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<span class="kw">let </span><font color="Maroon" title="c1">x1</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">1 <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>  ) 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#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">x1</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&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="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> ; <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">1 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c2">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="c2">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>  ) 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#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">x1</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&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="c3">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default">1 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c2">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="c2">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>  ) 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#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">x1</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>,<font color="Maroon" title="c3">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">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>; <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="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>  ) implies <font color="Maroon" title="c4">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#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">x1</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>,<font color="Maroon" title="c3">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="E1:115"/><span class="lab"><font color="Green" title="E52">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="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>  )
 ; <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="c4">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#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">x1</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>,<font color="Maroon" title="c3">f</font></span>)</span> is  <a href="circcmb3.html#V6" title="CIRCCMB3:attr.6">with_nonpair_inputs</a> </span><br/>

<a NAME="E2:115"/><span class="lab"><font color="Green" title="E53">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#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">x1</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&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>
 
<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="115_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="E1:115_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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:115"><span class="lab"><font color="Green" title="E52">A1</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_1_1"><span class="kw">suppose </span></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>
 ; <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="c4">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#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">x1</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>,<font color="Maroon" title="c3">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">hence </span><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#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#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">x1</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:115"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="ref" href="circcmb3.html#T41" target="_self" title="CIRCCMB3:th.41">Th41</a></span>; <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="115_1_2"><span class="kw">suppose </span></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> 
 ; <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="c4">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#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">x1</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>,<font color="Maroon" title="c3">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="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="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">x1</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a></span>;<br/><span class="kw">hence </span><a NAME="E4:115_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#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#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">x1</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&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> 
 ; <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>

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