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<span class="kw">set </span><font color="Maroon" title="c1">f0</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#K6" title="MARGREL1:func.6">FALSE</a></span>)</span>);<br/>
<span class="kw">set </span><font color="Maroon" title="c2">g0</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#K6" title="MARGREL1:func.6">FALSE</a></span>)</span>);<br/>
<span class="kw">set </span><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#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#K6" title="MARGREL1:func.6">FALSE</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">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="circcomb.html#NM2" title="CIRCCOMB:NM.2">FinSeqLen</a> of <font color="Maroon" title="c4">n</font><br/>  for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds <br/> ( <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="Olive" title="b1">x</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Olive" title="b3">a</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Olive" title="b4">b</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&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="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>)</span>)</span> <a href="circcomb.html#K2" title="CIRCCOMB:func.2">+*</a> <span class="p1">(<span class="default"><a href="gfacirc1.html#K13" title="GFACIRC1:func.13">BitGFA0Str</a> (<font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">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="Olive" title="b1">x</font>,<font color="Olive" title="b2">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="Olive" title="b1">x</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Olive" title="b3">a</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Olive" title="b4">b</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&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="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>)</span>)</span> <a href="circcomb.html#K3" title="CIRCCOMB:func.3">+*</a> <span class="p1">(<span class="default"><a href="gfacirc1.html#K14" title="GFACIRC1:func.14">BitGFA0Circ</a> (<font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">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="Olive" title="b1">x</font>,<font color="Olive" title="b2">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="Olive" title="b1">x</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Olive" title="b3">a</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Olive" title="b4">b</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&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="Olive" title="b3">a</font>,<font color="Olive" title="b4">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="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>)</span>)</span>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be   <a href="circcomb.html#NM2" title="CIRCCOMB:NM.2">FinSeqLen</a> of <font color="Maroon" title="c4">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">a</font>, <font color="Olive" title="b2">b</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds <br/> ( <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="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Olive" title="b1">a</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Olive" title="b2">b</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&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#K2" title="CIRCCOMB:func.2">+*</a> <span class="p1">(<span class="default"><a href="gfacirc1.html#K13" title="GFACIRC1:func.13">BitGFA0Str</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">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="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Olive" title="b1">a</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Olive" title="b2">b</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&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#K3" title="CIRCCOMB:func.3">+*</a> <span class="p1">(<span class="default"><a href="gfacirc1.html#K14" title="GFACIRC1:func.14">BitGFA0Circ</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">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="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Olive" title="b1">a</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Olive" title="b2">b</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&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="Olive" title="b1">a</font>,<font color="Olive" title="b2">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>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</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"> ( <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="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c8">b</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&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#K2" title="CIRCCOMB:func.2">+*</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="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c8">b</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&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#K3" title="CIRCCOMB:func.3">+*</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="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c8">b</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&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>) )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c9">p</font> = <font color="Maroon" title="c5">x</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">q</font> = <font color="Maroon" title="c6">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c8">b</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>;<br/>
<span class="kw">consider </span><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#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="E2:9"/><span class="lab"><font color="Green" title="E8">A1</font></span>: 
<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="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c8">b</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&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>
 <span class="kw">and </span><br/><a NAME="E3:9"/><span class="lab"><font color="Green" title="E9">A2</font></span>: 
<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="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c8">b</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&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>
 <span class="kw">and </span><br/><a NAME="E4:9"/><span class="lab"><font color="Green" title="E10">A3</font></span>: 
<font color="Maroon" title="c11">f</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#K6" title="MARGREL1:func.6">FALSE</a></span>)</span>)
 <span class="kw">and </span><br/><a NAME="E5:9"/><span class="lab"><font color="Green" title="E11">A4</font></span>: 
<font color="Maroon" title="c12">g</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#K6" title="MARGREL1:func.6">FALSE</a></span>)</span>)
 <span class="kw">and </span><br/><a NAME="E6:9"/><span class="lab"><font color="Green" title="E12">A5</font></span>: 
<font color="Maroon" title="c13">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> <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#K6" title="MARGREL1:func.6">FALSE</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E7:9"/><span class="lab"><font color="Green" title="E13">A6</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="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#K2" title="CIRCCOMB:func.2">+*</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="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p4"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&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="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p4"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c8">b</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&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#K3" title="CIRCCOMB:func.3">+*</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="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p4"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&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="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p4"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c8">b</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&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="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&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="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c8">b</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&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>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gfacirc2.html#D2" target="_self" onmouseover="rs('gfacirc2/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<a NAME="E8:9"/><span class="lab"><font color="Green" title="E14">A7</font></span>: 
<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="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c8">b</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="txt" href="#E6:9"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="txt" href="#E7:9"><span class="lab"><font color="Green" title="E13">A6</font></span></a>, <a class="ref" href="gfacirc2.html#T1" target="_self" onmouseover="rs('gfacirc2/T1')" onmouseout="rh()">Th1</a></span>;<br/>
<a NAME="E9:9"/><span class="lab"><font color="Green" title="E15">A8</font></span>: 
<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="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c8">b</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="txt" href="#E6:9"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="txt" href="#E7:9"><span class="lab"><font color="Green" title="E13">A6</font></span></a>, <a class="ref" href="gfacirc2.html#T1" target="_self" onmouseover="rs('gfacirc2/T1')" onmouseout="rh()">Th1</a></span>;<br/>
<a NAME="E10:9"/><span class="lab"><font color="Green" title="E16">A9</font></span>: 
<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="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c8">b</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="txt" href="#E6:9"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="txt" href="#E7:9"><span class="lab"><font color="Green" title="E13">A6</font></span></a>, <a class="ref" href="gfacirc2.html#T1" target="_self" onmouseover="rs('gfacirc2/T1')" onmouseout="rh()">Th1</a></span>;<br/>
<a NAME="E11:9"/><span class="lab"><font color="Green" title="E17">A10</font></span>: 
<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="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c8">b</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="txt" href="#E6:9"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="txt" href="#E7:9"><span class="lab"><font color="Green" title="E13">A6</font></span></a>, <a class="ref" href="gfacirc2.html#T1" target="_self" onmouseover="rs('gfacirc2/T1')" onmouseout="rh()">Th1</a></span>;<br/>
<a NAME="E12:9"/><span class="lab"><font color="Green" title="E18">A11</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" onmouseover="rs('card_1/D7')" onmouseout="rh()">CARD_1:def 7</a></span>;<br/>
<a NAME="E13:9"/><span class="lab"><font color="Green" title="E19">A12</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" onmouseover="rs('card_1/D7')" onmouseout="rh()">CARD_1:def 7</a></span>;<br/>
<a NAME="E14:9"/><span class="lab"><font color="Green" title="E20">A13</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:9"><span class="lab"><font color="Green" title="E18">A11</font></span></a>, <a class="ref" href="finseq_1.html#T42" onmouseover="rs('finseq_1/T42')" onmouseout="rh()">FINSEQ_1:42</a></span>;<br/>
<a NAME="E15:9"/><span class="lab"><font color="Green" title="E21">A14</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c8">b</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:9"><span class="lab"><font color="Green" title="E19">A12</font></span></a>, <a class="ref" href="finseq_1.html#T42" onmouseover="rs('finseq_1/T42')" onmouseout="rh()">FINSEQ_1:42</a></span>;<br/>
<a NAME="E16:9"/><span class="lab"><font color="Green" title="E22">A15</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <a href="facirc_2.html#NK3" title="FACIRC_2:NK.3">&lt;*&gt;</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T34" onmouseover="rs('finseq_1/T34')" onmouseout="rh()">FINSEQ_1:34</a></span>;<br/>
<a NAME="E17:9"/><span class="lab"><font color="Green" title="E23">A16</font></span>: 
<font color="Maroon" title="c6">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <a href="facirc_2.html#NK3" title="FACIRC_2:NK.3">&lt;*&gt;</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T34" onmouseover="rs('finseq_1/T34')" onmouseout="rh()">FINSEQ_1:34</a></span>;<br/>
<a NAME="E18:9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A17</font></span>: 
<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="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c8">b</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:9"><span class="lab"><font color="Green" title="E22">A15</font></span></a>, <a class="ref" href="gfacirc2.html#T5" target="_self" onmouseover="rs('gfacirc2/T5')" onmouseout="rh()">Th5</a></span>;<br/>
<a NAME="E19:9"/><span class="lab"><font color="Green" title="E25">A18</font></span>: 
<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="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c8">b</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:9"><span class="lab"><font color="Green" title="E22">A15</font></span></a>, <a class="txt" href="#E17:9"><span class="lab"><font color="Green" title="E23">A16</font></span></a>, <a class="ref" href="gfacirc2.html#T5" target="_self" onmouseover="rs('gfacirc2/T5')" onmouseout="rh()">Th5</a></span>;<br/>
<a NAME="E20:9"/>
<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="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c8">b</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:9"><span class="lab"><font color="Green" title="E22">A15</font></span></a>, <a class="txt" href="#E17:9"><span class="lab"><font color="Green" title="E23">A16</font></span></a>, <a class="ref" href="gfacirc2.html#T5" target="_self" onmouseover="rs('gfacirc2/T5')" onmouseout="rh()">Th5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:9"/>
( <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="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c8">b</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&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#K2" title="CIRCCOMB:func.2">+*</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="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c8">b</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&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#K3" title="CIRCCOMB:func.3">+*</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="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c8">b</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&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>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="txt" href="#E7:9"><span class="lab"><font color="Green" title="E13">A6</font></span></a>, <a class="txt" href="#E8:9"><span class="lab"><font color="Green" title="E14">A7</font></span></a>, <a class="txt" href="#E9:9"><span class="lab"><font color="Green" title="E15">A8</font></span></a>, <a class="txt" href="#E10:9"><span class="lab"><font color="Green" title="E16">A9</font></span></a>, <a class="txt" href="#E11:9"><span class="lab"><font color="Green" title="E17">A10</font></span></a>, <a class="txt" href="#E14:9"><span class="lab"><font color="Green" title="E20">A13</font></span></a>, <a class="txt" href="#E15:9"><span class="lab"><font color="Green" title="E21">A14</font></span></a>, <a class="txt" href="#E18:9"><span class="lab"><font color="Green" title="E24">A17</font></span></a>, <a class="txt" href="#E19:9"><span class="lab"><font color="Green" title="E25">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>
