols</a> <font color="Maroon" title="c1">Al</font> st <br/>( <font color="Olive" title="b1">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font> &amp;  <a href="qc_lang1.html#K7" title="QC_LANG1:func.7">the_arity_of</a> <font color="Olive" title="b1">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E8:2"/>
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2"><span class="lab"><font color="Green" title="E4">A2</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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