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<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of 1 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">X</font>,<font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">S</font> be  <a href="circcmb3.html#V6" target="_self">with_nonpair_inputs</a>  <a href="circcmb3.html#M1" target="_self">Signature</a> of <font color="Maroon">X</font>;<br/>







<b>assume </b><a NAME="E1:115"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> or  not <font color="Maroon">x1</font> is <a href="facirc_1.html#V1">pair</a> )
 ;<br/>

<a NAME="E2:115"/><i><font color="Green">E36</font></i>: 
not  <a href="circcmb3.html#K5" target="_self">Output</a> <span class="p1">(<span class="default"><a href="circcmb3.html#K6" target="_self">1GateCircStr</a> <span class="p2"><a href="facirc_1.html#K5">&lt;*</a><span class="default"><font color="Maroon">x1</font></span><a href="facirc_1.html#K5">*&gt;</a></span>,<font color="Maroon">f</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="facirc_1.html#D2">FACIRC_1:def 2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:115_1"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> or  not <font color="Maroon">x1</font> is <a href="facirc_1.html#V1">pair</a> )
 </span><b>by </b><i><a class="ref" href="circcmb3.html#T5" target="_self">Th5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_1_1"><b>suppose </b></a><a NAME="E1:115_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:115_1_1"/>
not  <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="circcmb3.html#K8" target="_self">+*</a> <span class="p2">(<span class="default"><a href="circcmb3.html#K6" target="_self">1GateCircStr</a> <span class="p3"><a href="facirc_1.html#K5">&lt;*</a><span class="default"><font color="Maroon">x1</font></span><a href="facirc_1.html#K5">*&gt;</a></span>,<font color="Maroon">f</font></span>)</span></span>)</span> is <a href="facirc_1.html#V2">with_pair</a>
 <b>by </b><i><a class="ref" href="circcmb3.html#T8" target="_self">Th8</a>, <a class="ref" href="circcmb3.html#D1" target="_self">Def1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="circcmb3.html#D11" target="_self">CIRCCMB3:def 11</a><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_1_2"><b>suppose </b></a><a NAME="E1:115_1_2"/>
not <font color="Maroon">x1</font> is <a href="facirc_1.html#V1">pair</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">x1</font> as  non <a href="facirc_1.html#V1">pair</a>  <a href="hidden.html#M1">set</a>  ;<br/><a NAME="E3:115_1_2"/>
(  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="facirc_1.html#K5">&lt;*</a><span class="default"><font color="Maroon">x1</font></span><a href="facirc_1.html#K5">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a</font></span><a href="tarski.html#K1">}</a></span> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a</font></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a</font></span><a href="tarski.html#K1">}</a></span> <a href="subset_1.html#R1">misses</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">S</font> )
 <b>by </b><i><a class="ref" href="facirc_1.html#T5">FACIRC_1:5</a>, <a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a>, <a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/><a NAME="E4:115_1_2"/><b>then </b>
 <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="circcmb3.html#K8" target="_self">+*</a> <span class="p2">(<span class="default"><a href="circcmb3.html#K6" target="_self">1GateCircStr</a> <span class="p3"><a href="facirc_1.html#K5">&lt;*</a><span class="default"><font color="Maroon">x1</font></span><a href="facirc_1.html#K5">*&gt;</a></span>,<font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">S</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="circcmb3.html#T8" target="_self">Th8</a>, </i>;<br/><b>hence </b><a NAME="E5:115_1_2"/>
not  <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="circcmb3.html#K8" target="_self">+*</a> <span class="p2">(<span class="default"><a href="circcmb3.html#K6" target="_self">1GateCircStr</a> <span class="p3"><a href="facirc_1.html#K5">&lt;*</a><span class="default"><font color="Maroon">x1</font></span><a href="facirc_1.html#K5">*&gt;</a></span>,<font color="Maroon">f</font></span>)</span></span>)</span> is <a href="facirc_1.html#V2">with_pair</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="circcmb3.html#D11" target="_self">CIRCCMB3:def 11</a><br/></div><b>end;</b></div></div><b>end;</b></div>

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