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<span class="kw">consider </span><font color="Maroon" title="c5">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E2:13_1_1"/><span class="lab"><font color="Green" title="E12">A3</font></span>: 
<font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">i</font> = <font color="Maroon" title="c5">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">o</font> =  <a href="fscirc_1.html#K1" title="FSCIRC_1:func.1">BitSubtracterOutput</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="fscirc_2.html#K3" title="FSCIRC_2:func.3">-BitBorrowOutput</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>);<br/>
<a NAME="E4:13_1_1"/><span class="lab"><font color="Green" title="E13">A4</font></span>: 
 <a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="fscirc_2.html#K1" title="FSCIRC_2:func.1">-BitSubtracterStr</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="fscirc_2.html#K1" title="FSCIRC_2:func.1">-BitSubtracterStr</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:13"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="ref" href="fscirc_2.html#T8" target="_self" title="FSCIRC_2:th.8">Th8</a></span>;<br/>
<a NAME="E5:13_1_1"/><span class="lab"><font color="Green" title="E14">A5</font></span>: 
 <a href="fscirc_1.html#K1" title="FSCIRC_1:func.1">BitSubtracterOutput</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="fscirc_2.html#K3" title="FSCIRC_2:func.3">-BitBorrowOutput</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <span class="p1">(<span class="default"><a href="fscirc_1.html#K8" title="FSCIRC_1:func.8">BitSubtracterWithBorrowStr</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="fscirc_2.html#K3" title="FSCIRC_2:func.3">-BitBorrowOutput</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:13_1_1"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="facirc_1.html#T21" title="FACIRC_1:th.21">FACIRC_1:21</a></span>;<br/>
<a NAME="E6:13_1_1"/><span class="lab"><font color="Green" title="E15">A6</font></span>: 
<font color="Maroon" title="c1">k</font> <a href="fscirc_2.html#K1" title="FSCIRC_2:func.1">-BitSubtracterStr</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="fscirc_2.html#K1" title="FSCIRC_2:func.1">-BitSubtracterStr</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>)</span>)</span> <a href="circcomb.html#K2" title="CIRCCOMB:func.2">+*</a> <span class="p1">(<span class="default"><a href="fscirc_1.html#K8" title="FSCIRC_1:func.8">BitSubtracterWithBorrowStr</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="fscirc_2.html#K3" title="FSCIRC_2:func.3">-BitBorrowOutput</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:13_1_1"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="fscirc_2.html#T7" target="_self" title="FSCIRC_2:th.7">Th7</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">o</font> =  <a href="fscirc_1.html#K1" title="FSCIRC_1:func.1">BitSubtracterOutput</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="fscirc_2.html#K3" title="FSCIRC_2:func.3">-BitBorrowOutput</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>) as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="fscirc_1.html#K8" title="FSCIRC_1:func.8">BitSubtracterWithBorrowStr</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="fscirc_2.html#K3" title="FSCIRC_2:func.3">-BitBorrowOutput</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:13_1_1"><span class="lab"><font color="Green" title="E14">A5</font></span></a></span>;<br/>
<a NAME="E8:13_1_1"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="fscirc_1.html#K8" title="FSCIRC_1:func.8">BitSubtracterWithBorrowStr</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="fscirc_2.html#K3" title="FSCIRC_2:func.3">-BitBorrowOutput</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="fscirc_2.html#K1" title="FSCIRC_2:func.1">-BitSubtracterStr</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="fscirc_2.html#K1" title="FSCIRC_2:func.1">-BitSubtracterStr</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:13_1_1"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="ref" href="circcomb.html#D2" title="CIRCCOMB:def.2">CIRCCOMB:def 2</a></span>;<br/>
<a NAME="E9:13_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="fscirc_2.html#K1" title="FSCIRC_2:func.1">-BitSubtracterStr</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E10:13_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="fscirc_2.html#K1" title="FSCIRC_2:func.1">-BitSubtracterStr</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:13_1_1"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="txt" href="#E6:13_1_1"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="ref" href="circcomb.html#T15" title="CIRCCOMB:th.15">CIRCCOMB:15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:13_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being    <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"><font color="Maroon" title="c2">n</font> <a href="fscirc_2.html#K1" title="FSCIRC_2:func.1">-BitSubtracterStr</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>)</span>)</span> ex <font color="Olive" title="b2">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fscirc_1.html#K1" title="FSCIRC_1:func.1">BitSubtracterOutput</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">i</font> <a href="fscirc_2.html#K3" title="FSCIRC_2:func.3">-BitBorrowOutput</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:13_1_1"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="txt" href="#E4:13_1_1"><span class="lab"><font color="Green" title="E13">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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