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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></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">c<sub>4</sub></font> be   <a href="funct_2.html#NM1">Function</a> of 2 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></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">c<sub>3</sub></font>;<br/>







<b>assume </b><a NAME="E1:118"/><i><font color="Green">E56</font></i>: 
( ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> or  not <font color="Maroon">c<sub>1</sub></font> is <a href="facirc_1.html#V1">pair</a> ) &amp; ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> or  not <font color="Maroon">c<sub>2</sub></font> is <a href="facirc_1.html#V1">pair</a> ) )
 ;<br/>

<a NAME="E2:118"/><i><font color="Green">E57</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#K6">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">c<sub>5</sub></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="118_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:118_1"/>
( ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> ) or ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">c<sub>5</sub></font> ) or ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> &amp; not <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">c<sub>5</sub></font> ) or ( not <font color="Maroon">c<sub>1</sub></font> is <a href="facirc_1.html#V1">pair</a> &amp; not <font color="Maroon">c<sub>2</sub></font> is <a href="facirc_1.html#V1">pair</a> ) )
 </span><b>by </b><i><a class="txt" href="circcmb3.html#E1:118"><i><font color="Green">E56</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="118_1_1"><b>suppose </b></a><a NAME="E1:118_1_1"/>
( ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> ) or ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">c<sub>5</sub></font> ) or ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> &amp; not <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">c<sub>5</sub></font> ) )
 ;<br/><div class="add"><a NAME="E2:118_1_1"/><b>then </b>
(  <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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#K6">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">c<sub>5</sub></font> or ( not <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span> is <a href="facirc_1.html#V2">with_pair</a> &amp;  <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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#K6">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> ) or ( not <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> is <a href="facirc_1.html#V2">with_pair</a> &amp;  <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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#K6">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="circcmb3.html#E1:118"><i><font color="Green">E56</font></i></a>, <a class="txt" href="circcmb3.html#E2:118"><i><font color="Green">E57</font></i></a>, <a class="txt" href="circcmb3.html#E67"><i><font color="Green">Lemma53</font></i></a>, <a class="ref" href="circcmb3.html#T42" target="_self">Th42</a>, <a class="ref" href="circcmb3.html#T43" target="_self">Th43</a>, <a class="ref" href="circcmb3.html#T44" target="_self">Th44</a></i>;<br/><b>hence </b><a NAME="E3:118_1_1"/>
not  <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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#K6">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> is <a href="facirc_1.html#V2">with_pair</a>
 <b>by </b><i><a class="txt" href="circcmb3.html#E68"><i><font color="Green">Lemma54</font></i></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="118_1_2"><b>suppose </b></a><a NAME="E1:118_1_2"/>
( not <font color="Maroon">c<sub>1</sub></font> is <a href="facirc_1.html#V1">pair</a> &amp; not <font color="Maroon">c<sub>2</sub></font> is <a href="facirc_1.html#V1">pair</a> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>2</sub></font> as  non <a href="facirc_1.html#V1">pair</a>  <a href="hidden.html#M1">set</a>  ;<br/><a NAME="E3:118_1_2"/>
(  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="facirc_1.html#K6">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K2">}</a></span> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K2">}</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#K2">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K2">}</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">c<sub>5</sub></font> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K2">}</a></span> <a href="subset_1.html#R1">misses</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="facirc_1.html#T5">FACIRC_1:5</a>, <a class="ref" href="finseq_2.html#T147">FINSEQ_2:147</a>, <a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/><a NAME="E4:118_1_2"/><b>then </b>
 <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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#K6">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<font color="Maroon">c<sub>4</sub></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">c<sub>5</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="txt" href="circcmb3.html#E2:118"><i><font color="Green">E57</font></i></a>, <a class="ref" href="circcmb3.html#T40" target="_self">Th40</a></i>;<br/><b>hence </b><a NAME="E5:118_1_2"/>
not  <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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#K6">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<font color="Maroon">c<sub>4</sub></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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