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<span class="kw">let </span><font color="Maroon" title="c1">x1</font>, <font color="Maroon" title="c2">x2</font>, <font color="Maroon" title="c3">x3</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">X</font> being 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/> for <font color="Olive" title="b2">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default">3 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Olive" title="b1">X</font></span>)</span>,<font color="Olive" title="b1">X</font><br/> for <font color="Olive" title="b3">S</font> being  <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="Olive" title="b1">X</font>  st ( <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="Olive" title="b3">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>  ) &amp; ( <font color="Maroon" title="c2">x2</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="Olive" title="b3">S</font> or  not <font color="Maroon" title="c2">x2</font> is  <a href="facirc_1.html#V1" title="FACIRC_1:attr.1">pair</a>  ) &amp; ( <font color="Maroon" title="c3">x3</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="Olive" title="b3">S</font> or  not <font color="Maroon" title="c3">x3</font> is  <a href="facirc_1.html#V1" title="FACIRC_1:attr.1">pair</a>  ) holds <br/><font color="Olive" title="b3">S</font> <a href="circcmb3.html#K6" title="CIRCCMB3:func.6">+*</a> <span class="p1">(<span class="default"><a href="circcmb3.html#K4" title="CIRCCMB3:func.4">1GateCircStr</a> <span class="p2"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font>,<font color="Maroon" title="c3">x3</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>,<font color="Olive" title="b2">f</font></span>)</span> is  <a href="circcmb3.html#V6" title="CIRCCMB3:attr.6">with_nonpair_inputs</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default">3 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c4">X</font></span>)</span>,<font color="Maroon" title="c4">X</font><br/> for <font color="Olive" title="b2">S</font> being  <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="c4">X</font>  st ( <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="Olive" title="b2">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>  ) &amp; ( <font color="Maroon" title="c2">x2</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="Olive" title="b2">S</font> or  not <font color="Maroon" title="c2">x2</font> is  <a href="facirc_1.html#V1" title="FACIRC_1:attr.1">pair</a>  ) &amp; ( <font color="Maroon" title="c3">x3</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="Olive" title="b2">S</font> or  not <font color="Maroon" title="c3">x3</font> is  <a href="facirc_1.html#V1" title="FACIRC_1:attr.1">pair</a>  ) holds <br/><font color="Olive" title="b2">S</font> <a href="circcmb3.html#K6" title="CIRCCMB3:func.6">+*</a> <span class="p1">(<span class="default"><a href="circcmb3.html#K4" title="CIRCCMB3:func.4">1GateCircStr</a> <span class="p2"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font>,<font color="Maroon" title="c3">x3</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>,<font color="Olive" title="b1">f</font></span>)</span> is  <a href="circcmb3.html#V6" title="CIRCCMB3:attr.6">with_nonpair_inputs</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c5">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default">3 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c4">X</font></span>)</span>,<font color="Maroon" title="c4">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">S</font> being  <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="c4">X</font>  st ( <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="Olive" title="b1">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>  ) &amp; ( <font color="Maroon" title="c2">x2</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="Olive" title="b1">S</font> or  not <font color="Maroon" title="c2">x2</font> is  <a href="facirc_1.html#V1" title="FACIRC_1:attr.1">pair</a>  ) &amp; ( <font color="Maroon" title="c3">x3</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="Olive" title="b1">S</font> or  not <font color="Maroon" title="c3">x3</font> is  <a href="facirc_1.html#V1" title="FACIRC_1:attr.1">pair</a>  ) holds <br/><font color="Olive" title="b1">S</font> <a href="circcmb3.html#K6" title="CIRCCMB3:func.6">+*</a> <span class="p1">(<span class="default"><a href="circcmb3.html#K4" title="CIRCCMB3:func.4">1GateCircStr</a> <span class="p2"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font>,<font color="Maroon" title="c3">x3</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>,<font color="Maroon" title="c5">f</font></span>)</span> is  <a href="circcmb3.html#V6" title="CIRCCMB3:attr.6">with_nonpair_inputs</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c6">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="c4">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <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="c6">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>  ) &amp; ( <font color="Maroon" title="c2">x2</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="c6">S</font> or  not <font color="Maroon" title="c2">x2</font> is  <a href="facirc_1.html#V1" title="FACIRC_1:attr.1">pair</a>  ) &amp; ( <font color="Maroon" title="c3">x3</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="c6">S</font> or  not <font color="Maroon" title="c3">x3</font> is  <a href="facirc_1.html#V1" title="FACIRC_1:attr.1">pair</a>  ) implies <font color="Maroon" title="c6">S</font> <a href="circcmb3.html#K6" title="CIRCCMB3:func.6">+*</a> <span class="p1">(<span class="default"><a href="circcmb3.html#K4" title="CIRCCMB3:func.4">1GateCircStr</a> <span class="p2"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font>,<font color="Maroon" title="c3">x3</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>,<font color="Maroon" title="c5">f</font></span>)</span> is  <a href="circcmb3.html#V6" title="CIRCCMB3:attr.6">with_nonpair_inputs</a>  )</span><br/>







<span class="kw">assume </span><a NAME="E1:121"/><span class="lab"><font color="Green" title="E54">A1</font></span>: 
( ( <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="c6">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>  ) &amp; ( <font color="Maroon" title="c2">x2</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="c6">S</font> or  not <font color="Maroon" title="c2">x2</font> is  <a href="facirc_1.html#V1" title="FACIRC_1:attr.1">pair</a>  ) &amp; ( <font color="Maroon" title="c3">x3</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="c6">S</font> or  not <font color="Maroon" title="c3">x3</font> is  <a href="facirc_1.html#V1" title="FACIRC_1:attr.1">pair</a>  ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">S</font> <a href="circcmb3.html#K6" title="CIRCCMB3:func.6">+*</a> <span class="p1">(<span class="default"><a href="circcmb3.html#K4" title="CIRCCMB3:func.4">1GateCircStr</a> <span class="p2"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font>,<font color="Maroon" title="c3">x3</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>,<font color="Maroon" title="c5">f</font></span>)</span> is  <a href="circcmb3.html#V6" title="CIRCCMB3:attr.6">with_nonpair_inputs</a> </span><br/>

<a NAME="E2:121"/><span class="lab"><font color="Green" title="E55">A2</font></span>: 
not  <a href="circcmb3.html#K3" title="CIRCCMB3:func.3">Output</a> <span class="p1">(<span class="default"><a href="circcmb3.html#K4" title="CIRCCMB3:func.4">1GateCircStr</a> <span class="p2"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font>,<font color="Maroon" title="c3">x3</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>,<font color="Maroon" title="c5">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="c6">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="facirc_1.html#D2" title="FACIRC_1:def.2">FACIRC_1:def 2</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="121_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:121_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="c6">S</font> &amp; <font color="Maroon" title="c2">x2</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="c6">S</font> &amp; <font color="Maroon" title="c3">x3</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="c6">S</font> ) or ( <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="c6">S</font> &amp; not <font color="Maroon" title="c2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon" title="c6">S</font> &amp; <font color="Maroon" title="c3">x3</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="c6">S</font> ) or ( <font color="Maroon" title="c2">x2</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="c6">S</font> &amp; not <font color="Maroon" title="c1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon" title="c6">S</font> &amp; <font color="Maroon" title="c3">x3</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="c6">S</font> ) or ( <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="c6">S</font> &amp; <font color="Maroon" title="c2">x2</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="c6">S</font> &amp; not <font color="Maroon" title="c3">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon" title="c6">S</font> ) or ( <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="c6">S</font> &amp; not <font color="Maroon" title="c2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon" title="c6">S</font> &amp; not <font color="Maroon" title="c3">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon" title="c6">S</font> ) or ( <font color="Maroon" title="c2">x2</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="c6">S</font> &amp; not <font color="Maroon" title="c1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon" title="c6">S</font> &amp; not <font color="Maroon" title="c3">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon" title="c6">S</font> ) or ( not <font color="Maroon" title="c1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon" title="c6">S</font> &amp; not <font color="Maroon" title="c2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon" title="c6">S</font> &amp; <font color="Maroon" title="c3">x3</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="c6">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>  &amp; not <font color="Maroon" title="c2">x2</font> is  <a href="facirc_1.html#V1" title="FACIRC_1:attr.1">pair</a>  &amp; not <font color="Maroon" title="c3">x3</font> is  <a href="facirc_1.html#V1" title="FACIRC_1:attr.1">pair</a>  ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:121"><span class="lab"><font color="Green" title="E54">A1</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="121_1_1"><span class="kw">suppose </span></a><a NAME="E1:121_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="c6">S</font> &amp; <font color="Maroon" title="c2">x2</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="c6">S</font> &amp; <font color="Maroon" title="c3">x3</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="c6">S</font> ) or ( <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="c6">S</font> &amp; not <font color="Maroon" title="c2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon" title="c6">S</font> &amp; <font color="Maroon" title="c3">x3</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="c6">S</font> ) or ( <font color="Maroon" title="c2">x2</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="c6">S</font> &amp; not <font color="Maroon" title="c1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon" title="c6">S</font> &amp; <font color="Maroon" title="c3">x3</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="c6">S</font> ) or ( <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="c6">S</font> &amp; <font color="Maroon" title="c2">x2</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="c6">S</font> &amp; not <font color="Maroon" title="c3">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon" title="c6">S</font> ) or ( <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="c6">S</font> &amp; not <font color="Maroon" title="c2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon" title="c6">S</font> &amp; not <font color="Maroon" title="c3">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon" title="c6">S</font> ) or ( <font color="Maroon" title="c2">x2</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="c6">S</font> &amp; not <font color="Maroon" title="c1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon" title="c6">S</font> &amp; not <font color="Maroon" title="c3">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon" title="c6">S</font> ) or ( not <font color="Maroon" title="c1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon" title="c6">S</font> &amp; not <font color="Maroon" title="c2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon" title="c6">S</font> &amp; <font color="Maroon" title="c3">x3</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="c6">S</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">S</font> <a href="circcmb3.html#K6" title="CIRCCMB3:func.6">+*</a> <span class="p1">(<span class="default"><a href="circcmb3.html#K4" title="CIRCCMB3:func.4">1GateCircStr</a> <span class="p2"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font>,<font color="Maroon" title="c3">x3</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>,<font color="Maroon" title="c5">f</font></span>)</span> is  <a href="circcmb3.html#V6" title="CIRCCMB3:attr.6">with_nonpair_inputs</a> </span><br/><div class="add"><a NAME="E2:121_1_1"/><span class="kw">then </span>
(  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="circcmb3.html#K6" title="CIRCCMB3:func.6">+*</a> <span class="p2">(<span class="default"><a href="circcmb3.html#K4" title="CIRCCMB3:func.4">1GateCircStr</a> <span class="p3"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font>,<font color="Maroon" title="c3">x3</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>,<font color="Maroon" title="c5">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c6">S</font> or ( not <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">x2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is  <a href="facirc_1.html#V2" title="FACIRC_1:attr.2">with_pair</a>  &amp;  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="circcmb3.html#K6" title="CIRCCMB3:func.6">+*</a> <span class="p2">(<span class="default"><a href="circcmb3.html#K4" title="CIRCCMB3:func.4">1GateCircStr</a> <span class="p3"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font>,<font color="Maroon" title="c3">x3</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>,<font color="Maroon" title="c5">f</font></span>)</span></span>)</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="c2">x2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c6">S</font></span>)</span> ) or ( not <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is  <a href="facirc_1.html#V2" title="FACIRC_1:attr.2">with_pair</a>  &amp;  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="circcmb3.html#K6" title="CIRCCMB3:func.6">+*</a> <span class="p2">(<span class="default"><a href="circcmb3.html#K4" title="CIRCCMB3:func.4">1GateCircStr</a> <span class="p3"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font>,<font color="Maroon" title="c3">x3</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>,<font color="Maroon" title="c5">f</font></span>)</span></span>)</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="c1">x1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c6">S</font></span>)</span> ) or ( not <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x3</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is  <a href="facirc_1.html#V2" title="FACIRC_1:attr.2">with_pair</a>  &amp;  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="circcmb3.html#K6" title="CIRCCMB3:func.6">+*</a> <span class="p2">(<span class="default"><a href="circcmb3.html#K4" title="CIRCCMB3:func.4">1GateCircStr</a> <span class="p3"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font>,<font color="Maroon" title="c3">x3</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>,<font color="Maroon" title="c5">f</font></span>)</span></span>)</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="c3">x3</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c6">S</font></span>)</span> ) or ( not <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> is  <a href="facirc_1.html#V2" title="FACIRC_1:attr.2">with_pair</a>  &amp;  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="circcmb3.html#K6" title="CIRCCMB3:func.6">+*</a> <span class="p2">(<span class="default"><a href="circcmb3.html#K4" title="CIRCCMB3:func.4">1GateCircStr</a> <span class="p3"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font>,<font color="Maroon" title="c3">x3</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>,<font color="Maroon" title="c5">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c6">S</font></span>)</span> ) or ( not <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">x2</font>,<font color="Maroon" title="c3">x3</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> is  <a href="facirc_1.html#V2" title="FACIRC_1:attr.2">with_pair</a>  &amp;  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="circcmb3.html#K6" title="CIRCCMB3:func.6">+*</a> <span class="p2">(<span class="default"><a href="circcmb3.html#K4" title="CIRCCMB3:func.4">1GateCircStr</a> <span class="p3"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font>,<font color="Maroon" title="c3">x3</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>,<font color="Maroon" title="c5">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">x2</font>,<font color="Maroon" title="c3">x3</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c6">S</font></span>)</span> ) or ( not <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c3">x3</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> is  <a href="facirc_1.html#V2" title="FACIRC_1:attr.2">with_pair</a>  &amp;  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="circcmb3.html#K6" title="CIRCCMB3:func.6">+*</a> <span class="p2">(<span class="default"><a href="circcmb3.html#K4" title="CIRCCMB3:func.4">1GateCircStr</a> <span class="p3"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font>,<font color="Maroon" title="c3">x3</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>,<font color="Maroon" title="c5">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c3">x3</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c6">S</font></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:121"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="txt" href="#E2:121"><span class="lab"><font color="Green" title="E55">A2</font></span></a>, <a class="ref" href="circcmb3.html#T45" target="_self" title="CIRCCMB3:th.45">Th45</a>, <a class="ref" href="circcmb3.html#T46" target="_self" title="CIRCCMB3:th.46">Th46</a>, <a class="ref" href="circcmb3.html#T47" target="_self" title="CIRCCMB3:th.47">Th47</a>, <a class="ref" href="circcmb3.html#T48" target="_self" title="CIRCCMB3:th.48">Th48</a>, <a class="ref" href="circcmb3.html#T49" target="_self" title="CIRCCMB3:th.49">Th49</a>, <a class="ref" href="circcmb3.html#T50" target="_self" title="CIRCCMB3:th.50">Th50</a>, <a class="ref" href="circcmb3.html#T51" target="_self" title="CIRCCMB3:th.51">Th51</a></span>;<br/><span class="kw">hence </span><a NAME="E3:121_1_1"/>
not  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="circcmb3.html#K6" title="CIRCCMB3:func.6">+*</a> <span class="p2">(<span class="default"><a href="circcmb3.html#K4" title="CIRCCMB3:func.4">1GateCircStr</a> <span class="p3"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font>,<font color="Maroon" title="c3">x3</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>,<font color="Maroon" title="c5">f</font></span>)</span></span>)</span> is  <a href="facirc_1.html#V2" title="FACIRC_1:attr.2">with_pair</a> 
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="circcmb3.html#D11" target="_self" title="CIRCCMB3:def.11">CIRCCMB3:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="121_1_2"><span class="kw">suppose </span></a><a NAME="E1:121_1_2"/>
( not <font color="Maroon" title="c1">x1</font> is  <a href="facirc_1.html#V1" title="FACIRC_1:attr.1">pair</a>  &amp; not <font color="Maroon" title="c2">x2</font> is  <a href="facirc_1.html#V1" title="FACIRC_1:attr.1">pair</a>  &amp; not <font color="Maroon" title="c3">x3</font> is  <a href="facirc_1.html#V1" title="FACIRC_1:attr.1">pair</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">S</font> <a href="circcmb3.html#K6" title="CIRCCMB3:func.6">+*</a> <span class="p1">(<span class="default"><a href="circcmb3.html#K4" title="CIRCCMB3:func.4">1GateCircStr</a> <span class="p2"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font>,<font color="Maroon" title="c3">x3</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>,<font color="Maroon" title="c5">f</font></span>)</span> is  <a href="circcmb3.html#V6" title="CIRCCMB3:attr.6">with_nonpair_inputs</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">a</font> = <font color="Maroon" title="c1">x1</font>, <font color="Maroon" title="c8">b</font> = <font color="Maroon" title="c2">x2</font>, <font color="Maroon" title="c9">c</font> = <font color="Maroon" title="c3">x3</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="E3:121_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font>,<font color="Maroon" title="c3">x3</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T148" title="FINSEQ_2:th.148">FINSEQ_2:148</a></span>;<br/><span class="kw">hence </span><a NAME="E4:121_1_2"/>
not  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="circcmb3.html#K6" title="CIRCCMB3:func.6">+*</a> <span class="p2">(<span class="default"><a href="circcmb3.html#K4" title="CIRCCMB3:func.4">1GateCircStr</a> <span class="p3"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font>,<font color="Maroon" title="c3">x3</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>,<font color="Maroon" title="c5">f</font></span>)</span></span>)</span> is  <a href="facirc_1.html#V2" title="FACIRC_1:attr.2">with_pair</a> 
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="circcmb3.html#D11" target="_self" title="CIRCCMB3:def.11">CIRCCMB3:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
