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<span class="kw">let </span><font color="Maroon" title="c2">C</font> be   non  <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>  <a href="catalg_1.html#NM1" title="CATALG_1:NM.1">Signature</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">o</font> being   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c2">C</font> holds <br/> ( ( <font color="Olive" title="b1">o</font> is  <a href="abcmiz_a.html#V4" title="ABCMIZ_A:attr.4">nullary</a>  implies  not <font color="Olive" title="b1">o</font> is  <a href="abcmiz_a.html#V5" title="ABCMIZ_A:attr.5">unary</a>  ) &amp; ( <font color="Olive" title="b1">o</font> is  <a href="abcmiz_a.html#V4" title="ABCMIZ_A:attr.4">nullary</a>  implies  not <font color="Olive" title="b1">o</font> is  <a href="abcmiz_a.html#V6" title="ABCMIZ_A:attr.6">binary</a>  ) &amp; ( <font color="Olive" title="b1">o</font> is  <a href="abcmiz_a.html#V5" title="ABCMIZ_A:attr.5">unary</a>  implies  not <font color="Olive" title="b1">o</font> is  <a href="abcmiz_a.html#V6" title="ABCMIZ_A:attr.6">binary</a>  ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">o</font> be   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c2">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c3">o</font> is  <a href="abcmiz_a.html#V4" title="ABCMIZ_A:attr.4">nullary</a>  implies  not <font color="Maroon" title="c3">o</font> is  <a href="abcmiz_a.html#V5" title="ABCMIZ_A:attr.5">unary</a>  ) &amp; ( <font color="Maroon" title="c3">o</font> is  <a href="abcmiz_a.html#V4" title="ABCMIZ_A:attr.4">nullary</a>  implies  not <font color="Maroon" title="c3">o</font> is  <a href="abcmiz_a.html#V6" title="ABCMIZ_A:attr.6">binary</a>  ) &amp; ( <font color="Maroon" title="c3">o</font> is  <a href="abcmiz_a.html#V5" title="ABCMIZ_A:attr.5">unary</a>  implies  not <font color="Maroon" title="c3">o</font> is  <a href="abcmiz_a.html#V6" title="ABCMIZ_A:attr.6">binary</a>  ) )</span><br/>



<span class="kw">thus </span><a NAME="E1:60"/>
( <font color="Maroon" title="c3">o</font> is  <a href="abcmiz_a.html#V4" title="ABCMIZ_A:attr.4">nullary</a>  implies  not <font color="Maroon" title="c3">o</font> is  <a href="abcmiz_a.html#V5" title="ABCMIZ_A:attr.5">unary</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c3">o</font> is  <a href="abcmiz_a.html#V4" title="ABCMIZ_A:attr.4">nullary</a>  implies  not <font color="Maroon" title="c3">o</font> is  <a href="abcmiz_a.html#V6" title="ABCMIZ_A:attr.6">binary</a>  ) &amp; ( <font color="Maroon" title="c3">o</font> is  <a href="abcmiz_a.html#V5" title="ABCMIZ_A:attr.5">unary</a>  implies  not <font color="Maroon" title="c3">o</font> is  <a href="abcmiz_a.html#V6" title="ABCMIZ_A:attr.6">binary</a>  ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:60_1"/>
 <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c3">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="abcmiz_a.html#D13" target="_self" title="ABCMIZ_A:def.13">ABCMIZ_A:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c3">o</font> is  <a href="abcmiz_a.html#V5" title="ABCMIZ_A:attr.5">unary</a> </span><br/>

<span class="kw">hence </span><a NAME="E2:60_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c3">o</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="abcmiz_a.html#D14" target="_self" title="ABCMIZ_A:def.14">ABCMIZ_A:def 14</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E2:60"/>
( <font color="Maroon" title="c3">o</font> is  <a href="abcmiz_a.html#V4" title="ABCMIZ_A:attr.4">nullary</a>  implies  not <font color="Maroon" title="c3">o</font> is  <a href="abcmiz_a.html#V6" title="ABCMIZ_A:attr.6">binary</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">o</font> is  <a href="abcmiz_a.html#V5" title="ABCMIZ_A:attr.5">unary</a>  implies  not <font color="Maroon" title="c3">o</font> is  <a href="abcmiz_a.html#V6" title="ABCMIZ_A:attr.6">binary</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:60_2"/>
 <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c3">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="abcmiz_a.html#D13" target="_self" title="ABCMIZ_A:def.13">ABCMIZ_A:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c3">o</font> is  <a href="abcmiz_a.html#V6" title="ABCMIZ_A:attr.6">binary</a> </span><br/>

<span class="kw">hence </span><a NAME="E2:60_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c3">o</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 2
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="abcmiz_a.html#D15" target="_self" title="ABCMIZ_A:def.15">ABCMIZ_A:def 15</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E3:60"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c3">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="abcmiz_a.html#D14" target="_self" title="ABCMIZ_A:def.14">ABCMIZ_A:def 14</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c3">o</font> is  <a href="abcmiz_a.html#V6" title="ABCMIZ_A:attr.6">binary</a> </span><br/>

<span class="kw">hence </span><a NAME="E4:60"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c3">o</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 2
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="abcmiz_a.html#D15" target="_self" title="ABCMIZ_A:def.15">ABCMIZ_A:def 15</a> <a class="txt" 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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