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<b>let </b><font color="Maroon" title="c1">x1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of 1 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c2">X</font>;<br/><b>let </b><font color="Maroon" title="c4">S</font> be  <a href="circcmb3.html#V6" title="CIRCCMB3:attr.6">with_nonpair_inputs</a>  <a href="circcmb3.html#M1" title="CIRCCMB3:mode.1">Signature</a> of <font color="Maroon" title="c2">X</font>;<br/>







<b>assume </b><a NAME="E1:115"/><i><font color="Green" title="E53">A1</font></i>: 
( <font color="Maroon" title="c1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">S</font> or  not <font color="Maroon" title="c1">x1</font> is <a href="facirc_1.html#V1" title="FACIRC_1:attr.1">pair</a> )
 ;<br/>

<a NAME="E2:115"/><i><font color="Green" title="E54">A2</font></i>: 
not  <a href="circcmb3.html#K5" title="CIRCCMB3:func.5">Output</a> <span class="p1">(<span class="default"><a href="circcmb3.html#K6" title="CIRCCMB3:func.6">1GateCircStr</a> <span class="p2"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c1">x1</font></span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span>,<font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c4">S</font>
 
<b>by </b><i><a class="ref" href="facirc_1.html#D2" title="FACIRC_1:def.2">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" title="c1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">S</font> or  not <font color="Maroon" title="c1">x1</font> is <a href="facirc_1.html#V1" title="FACIRC_1:attr.1">pair</a> )
 </span><b>by </b><i><a class="txt" href="circcmb3.html#E1:115"><i><font color="Green" title="E53">A1</font></i></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" title="c1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">S</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:115_1_1"/>
not  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="circcmb3.html#K8" title="CIRCCMB3:func.8">+*</a> <span class="p2">(<span class="default"><a href="circcmb3.html#K6" title="CIRCCMB3:func.6">1GateCircStr</a> <span class="p3"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c1">x1</font></span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span>,<font color="Maroon" title="c3">f</font></span>)</span></span>)</span> is <a href="facirc_1.html#V2" title="FACIRC_1:attr.2">with_pair</a>
 <b>by </b><i><a class="txt" href="circcmb3.html#E2:115"><i><font color="Green" title="E54">A2</font></i></a>, <a class="ref" href="circcmb3.html#T41" target="_self" title="CIRCCMB3:th.41">Th41</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="circcmb3.html#D11" target="_self" title="CIRCCMB3:def.11">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" title="c1">x1</font> is <a href="facirc_1.html#V1" title="FACIRC_1:attr.1">pair</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c5">a</font> = <font color="Maroon" title="c1">x1</font> as  non <a href="facirc_1.html#V1" title="FACIRC_1:attr.1">pair</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><a NAME="E3:115_1_2"/>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c1">x1</font></span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">S</font> &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon" title="c4">S</font> )
 <b>by </b><i><a class="ref" href="facirc_1.html#T5" title="FACIRC_1:th.5">FACIRC_1:5</a>, <a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a>, <a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/><a NAME="E4:115_1_2"/><b>then </b>
 <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="circcmb3.html#K8" title="CIRCCMB3:func.8">+*</a> <span class="p2">(<span class="default"><a href="circcmb3.html#K6" title="CIRCCMB3:func.6">1GateCircStr</a> <span class="p3"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c1">x1</font></span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span>,<font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c4">S</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="circcmb3.html#E2:115"><i><font color="Green" title="E54">A2</font></i></a>, <a class="ref" href="circcmb3.html#T40" target="_self" title="CIRCCMB3:th.40">Th40</a></i>;<br/><b>hence </b><a NAME="E5:115_1_2"/>
not  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="circcmb3.html#K8" title="CIRCCMB3:func.8">+*</a> <span class="p2">(<span class="default"><a href="circcmb3.html#K6" title="CIRCCMB3:func.6">1GateCircStr</a> <span class="p3"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c1">x1</font></span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span>,<font color="Maroon" title="c3">f</font></span>)</span></span>)</span> is <a href="facirc_1.html#V2" title="FACIRC_1:attr.2">with_pair</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="circcmb3.html#D11" target="_self" title="CIRCCMB3:def.11">CIRCCMB3:def 11</a><br/></div><b>end;</b></div></div><b>end;</b></div>

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