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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="facirc_1.html#V2" title="FACIRC_1:attr.2">nonpair-yielding</a>  <a href="circcomb.html#NM2" title="CIRCCOMB:NM.2">FinSeqLen</a> of <font color="Maroon" title="c1">n</font><br/>  for <font color="Olive" title="b3">s</font> being   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="gfacirc2.html#K6" title="GFACIRC2:func.6">-BitGFA1Circ</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>)</span>)</span> holds   <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Olive" title="b3">s</font>,<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>) is  <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c3">g</font> be   <a href="facirc_1.html#V2" title="FACIRC_1:attr.2">nonpair-yielding</a>  <a href="circcomb.html#NM2" title="CIRCCOMB:NM.2">FinSeqLen</a> of <font color="Maroon" title="c1">n</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">s</font> being   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="gfacirc2.html#K6" title="GFACIRC2:func.6">-BitGFA1Circ</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font>)</span>)</span> holds   <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>) is  <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a> </span><br/>



<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ,  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>) <span class="kw">-&gt; </span>   <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a>  =  <a href="gfacirc1.html#K25" title="GFACIRC1:func.25">BitGFA1Str</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">$2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">$2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,$1);<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ,  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>) <span class="kw">-&gt; </span>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> over  <a href="gfacirc1.html#K25" title="GFACIRC1:func.25">BitGFA1Str</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">$2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">$2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,$1) =  <a href="gfacirc1.html#K26" title="GFACIRC1:func.26">BitGFA1Circ</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">$2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">$2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,$1);<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>3</sub></font>(   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ,  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <span class="p1">(<span class="default"><a href="gfacirc1.html#K19" title="GFACIRC1:func.19">GFA1CarryStr</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default">$2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default">$2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,$1)</span>)</span> =  <a href="gfacirc1.html#K21" title="GFACIRC1:func.21">GFA1CarryOutput</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">$2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">$2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,$1);<br/>
<span class="kw">set </span><font color="Maroon" title="c4">S0</font> =  <a href="circcomb.html#K5" title="CIRCCOMB:func.5">1GateCircStr</a> (<a href="facirc_2.html#NK3" title="FACIRC_2:NK.3">&lt;*&gt;</a>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a></span>)</span>);<br/>
<span class="kw">set </span><font color="Maroon" title="c5">A0</font> =  <a href="circcomb.html#K7" title="CIRCCOMB:func.7">1GateCircuit</a> (<a href="facirc_2.html#NK3" title="FACIRC_2:NK.3">&lt;*&gt;</a>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a></span>)</span>);<br/>
<span class="kw">set </span><font color="Maroon" title="c6">h0</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="facirc_2.html#NK3" title="FACIRC_2:NK.3">&lt;*&gt;</a>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">N</font> = (1,2) <a href="funct_7.html#K13" title="FUNCT_7:func.13">followed_by</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<font color="Maroon" title="c1">n</font>,<a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>)</span>)</span>;<br/>
<a NAME="E1:42"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
<span class="p1">(<span class="default">(1,2) <a href="funct_7.html#K13" title="FUNCT_7:func.13">followed_by</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<font color="Maroon" title="c1">n</font>,<a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>)</span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T122" onmouseover="rs('funct_7/T122')" onmouseout="rh()">FUNCT_7:122</a></span>;<br/>
<a NAME="E2:42"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
<span class="p1">(<span class="default">(1,2) <a href="funct_7.html#K13" title="FUNCT_7:func.13">followed_by</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<font color="Maroon" title="c1">n</font>,<a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>)</span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T123" onmouseover="rs('funct_7/T123')" onmouseout="rh()">FUNCT_7:123</a></span>;<br/>
<a NAME="E3:42"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
<span class="p1">(<span class="default">(1,2) <a href="funct_7.html#K13" title="FUNCT_7:func.13">followed_by</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<font color="Maroon" title="c1">n</font>,<a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>)</span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T124" onmouseover="rs('funct_7/T124')" onmouseout="rh()">FUNCT_7:124</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>4</sub></font>(  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  = <span class="p1">(<span class="default">(1,2) <a href="funct_7.html#K13" title="FUNCT_7:func.13">followed_by</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<font color="Maroon" title="c1">n</font>,<a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>)</span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> $1;<br/>
<a NAME="E4:42"/><span class="lab"><font color="Green" title="E30">A4</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">n</font>) is   <a href="msualg_1.html#V3" title="MSUALG_1:attr.3">strict</a>   <a href="circcomb.html#V4" title="CIRCCOMB:attr.4">gate`2=den</a>   <a href="circcomb.html#V6" title="CIRCCOMB:attr.6">Boolean</a>  <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">n</font>)
 
;<br/>
<div><span class="lab"><font color="Green" title="E31">A5</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">s</font> being   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <span class="p1">(<span class="default"><a href="circcomb.html#K7" title="CIRCCOMB:func.7">1GateCircuit</a> (<a href="facirc_2.html#NK3" title="FACIRC_2:NK.3">&lt;*&gt;</a>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a></span>)</span>)</span>)</span> holds   <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Olive" title="b1">s</font>,<font color="Maroon">H<sub>4</sub></font>( <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> )) is  <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a> </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">s</font> be   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <span class="p1">(<span class="default"><a href="circcomb.html#K7" title="CIRCCOMB:func.7">1GateCircuit</a> (<a href="facirc_2.html#NK3" title="FACIRC_2:NK.3">&lt;*&gt;</a>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a></span>)</span>)</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c8">s</font>,<font color="Maroon">H<sub>4</sub></font>( <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> )) is  <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a> </span><br/><a NAME="E1:42_1"/>
 <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c8">s</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <font color="Maroon" title="c8">s</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="facirc_1.html#T14" onmouseover="rs('facirc_1/T14')" onmouseout="rh()">FACIRC_1:14</a></span>;<br/><span class="kw">hence </span><a NAME="E2:42_1"/>
 <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c8">s</font>,<font color="Maroon">H<sub>4</sub></font>( <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> )) is  <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="circcmb2.html#T2" onmouseover="rs('circcmb2/T2')" onmouseout="rh()">CIRCCMB2:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">deffunc </span><font color="Maroon">H<sub>5</sub></font>(  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <span class="p1">(<span class="default">$1 <a href="gfacirc2.html#K5" title="GFACIRC2:func.5">-BitGFA1Str</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font>)</span>)</span> = $1 <a href="gfacirc2.html#K7" title="GFACIRC2:func.7">-BitGFA1CarryOutput</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font>);<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">h</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E7:42"/><span class="lab"><font color="Green" title="E32">A6</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive" title="b1">n</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="pboole.html#S5" onmouseover="rs('pboole/S5')" onmouseout="rh()">PBOOLE:sch 5</a>();<br/></span>
<a NAME="E8:42"/><span class="lab"><font color="Green" title="E33">A7</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b3">A</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>  <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">n</font>)  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">n</font>) holds <br/> for <font color="Olive" title="b4">s</font> being   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Olive" title="b3">A</font> holds   <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Olive" title="b4">s</font>,<font color="Maroon">H<sub>4</sub></font>(1)) is  <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c9">f1</font> =  <a href="gfacirc1.html#K3" title="GFACIRC1:func.3">and2c</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">f0</font> =  <a href="gfacirc1.html#K4" title="GFACIRC1:func.4">xor2c</a> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c11">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b2">A</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>  <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c11">n</font>)  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">n</font> &amp; <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c11">n</font>) holds <br/> for <font color="Olive" title="b3">s</font> being   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Olive" title="b2">A</font> holds   <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Olive" title="b3">s</font>,<font color="Maroon">H<sub>4</sub></font>(1)) is  <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>  <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c11">n</font>)  st <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">n</font> &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c11">n</font>) holds <br/> for <font color="Olive" title="b2">s</font> being   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Olive" title="b1">A</font> holds   <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Olive" title="b2">s</font>,<font color="Maroon">H<sub>4</sub></font>(1)) is  <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c13">A</font> be   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>  <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c11">n</font>); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">n</font> &amp; <font color="Maroon" title="c13">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c11">n</font>) implies  for <font color="Olive" title="b1">s</font> being   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c13">A</font> holds   <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Olive" title="b1">s</font>,<font color="Maroon">H<sub>4</sub></font>(1)) is  <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a>  )</span><br/>





<span class="kw">assume </span><a NAME="E1:42_2"/><span class="lab"><font color="Green" title="E34">A8</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">n</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c13">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c11">n</font>) or  for <font color="Olive" title="b1">s</font> being   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c13">A</font> holds   <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Olive" title="b1">s</font>,<font color="Maroon">H<sub>4</sub></font>(1)) is  <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a>  )</span><br/>

<a NAME="E2:42_2"/>
<font color="Maroon" title="c11">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E3:42_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A9</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Maroon" title="c11">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:42"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="txt" href="#E1:42_2"><span class="lab"><font color="Green" title="E34">A8</font></span></a></span>;<br/>
<a NAME="E4:42_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A10</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>,<a href="gfacirc1.html#K3" title="GFACIRC1:func.3">and2c</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="gfacirc2.html#E37"><span class="lab"><font color="Green" title="E25">Lm4</font></span></a></span>;<br/>
<a NAME="E5:42_2"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>,<a href="gfacirc1.html#K4" title="GFACIRC1:func.4">xor2c</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_2"><span class="lab"><font color="Green" title="E35">A9</font></span></a>, <a class="txt" href="gfacirc2.html#E37"><span class="lab"><font color="Green" title="E25">Lm4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:42_2"/>
(  not <font color="Maroon" title="c13">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c11">n</font>) or  for <font color="Olive" title="b1">s</font> being   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c13">A</font> holds   <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Olive" title="b1">s</font>,<font color="Maroon">H<sub>4</sub></font>(1)) is  <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E4:42_2"><span class="lab"><font color="Green" title="E36">A10</font></span></a>, <a class="ref" href="gfacirc1.html#T72" onmouseover="rs('gfacirc1/T72')" onmouseout="rh()">GFACIRC1:72</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">set </span><font color="Maroon" title="c9">Sn</font> = <font color="Maroon" title="c1">n</font> <a href="gfacirc2.html#K5" title="GFACIRC2:func.5">-BitGFA1Str</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c10">An</font> = <font color="Maroon" title="c1">n</font> <a href="gfacirc2.html#K6" title="GFACIRC2:func.6">-BitGFA1Circ</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c11">o0</font> = <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="gfacirc2.html#K7" title="GFACIRC2:func.7">-BitGFA1CarryOutput</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font>);<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">f1</font>, <font color="Maroon" title="c13">g1</font>, <font color="Maroon" title="c14">h1</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E10:42"/><span class="lab"><font color="Green" title="E34">A11</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="gfacirc2.html#K5" title="GFACIRC2:func.5">-BitGFA1Str</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E11:42"/><span class="lab"><font color="Green" title="E35">A12</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="gfacirc2.html#K6" title="GFACIRC2:func.6">-BitGFA1Circ</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E12:42"/><span class="lab"><font color="Green" title="E36">A13</font></span>: 
<font color="Maroon" title="c12">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circcomb.html#K5" title="CIRCCOMB:func.5">1GateCircStr</a> (<a href="facirc_2.html#NK3" title="FACIRC_2:NK.3">&lt;*&gt;</a>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a></span>)</span>)
 <span class="kw">and </span><br/><a NAME="E13:42"/><span class="lab"><font color="Green" title="E37">A14</font></span>: 
<font color="Maroon" title="c13">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circcomb.html#K7" title="CIRCCOMB:func.7">1GateCircuit</a> (<a href="facirc_2.html#NK3" title="FACIRC_2:NK.3">&lt;*&gt;</a>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a></span>)</span>)
 <span class="kw">and </span><br/><a NAME="E14:42"/><span class="lab"><font color="Green" title="E38">A15</font></span>: 
<font color="Maroon" title="c14">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="facirc_2.html#NK3" title="FACIRC_2:NK.3">&lt;*&gt;</a>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E15:42"/><span class="lab"><font color="Green" title="E39">A16</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">S</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> <br/>  for <font color="Olive" title="b3">A</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> over <font color="Olive" title="b2">S</font><br/>  for <font color="Olive" title="b4">z</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> holds <br/>( <font color="Maroon" title="c12">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">S</font> <a href="circcomb.html#K2" title="CIRCCOMB:func.2">+*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b4">z</font>,<font color="Olive" title="b1">n</font>) &amp; <font color="Maroon" title="c13">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">A</font> <a href="circcomb.html#K3" title="CIRCCOMB:func.3">+*</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b4">z</font>,<font color="Olive" title="b1">n</font>) &amp; <font color="Maroon" title="c14">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b4">z</font>,<font color="Olive" title="b1">n</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gfacirc2.html#D6" target="_self" onmouseover="rs('gfacirc2/D6')" onmouseout="rh()">Def6</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds <br/><font color="Maroon" title="c14">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  implies <font color="Maroon" title="c14">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> )</span><br/><span class="kw">assume </span><a NAME="E1:42_3"/><span class="lab"><font color="Green" title="E40">A17</font></span>: 
<font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">j</font> = <font color="Maroon" title="c15">i</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/><span class="kw">thus </span><font color="Maroon" title="c14">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> = 
<font color="Maroon">H<sub>5</sub></font>(<font color="Maroon" title="c16">j</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:42"><span class="lab"><font color="Green" title="E36">A13</font></span></a>, <a class="txt" href="#E13:42"><span class="lab"><font color="Green" title="E37">A14</font></span></a>, <a class="txt" href="#E14:42"><span class="lab"><font color="Green" title="E38">A15</font></span></a>, <a class="txt" href="#E15:42"><span class="lab"><font color="Green" title="E39">A16</font></span></a>, <a class="ref" href="gfacirc2.html#T15" target="_self" onmouseover="rs('gfacirc2/T15')" onmouseout="rh()">Th15</a></span>
<br/>.= 
<font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:42"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="txt" href="#E1:42_3"><span class="lab"><font color="Green" title="E40">A17</font></span></a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E17:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A18</font></span>: 
<font color="Maroon" title="c14">h1</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c8">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T3" onmouseover="rs('pboole/T3')" onmouseout="rh()">PBOOLE:3</a></span>;<br/>
<a NAME="E18:42"/><span class="lab"><font color="Green" title="E41">A19</font></span>: 
 ex <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Maroon" title="c1">n</font> <a href="gfacirc2.html#K5" title="GFACIRC2:func.5">-BitGFA1Str</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>4</sub></font>(2) &amp; <font color="Maroon" title="c1">n</font> <a href="gfacirc2.html#K6" title="GFACIRC2:func.6">-BitGFA1Circ</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>4</sub></font>(2) &amp; <font color="Olive" title="b1">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circcomb.html#K5" title="CIRCCOMB:func.5">1GateCircStr</a> (<a href="facirc_2.html#NK3" title="FACIRC_2:NK.3">&lt;*&gt;</a>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a></span>)</span>) &amp; <font color="Olive" title="b2">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circcomb.html#K7" title="CIRCCOMB:func.7">1GateCircuit</a> (<a href="facirc_2.html#NK3" title="FACIRC_2:NK.3">&lt;*&gt;</a>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a></span>)</span>) &amp; <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="gfacirc2.html#K7" title="GFACIRC2:func.7">-BitGFA1CarryOutput</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font>) &amp; (  for <font color="Olive" title="b3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b4">S</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> <br/>  for <font color="Olive" title="b5">A1</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> over <font color="Olive" title="b4">S</font><br/>  for <font color="Olive" title="b6">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b7">A2</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> over <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b6">x</font>,<font color="Olive" title="b3">n</font>)  st <font color="Olive" title="b4">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font> &amp; <font color="Olive" title="b5">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font> &amp; <font color="Olive" title="b6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font> &amp; <font color="Olive" title="b7">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b6">x</font>,<font color="Olive" title="b3">n</font>) holds <br/>( <font color="Olive" title="b1">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">S</font> <a href="circcomb.html#K2" title="CIRCCOMB:func.2">+*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b6">x</font>,<font color="Olive" title="b3">n</font>) &amp; <font color="Olive" title="b2">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">A1</font> <a href="circcomb.html#K3" title="CIRCCOMB:func.3">+*</a> <font color="Olive" title="b7">A2</font> &amp; <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b6">x</font>,<font color="Olive" title="b3">n</font>) ) ) )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span>
<font color="Maroon" title="c12">f1</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">v</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Maroon" title="c1">n</font> <a href="gfacirc2.html#K5" title="GFACIRC2:func.5">-BitGFA1Str</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>4</sub></font>(2) &amp; <font color="Maroon" title="c1">n</font> <a href="gfacirc2.html#K6" title="GFACIRC2:func.6">-BitGFA1Circ</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>4</sub></font>(2) &amp; <font color="Maroon" title="c12">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circcomb.html#K5" title="CIRCCOMB:func.5">1GateCircStr</a> (<a href="facirc_2.html#NK3" title="FACIRC_2:NK.3">&lt;*&gt;</a>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a></span>)</span>) &amp; <font color="Olive" title="b1">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circcomb.html#K7" title="CIRCCOMB:func.7">1GateCircuit</a> (<a href="facirc_2.html#NK3" title="FACIRC_2:NK.3">&lt;*&gt;</a>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a></span>)</span>) &amp; <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="gfacirc2.html#K7" title="GFACIRC2:func.7">-BitGFA1CarryOutput</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font>) &amp; (  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b3">S</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> <br/>  for <font color="Olive" title="b4">A1</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> over <font color="Olive" title="b3">S</font><br/>  for <font color="Olive" title="b5">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b6">A2</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> over <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b5">x</font>,<font color="Olive" title="b2">n</font>)  st <font color="Olive" title="b3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font> &amp; <font color="Olive" title="b4">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font> &amp; <font color="Olive" title="b5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font> &amp; <font color="Olive" title="b6">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b5">x</font>,<font color="Olive" title="b2">n</font>) holds <br/>( <font color="Maroon" title="c12">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">S</font> <a href="circcomb.html#K2" title="CIRCCOMB:func.2">+*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b5">x</font>,<font color="Olive" title="b2">n</font>) &amp; <font color="Olive" title="b1">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">A1</font> <a href="circcomb.html#K3" title="CIRCCOMB:func.3">+*</a> <font color="Olive" title="b6">A2</font> &amp; <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b5">x</font>,<font color="Olive" title="b2">n</font>) ) ) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c13">g1</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">n</font> <a href="gfacirc2.html#K5" title="GFACIRC2:func.5">-BitGFA1Str</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>4</sub></font>(2) &amp; <font color="Maroon" title="c1">n</font> <a href="gfacirc2.html#K6" title="GFACIRC2:func.6">-BitGFA1Circ</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>4</sub></font>(2) &amp; <font color="Maroon" title="c12">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circcomb.html#K5" title="CIRCCOMB:func.5">1GateCircStr</a> (<a href="facirc_2.html#NK3" title="FACIRC_2:NK.3">&lt;*&gt;</a>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a></span>)</span>) &amp; <font color="Maroon" title="c13">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circcomb.html#K7" title="CIRCCOMB:func.7">1GateCircuit</a> (<a href="facirc_2.html#NK3" title="FACIRC_2:NK.3">&lt;*&gt;</a>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a></span>)</span>) &amp; <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="gfacirc2.html#K7" title="GFACIRC2:func.7">-BitGFA1CarryOutput</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font>) &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">S</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> <br/>  for <font color="Olive" title="b3">A1</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> over <font color="Olive" title="b2">S</font><br/>  for <font color="Olive" title="b4">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b5">A2</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> over <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b4">x</font>,<font color="Olive" title="b1">n</font>)  st <font color="Olive" title="b2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b3">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b5">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b4">x</font>,<font color="Olive" title="b1">n</font>) holds <br/>( <font color="Maroon" title="c12">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">S</font> <a href="circcomb.html#K2" title="CIRCCOMB:func.2">+*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b4">x</font>,<font color="Olive" title="b1">n</font>) &amp; <font color="Maroon" title="c13">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">A1</font> <a href="circcomb.html#K3" title="CIRCCOMB:func.3">+*</a> <font color="Olive" title="b5">A2</font> &amp; <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b4">x</font>,<font color="Olive" title="b1">n</font>) ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E1:42_4"/>
( <font color="Maroon" title="c1">n</font> <a href="gfacirc2.html#K5" title="GFACIRC2:func.5">-BitGFA1Str</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>4</sub></font>(2) &amp; <font color="Maroon" title="c1">n</font> <a href="gfacirc2.html#K6" title="GFACIRC2:func.6">-BitGFA1Circ</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon">H<sub>4</sub></font>(2) &amp; <font color="Maroon" title="c12">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circcomb.html#K5" title="CIRCCOMB:func.5">1GateCircStr</a> (<a href="facirc_2.html#NK3" title="FACIRC_2:NK.3">&lt;*&gt;</a>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a></span>)</span>) &amp; <font color="Maroon" title="c13">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circcomb.html#K7" title="CIRCCOMB:func.7">1GateCircuit</a> (<a href="facirc_2.html#NK3" title="FACIRC_2:NK.3">&lt;*&gt;</a>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a></span>)</span>) &amp; <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="gfacirc2.html#K7" title="GFACIRC2:func.7">-BitGFA1CarryOutput</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font>) &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">S</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> <br/>  for <font color="Olive" title="b3">A1</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> over <font color="Olive" title="b2">S</font><br/>  for <font color="Olive" title="b4">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b5">A2</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> over <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b4">x</font>,<font color="Olive" title="b1">n</font>)  st <font color="Olive" title="b2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b3">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b5">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b4">x</font>,<font color="Olive" title="b1">n</font>) holds <br/>( <font color="Maroon" title="c12">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">S</font> <a href="circcomb.html#K2" title="CIRCCOMB:func.2">+*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b4">x</font>,<font color="Olive" title="b1">n</font>) &amp; <font color="Maroon" title="c13">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">A1</font> <a href="circcomb.html#K3" title="CIRCCOMB:func.3">+*</a> <font color="Olive" title="b5">A2</font> &amp; <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b4">x</font>,<font color="Olive" title="b1">n</font>) ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E7:42"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="txt" href="#E10:42"><span class="lab"><font color="Green" title="E34">A11</font></span></a>, <a class="txt" href="#E11:42"><span class="lab"><font color="Green" title="E35">A12</font></span></a>, <a class="txt" href="#E12:42"><span class="lab"><font color="Green" title="E36">A13</font></span></a>, <a class="txt" href="#E13:42"><span class="lab"><font color="Green" title="E37">A14</font></span></a>, <a class="txt" href="#E15:42"><span class="lab"><font color="Green" title="E39">A16</font></span></a>, <a class="txt" href="#E17:42"><span class="lab"><font color="Green" title="E40">A18</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E19:42"/><span class="lab"><font color="Green" title="E42">A20</font></span>: 
(  <a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <span class="p1">(<span class="default"><a href="circcomb.html#K5" title="CIRCCOMB:func.5">1GateCircStr</a> (<a href="facirc_2.html#NK3" title="FACIRC_2:NK.3">&lt;*&gt;</a>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a></span>)</span>)</span>)</span> is   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a> &amp;  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <span class="p1">(<span class="default"><a href="circcomb.html#K5" title="CIRCCOMB:func.5">1GateCircStr</a> (<a href="facirc_2.html#NK3" title="FACIRC_2:NK.3">&lt;*&gt;</a>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a></span>)</span>)</span>)</span> is  <a href="facirc_1.html#NV2" title="FACIRC_1:NV.2">without_pairs</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="facirc_1.html#T38" onmouseover="rs('facirc_1/T38')" onmouseout="rh()">FACIRC_1:38</a>, <a class="ref" href="facirc_1.html#T39" onmouseover="rs('facirc_1/T39')" onmouseout="rh()">FACIRC_1:39</a></span>;<br/>
<a NAME="E20:42"/><span class="lab"><font color="Green" title="E43">A21</font></span>: 
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="gfacirc2.html#K7" title="GFACIRC2:func.7">-BitGFA1CarryOutput</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="facirc_2.html#NK3" title="FACIRC_2:NK.3">&lt;*&gt;</a>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gfacirc2.html#T16" target="_self" onmouseover="rs('gfacirc2/T16')" onmouseout="rh()">Th16</a></span>;<br/>
<a NAME="E21:42"/>
 <a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <span class="p1">(<span class="default"><a href="circcomb.html#K5" title="CIRCCOMB:func.5">1GateCircStr</a> (<a href="facirc_2.html#NK3" title="FACIRC_2:NK.3">&lt;*&gt;</a>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a></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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="facirc_2.html#NK3" title="FACIRC_2:NK.3">&lt;*&gt;</a>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="circcomb.html#T42" onmouseover="rs('circcomb/T42')" onmouseout="rh()">CIRCCOMB:42</a></span>;<br/>
<a NAME="E22:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A22</font></span>: 
( <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="gfacirc2.html#K7" title="GFACIRC2:func.7">-BitGFA1CarryOutput</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font>) &amp; <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="gfacirc2.html#K7" title="GFACIRC2:func.7">-BitGFA1CarryOutput</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <span class="p1">(<span class="default"><a href="circcomb.html#K5" title="CIRCCOMB:func.5">1GateCircStr</a> (<a href="facirc_2.html#NK3" title="FACIRC_2:NK.3">&lt;*&gt;</a>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">TRUE</a></span>)</span>)</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:42"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="txt" href="#E20:42"><span class="lab"><font color="Green" title="E43">A21</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E23:42"/><span class="lab"><font color="Green" title="E45">A23</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds   <a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">n</font>) is   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gfacirc1.html#T64" onmouseover="rs('gfacirc1/T64')" onmouseout="rh()">GFACIRC1:64</a></span>;<br/>
<a NAME="E24:42"/><span class="lab"><font color="Green" title="E46">A24</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> holds <br/><span class="p1">(<span class="default"><a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">n</font>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is  <a href="facirc_1.html#NV2" title="FACIRC_1:NV.2">without_pairs</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c15">f1</font> =  <a href="gfacirc1.html#K3" title="GFACIRC1:func.3">and2c</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">f0</font> =  <a href="gfacirc1.html#K4" title="GFACIRC1:func.4">xor2c</a> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c17">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font> holds <br/><span class="p1">(<span class="default"><a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c17">n</font>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is  <a href="facirc_1.html#NV2" title="FACIRC_1:NV.2">without_pairs</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c18">x</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font> implies <span class="p1">(<span class="default"><a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c18">x</font>,<font color="Maroon" title="c17">n</font>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is  <a href="facirc_1.html#NV2" title="FACIRC_1:NV.2">without_pairs</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:42_5"/><span class="lab"><font color="Green" title="E47">A25</font></span>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c18">x</font>,<font color="Maroon" title="c17">n</font>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is  <a href="facirc_1.html#NV2" title="FACIRC_1:NV.2">without_pairs</a> </span><br/>

<a NAME="E2:42_5"/>
<font color="Maroon" title="c17">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E3:42_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A26</font></span>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Maroon" title="c17">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:42"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="txt" href="#E1:42_5"><span class="lab"><font color="Green" title="E47">A25</font></span></a></span>;<br/>
<a NAME="E4:42_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A27</font></span>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>,<a href="gfacirc1.html#K3" title="GFACIRC1:func.3">and2c</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="gfacirc2.html#E37"><span class="lab"><font color="Green" title="E25">Lm4</font></span></a></span>;<br/>
<a NAME="E5:42_5"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>,<a href="gfacirc1.html#K4" title="GFACIRC1:func.4">xor2c</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_5"><span class="lab"><font color="Green" title="E48">A26</font></span></a>, <a class="txt" href="gfacirc2.html#E37"><span class="lab"><font color="Green" title="E25">Lm4</font></span></a></span>;<br/>
<a NAME="E6:42_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A28</font></span>: 
 <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c18">x</font>,<font color="Maroon" title="c17">n</font>) <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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c18">x</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_5"><span class="lab"><font color="Green" title="E49">A27</font></span></a>, <a class="ref" href="gfacirc1.html#T65" onmouseover="rs('gfacirc1/T65')" onmouseout="rh()">GFACIRC1:65</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c19">a</font> be   <a href="xtuple_0.html#V1" title="XTUPLE_0:attr.1">pair</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="facirc_1.html#D2" onmouseover="rs('facirc_1/D2')" onmouseout="rh()">FACIRC_1:def 2</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c19">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c18">x</font>,<font color="Maroon" title="c17">n</font>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<span class="kw">assume </span><a NAME="E7:42_5"/><span class="lab"><font color="Green" title="E51">A29</font></span>: 
<font color="Maroon" title="c19">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c18">x</font>,<font color="Maroon" title="c17">n</font>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E8:42_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A30</font></span>: 
<font color="Maroon" title="c19">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c18">x</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:42_5"><span class="lab"><font color="Green" title="E50">A28</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E9:42_5"/><span class="lab"><font color="Green" title="E53">A31</font></span>: 
 not <font color="Maroon" title="c19">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:42_5"><span class="lab"><font color="Green" title="E51">A29</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E10:42_5"/>
( <font color="Maroon" title="c19">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> or <font color="Maroon" title="c19">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> or <font color="Maroon" title="c19">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:42_5"><span class="lab"><font color="Green" title="E52">A30</font></span></a>, <a class="ref" href="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:42_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:42_5"><span class="lab"><font color="Green" title="E53">A31</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E25:42"/><span class="lab"><font color="Green" title="E47">A32</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> holds <br/>( <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">n</font>) &amp; <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">n</font>) &amp; <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">n</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">n</font>) )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c15">f1</font> =  <a href="gfacirc1.html#K3" title="GFACIRC1:func.3">and2c</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">f2</font> =  <a href="twoscomp.html#K2" title="TWOSCOMP:func.2">and2a</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c17">f3</font> =  <a href="twoscomp.html#NK2" title="TWOSCOMP:NK.2">and2</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c18">f0</font> =  <a href="gfacirc1.html#K4" title="GFACIRC1:func.4">xor2c</a> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c19">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">n</font> holds <br/>( <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c19">n</font>) &amp; <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c19">n</font>) &amp; <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c19">n</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c19">n</font>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c20">x</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">n</font> implies ( <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c20">x</font>,<font color="Maroon" title="c19">n</font>) &amp; <font color="Maroon" title="c20">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c20">x</font>,<font color="Maroon" title="c19">n</font>) &amp; <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c20">x</font>,<font color="Maroon" title="c19">n</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c20">x</font>,<font color="Maroon" title="c19">n</font>) ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:42_6"/><span class="lab"><font color="Green" title="E48">A33</font></span>: 
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">n</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c20">x</font>,<font color="Maroon" title="c19">n</font>) &amp; <font color="Maroon" title="c20">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c20">x</font>,<font color="Maroon" title="c19">n</font>) &amp; <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c20">x</font>,<font color="Maroon" title="c19">n</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c20">x</font>,<font color="Maroon" title="c19">n</font>) )</span><br/>

<a NAME="E2:42_6"/>
<font color="Maroon" title="c19">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E3:42_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A34</font></span>: 
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Maroon" title="c19">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:42"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="txt" href="#E1:42_6"><span class="lab"><font color="Green" title="E48">A33</font></span></a></span>;<br/>
<a NAME="E4:42_6"/>
<font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Maroon" title="c19">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:42"><span class="lab"><font color="Green" title="E32">A6</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:42_6"/>
<font color="Maroon" title="c8">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c20">x</font>,<font color="Maroon" title="c19">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_6"><span class="lab"><font color="Green" title="E49">A34</font></span></a>, <a class="ref" href="gfacirc2.html#T21" target="_self" onmouseover="rs('gfacirc2/T21')" onmouseout="rh()">Th21</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c20">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c20">x</font>,<font color="Maroon" title="c19">n</font>) &amp; <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c20">x</font>,<font color="Maroon" title="c19">n</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c20">x</font>,<font color="Maroon" title="c19">n</font>) )</span><br/>

<a NAME="E6:42_6"/><span class="lab"><font color="Green" title="E50">A35</font></span>: 
<font color="Maroon" title="c20">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c19">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c19">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>,<a href="gfacirc1.html#K3" title="GFACIRC1:func.3">and2c</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_6"><span class="lab"><font color="Green" title="E49">A34</font></span></a>, <a class="txt" href="gfacirc2.html#E37"><span class="lab"><font color="Green" title="E25">Lm4</font></span></a></span>;<br/>
<a NAME="E7:42_6"/>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c19">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c19">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>,<a href="gfacirc1.html#K4" title="GFACIRC1:func.4">xor2c</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_6"><span class="lab"><font color="Green" title="E49">A34</font></span></a>, <a class="txt" href="gfacirc2.html#E37"><span class="lab"><font color="Green" title="E25">Lm4</font></span></a></span>;<br/>
<a NAME="E8:42_6"/><span class="kw">then </span>
 <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c20">x</font>,<font color="Maroon" title="c19">n</font>) <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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c20">x</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:42_6"><span class="lab"><font color="Green" title="E50">A35</font></span></a>, <a class="ref" href="gfacirc1.html#T65" onmouseover="rs('gfacirc1/T65')" onmouseout="rh()">GFACIRC1:65</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:42_6"/>
<font color="Maroon" title="c20">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c20">x</font>,<font color="Maroon" title="c19">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c20">x</font>,<font color="Maroon" title="c19">n</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c20">x</font>,<font color="Maroon" title="c19">n</font>)</span><br/>

<a NAME="E10:42_6"/><span class="lab"><font color="Green" title="E51">A36</font></span>: 
 <a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c20">x</font>,<font color="Maroon" title="c19">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>,<a href="gfacirc1.html#K4" title="GFACIRC1:func.4">xor2c</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></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="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4">(<span class="default"><a href="gfacirc1.html#K24" title="GFACIRC1:func.24">GFA1AdderOutput</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c19">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c19">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c20">x</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c19">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c19">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>,<a href="gfacirc1.html#K3" title="GFACIRC1:func.3">and2c</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c19">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c20">x</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>,<a href="twoscomp.html#K2" title="TWOSCOMP:func.2">and2a</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c20">x</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c19">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>,<a href="twoscomp.html#NK2" title="TWOSCOMP:NK.2">and2</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span></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"><span class="p2">(<span class="default"><a href="gfacirc1.html#K21" title="GFACIRC1:func.21">GFA1CarryOutput</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c19">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c19">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c20">x</font>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gfacirc1.html#T63" onmouseover="rs('gfacirc1/T63')" onmouseout="rh()">GFACIRC1:63</a></span>;<br/>
<a NAME="E11:42_6"/>
<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c20">x</font>,<font color="Maroon" title="c19">n</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c20">x</font>,<font color="Maroon" title="c19">n</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="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:42_6"/>
<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c20">x</font>,<font color="Maroon" title="c19">n</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c20">x</font>,<font color="Maroon" title="c19">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:42_6"><span class="lab"><font color="Green" title="E51">A36</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E26:42"/>
 for <font color="Olive" title="b1">s</font> being   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="gfacirc2.html#K6" title="GFACIRC2:func.6">-BitGFA1Circ</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font>)</span>)</span> holds   <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default"><font color="Maroon">H<sub>4</sub></font>( <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ) <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon">H<sub>4</sub></font>(2) <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon">H<sub>4</sub></font>(1)</span>)</span></span>)</span>) is  <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a> 
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="circcmb2.html#S22" onmouseover="rs('circcmb2/S22')" onmouseout="rh()">CIRCCMB2:sch 22</a>(<a class="txt" href="#E4:42"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="txt" href="#E5:42"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E8:42"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="txt" href="#E18:42"><span class="lab"><font color="Green" title="E41">A19</font></span></a>, <a class="txt" href="#E19:42"><span class="lab"><font color="Green" title="E42">A20</font></span></a>, <a class="txt" href="#E22:42"><span class="lab"><font color="Green" title="E44">A22</font></span></a>, <a class="txt" href="#E23:42"><span class="lab"><font color="Green" title="E45">A23</font></span></a>, <a class="txt" href="#E24:42"><span class="lab"><font color="Green" title="E46">A24</font></span></a>, <a class="txt" href="#E25:42"><span class="lab"><font color="Green" title="E47">A32</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E27:42"/>
 for <font color="Olive" title="b1">s</font> being   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="gfacirc2.html#K6" title="GFACIRC2:func.6">-BitGFA1Circ</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font>)</span>)</span> holds   <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>) is  <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E3:42"><span class="lab"><font color="Green" title="E29">A3</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
