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<b>set </b><font color="Maroon" title="c1">f0</font> =  <a href="circcomb.html#K6" title="CIRCCOMB:func.6">1GateCircStr</a> <a href="facirc_2.html#K3" title="FACIRC_2:func.3">&lt;*&gt;</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="circcomb.html#K9" title="CIRCCOMB:func.9">BOOLEAN</a> </span>)</span> <a href="margrel1.html#K1" title="MARGREL1:func.1">--&gt;</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c2">g0</font> =  <a href="circcomb.html#K8" title="CIRCCOMB:func.8">1GateCircuit</a> <a href="facirc_2.html#K3" title="FACIRC_2:func.3">&lt;*&gt;</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="circcomb.html#K9" title="CIRCCOMB:func.9">BOOLEAN</a> </span>)</span> <a href="margrel1.html#K1" title="MARGREL1:func.1">--&gt;</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c3">h0</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="facirc_2.html#K3" title="FACIRC_2:func.3">&lt;*&gt;</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="circcomb.html#K9" title="CIRCCOMB:func.9">BOOLEAN</a> </span>)</span> <a href="margrel1.html#K1" title="MARGREL1:func.1">--&gt;</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<b>let </b><font color="Maroon" title="c4">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be    <a href="circcomb.html#M1" title="CIRCCOMB:mode.1">FinSeqLen</a> of <font color="Maroon" title="c4">n</font>;<br/><b>let </b><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>





<b>set </b><font color="Maroon" title="c9">p</font> = <font color="Maroon" title="c5">x</font> <a href="facirc_1.html#K8" title="FACIRC_1:func.8">^</a> <span class="p1"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span>;<br/>
<b>set </b><font color="Maroon" title="c10">q</font> = <font color="Maroon" title="c6">y</font> <a href="facirc_1.html#K8" title="FACIRC_1:func.8">^</a> <span class="p1"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c8">b</font></span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span>;<br/>
<b>consider </b><font color="Maroon" title="c11">f</font>, <font color="Maroon" title="c12">g</font>, <font color="Maroon" title="c13">h</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E2:9"/><i><font color="Green" title="E8">A1</font></i>: 
( <font color="Maroon" title="c4">n</font> <a href="gfacirc2.html#K1" title="GFACIRC2:func.1">-BitGFA0Str</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="facirc_1.html#K8" title="FACIRC_1:func.8">^</a> <span class="p2"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="facirc_1.html#K8" title="FACIRC_1:func.8">^</a> <span class="p2"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c8">b</font></span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c4">n</font> <a href="gfacirc2.html#K2" title="GFACIRC2:func.2">-BitGFA0Circ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="facirc_1.html#K8" title="FACIRC_1:func.8">^</a> <span class="p2"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="facirc_1.html#K8" title="FACIRC_1:func.8">^</a> <span class="p2"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c8">b</font></span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> )
 <b>and </b><br/><a NAME="E3:9"/><i><font color="Green" title="E9">A2</font></i>: 
( <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circcomb.html#K6" title="CIRCCOMB:func.6">1GateCircStr</a> <a href="facirc_2.html#K3" title="FACIRC_2:func.3">&lt;*&gt;</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="circcomb.html#K9" title="CIRCCOMB:func.9">BOOLEAN</a> </span>)</span> <a href="margrel1.html#K1" title="MARGREL1:func.1">--&gt;</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span>)</span> &amp; <font color="Maroon" title="c12">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circcomb.html#K8" title="CIRCCOMB:func.8">1GateCircuit</a> <a href="facirc_2.html#K3" title="FACIRC_2:func.3">&lt;*&gt;</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="circcomb.html#K9" title="CIRCCOMB:func.9">BOOLEAN</a> </span>)</span> <a href="margrel1.html#K1" title="MARGREL1:func.1">--&gt;</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span>)</span> &amp; <font color="Maroon" title="c13">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">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#K3" title="FACIRC_2:func.3">&lt;*&gt;</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="circcomb.html#K9" title="CIRCCOMB:func.9">BOOLEAN</a> </span>)</span> <a href="margrel1.html#K1" title="MARGREL1:func.1">--&gt;</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>and </b><br/><a NAME="E4:9"/><i><font color="Green" title="E10">A3</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">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#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Olive" title="b2">S</font><br/> for <font color="Olive" title="b4">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f</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="c12">g</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="c13">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="c11">f</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#K3" title="CIRCCOMB:func.3">+*</a> <span class="p1">(<span class="default"><a href="gfacirc1.html#K13" title="GFACIRC1:func.13">BitGFA0Str</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="facirc_1.html#K8" title="FACIRC_1:func.8">^</a> <span class="p4"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<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></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="facirc_1.html#K8" title="FACIRC_1:func.8">^</a> <span class="p4"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c8">b</font></span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<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></span>)</span>,<font color="Olive" title="b4">z</font></span>)</span> &amp; <font color="Maroon" title="c12">g</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#K4" title="CIRCCOMB:func.4">+*</a> <span class="p1">(<span class="default"><a href="gfacirc1.html#K14" title="GFACIRC1:func.14">BitGFA0Circ</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="facirc_1.html#K8" title="FACIRC_1:func.8">^</a> <span class="p4"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<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></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="facirc_1.html#K8" title="FACIRC_1:func.8">^</a> <span class="p4"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c8">b</font></span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<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></span>)</span>,<font color="Olive" title="b4">z</font></span>)</span> &amp; <font color="Maroon" title="c13">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>  <a href="gfacirc1.html#K9" title="GFACIRC1:func.9">GFA0CarryOutput</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="facirc_1.html#K8" title="FACIRC_1:func.8">^</a> <span class="p3"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<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></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="facirc_1.html#K8" title="FACIRC_1:func.8">^</a> <span class="p3"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c8">b</font></span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<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></span>)</span>,<font color="Olive" title="b4">z</font> )
 <b>by </b><i><a class="ref" href="gfacirc2.html#D2" target="_self" title="GFACIRC2:def.2">Def2</a></i>;<br/>
<a NAME="E5:9"/><i><font color="Green" title="E11">A4</font></i>: 
( <font color="Maroon" title="c4">n</font> <a href="gfacirc2.html#K3" title="GFACIRC2:func.3">-BitGFA0CarryOutput</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="facirc_1.html#K8" title="FACIRC_1:func.8">^</a> <span class="p2"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="facirc_1.html#K8" title="FACIRC_1:func.8">^</a> <span class="p2"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c8">b</font></span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="gfacirc2.html#K1" title="GFACIRC2:func.1">-BitGFA0Str</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="facirc_1.html#K8" title="FACIRC_1:func.8">^</a> <span class="p2"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="facirc_1.html#K8" title="FACIRC_1:func.8">^</a> <span class="p2"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c8">b</font></span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="gfacirc2.html#K2" title="GFACIRC2:func.2">-BitGFA0Circ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="facirc_1.html#K8" title="FACIRC_1:func.8">^</a> <span class="p2"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="facirc_1.html#K8" title="FACIRC_1:func.8">^</a> <span class="p2"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c8">b</font></span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="gfacirc2.html#K3" title="GFACIRC2:func.3">-BitGFA0CarryOutput</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="facirc_1.html#K8" title="FACIRC_1:func.8">^</a> <span class="p2"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="facirc_1.html#K8" title="FACIRC_1:func.8">^</a> <span class="p2"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c8">b</font></span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="gfacirc2.html#E3:9"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="gfacirc2.html#E4:9"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="gfacirc2.html#T1" target="_self" title="GFACIRC2:th.1">Th1</a></i>;<br/>
<a NAME="E6:9"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> )
 
<b>by </b><i><a class="ref" href="circcomb.html#D12" title="CIRCCOMB:def.12">CIRCCOMB:def 12</a></i>;<br/>
<a NAME="E7:9"/><b>then </b><i><font color="Green" title="E12">A5</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="facirc_1.html#K8" title="FACIRC_1:func.8">^</a> <span class="p2"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">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" title="c7">a</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="facirc_1.html#K8" title="FACIRC_1:func.8">^</a> <span class="p2"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c8">b</font></span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">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" title="c8">b</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></i>;<br/>
<a NAME="E8:9"/>
( <font color="Maroon" title="c5">x</font> <a href="facirc_1.html#K8" title="FACIRC_1:func.8">^</a> <a href="facirc_2.html#K3" title="FACIRC_2:func.3">&lt;*&gt;</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Maroon" title="c6">y</font> <a href="facirc_1.html#K8" title="FACIRC_1:func.8">^</a> <a href="facirc_2.html#K3" title="FACIRC_2:func.3">&lt;*&gt;</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></i>;<br/>
<a NAME="E9:9"/><b>then </b>
( <font color="Maroon" title="c4">n</font> <a href="gfacirc2.html#K1" title="GFACIRC2:func.1">-BitGFA0Str</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="facirc_1.html#K8" title="FACIRC_1:func.8">^</a> <span class="p2"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="facirc_1.html#K8" title="FACIRC_1:func.8">^</a> <span class="p2"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c8">b</font></span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="gfacirc2.html#K1" title="GFACIRC2:func.1">-BitGFA0Str</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> &amp; <font color="Maroon" title="c4">n</font> <a href="gfacirc2.html#K2" title="GFACIRC2:func.2">-BitGFA0Circ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="facirc_1.html#K8" title="FACIRC_1:func.8">^</a> <span class="p2"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="facirc_1.html#K8" title="FACIRC_1:func.8">^</a> <span class="p2"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c8">b</font></span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="gfacirc2.html#K2" title="GFACIRC2:func.2">-BitGFA0Circ</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> &amp; <font color="Maroon" title="c4">n</font> <a href="gfacirc2.html#K3" title="GFACIRC2:func.3">-BitGFA0CarryOutput</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="facirc_1.html#K8" title="FACIRC_1:func.8">^</a> <span class="p2"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="facirc_1.html#K8" title="FACIRC_1:func.8">^</a> <span class="p2"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c8">b</font></span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="gfacirc2.html#K3" title="GFACIRC2:func.3">-BitGFA0CarryOutput</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> )
 
<b>by </b><i><a class="ref" href="gfacirc2.html#T5" target="_self" title="GFACIRC2:th.5">Th5</a></i>;<br/>
<b>hence </b><a NAME="E10:9"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="gfacirc2.html#K1" title="GFACIRC2:func.1">-BitGFA0Str</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="facirc_1.html#K8" title="FACIRC_1:func.8">^</a> <span class="p2"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="facirc_1.html#K8" title="FACIRC_1:func.8">^</a> <span class="p2"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c8">b</font></span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="gfacirc2.html#K1" title="GFACIRC2:func.1">-BitGFA0Str</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="circcomb.html#K3" title="CIRCCOMB:func.3">+*</a> <span class="p1">(<span class="default"><a href="gfacirc1.html#K13" title="GFACIRC1:func.13">BitGFA0Str</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="gfacirc2.html#K3" title="GFACIRC2:func.3">-BitGFA0CarryOutput</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="gfacirc2.html#K2" title="GFACIRC2:func.2">-BitGFA0Circ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="facirc_1.html#K8" title="FACIRC_1:func.8">^</a> <span class="p2"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="facirc_1.html#K8" title="FACIRC_1:func.8">^</a> <span class="p2"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c8">b</font></span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="gfacirc2.html#K2" title="GFACIRC2:func.2">-BitGFA0Circ</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="circcomb.html#K4" title="CIRCCOMB:func.4">+*</a> <span class="p1">(<span class="default"><a href="gfacirc1.html#K14" title="GFACIRC1:func.14">BitGFA0Circ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="gfacirc2.html#K3" title="GFACIRC2:func.3">-BitGFA0CarryOutput</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="gfacirc2.html#K3" title="GFACIRC2:func.3">-BitGFA0CarryOutput</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="facirc_1.html#K8" title="FACIRC_1:func.8">^</a> <span class="p2"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="facirc_1.html#K8" title="FACIRC_1:func.8">^</a> <span class="p2"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c8">b</font></span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gfacirc1.html#K9" title="GFACIRC1:func.9">GFA0CarryOutput</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="gfacirc2.html#K3" title="GFACIRC2:func.3">-BitGFA0CarryOutput</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> )
 <b>by </b><i><a class="txt" href="gfacirc2.html#E2:9"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="gfacirc2.html#E4:9"><i><font color="Green" title="E10">A3</font></i></a>, <a class="txt" href="gfacirc2.html#E5:9"><i><font color="Green" title="E11">A4</font></i></a>, <a class="txt" href="gfacirc2.html#E7:9"><i><font color="Green" title="E12">A5</font></i></a></i>;<br/>


</div>
