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<span class="kw">let </span><font color="Maroon" title="c1">G1</font>, <font color="Maroon" title="c2">G2</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   non  <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>   <a href="msafree2.html#V2" title="MSAFREE2:attr.2">Circuit-like</a>   <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><br/>  for <font color="Olive" title="b3">C1</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>  <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c1">G1</font><br/>  for <font color="Olive" title="b4">C2</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>  <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font> <a href="circtrm1.html#R4" title="CIRCTRM1:pred.4">form_embedding_of</a> <font color="Olive" title="b3">C1</font>,<font color="Olive" title="b4">C2</font> &amp; <font color="Olive" title="b1">f</font> <a href="circtrm1.html#R3" title="CIRCTRM1:pred.3">preserves_inputs_of</a> <font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c2">G2</font> holds <br/> for <font color="Olive" title="b5">s2</font> being   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Olive" title="b4">C2</font><br/>  for <font color="Olive" title="b6">s1</font> being   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Olive" title="b3">C1</font>  st <font color="Olive" title="b6">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">s2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b1">f</font> holds <br/> for <font color="Olive" title="b7">n</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>  holds   <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Olive" title="b6">s1</font>,<font color="Olive" title="b7">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Olive" title="b5">s2</font>,<font color="Olive" title="b7">n</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b1">f</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c4">g</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">C1</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>  <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c1">G1</font><br/>  for <font color="Olive" title="b2">C2</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>  <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font> <a href="circtrm1.html#R4" title="CIRCTRM1:pred.4">form_embedding_of</a> <font color="Olive" title="b1">C1</font>,<font color="Olive" title="b2">C2</font> &amp; <font color="Maroon" title="c3">f</font> <a href="circtrm1.html#R3" title="CIRCTRM1:pred.3">preserves_inputs_of</a> <font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c2">G2</font> holds <br/> for <font color="Olive" title="b3">s2</font> being   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Olive" title="b2">C2</font><br/>  for <font color="Olive" title="b4">s1</font> being   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Olive" title="b1">C1</font>  st <font color="Olive" title="b4">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">s2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font> holds <br/> for <font color="Olive" title="b5">n</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>  holds   <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Olive" title="b4">s1</font>,<font color="Olive" title="b5">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Olive" title="b3">s2</font>,<font color="Olive" title="b5">n</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">C1</font> be   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>  <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c1">G1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">C2</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>  <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font> <a href="circtrm1.html#R4" title="CIRCTRM1:pred.4">form_embedding_of</a> <font color="Maroon" title="c5">C1</font>,<font color="Olive" title="b1">C2</font> &amp; <font color="Maroon" title="c3">f</font> <a href="circtrm1.html#R3" title="CIRCTRM1:pred.3">preserves_inputs_of</a> <font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c2">G2</font> holds <br/> for <font color="Olive" title="b2">s2</font> being   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Olive" title="b1">C2</font><br/>  for <font color="Olive" title="b3">s1</font> being   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c5">C1</font>  st <font color="Olive" title="b3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">s2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font> holds <br/> for <font color="Olive" title="b4">n</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>  holds   <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Olive" title="b3">s1</font>,<font color="Olive" title="b4">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Olive" title="b2">s2</font>,<font color="Olive" title="b4">n</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">C2</font> be   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>  <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c2">G2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font> <a href="circtrm1.html#R4" title="CIRCTRM1:pred.4">form_embedding_of</a> <font color="Maroon" title="c5">C1</font>,<font color="Maroon" title="c6">C2</font> &amp; <font color="Maroon" title="c3">f</font> <a href="circtrm1.html#R3" title="CIRCTRM1:pred.3">preserves_inputs_of</a> <font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c2">G2</font> implies  for <font color="Olive" title="b1">s2</font> being   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c6">C2</font><br/>  for <font color="Olive" title="b2">s1</font> being   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c5">C1</font>  st <font color="Olive" title="b2">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font> holds <br/> for <font color="Olive" title="b3">n</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>  holds   <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Olive" title="b2">s1</font>,<font color="Olive" title="b3">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Olive" title="b1">s2</font>,<font color="Olive" title="b3">n</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font> )</span><br/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:71"/><span class="lab"><font color="Green" title="E48">A1</font></span>: 
<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font> <a href="circtrm1.html#R4" title="CIRCTRM1:pred.4">form_embedding_of</a> <font color="Maroon" title="c5">C1</font>,<font color="Maroon" title="c6">C2</font>
 <span class="kw">and </span><br/><a NAME="E2:71"/><span class="lab"><font color="Green" title="E49">A2</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="circtrm1.html#R3" title="CIRCTRM1:pred.3">preserves_inputs_of</a> <font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c2">G2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">s2</font> being   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c6">C2</font><br/>  for <font color="Olive" title="b2">s1</font> being   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c5">C1</font>  st <font color="Olive" title="b2">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font> holds <br/> for <font color="Olive" title="b3">n</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>  holds   <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Olive" title="b2">s1</font>,<font color="Olive" title="b3">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Olive" title="b1">s2</font>,<font color="Olive" title="b3">n</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c7">s2</font> be   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c6">C2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">s1</font> being   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c5">C1</font>  st <font color="Olive" title="b1">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font> holds <br/> for <font color="Olive" title="b2">n</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>  holds   <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Olive" title="b1">s1</font>,<font color="Olive" title="b2">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c7">s2</font>,<font color="Olive" title="b2">n</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font></span><br/><span class="kw">let </span><font color="Maroon" title="c8">s1</font> be   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c5">C1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font> implies  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds   <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c8">s1</font>,<font color="Olive" title="b1">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c7">s2</font>,<font color="Olive" title="b1">n</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font> )</span><br/>



<span class="kw">assume </span><a NAME="E3:71"/><span class="lab"><font color="Green" title="E50">A3</font></span>: 
<font color="Maroon" title="c8">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds   <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c8">s1</font>,<font color="Olive" title="b1">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c7">s2</font>,<font color="Olive" title="b1">n</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font></span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <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">means </span> <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c8">s1</font>,$1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c7">s2</font>,$1)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font>;<br/>
<a NAME="E4:71"/>
 <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c8">s1</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>) <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c8">s1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="facirc_1.html#T11" title="FACIRC_1:th.11">FACIRC_1:11</a></span>;<br/>
<a NAME="E5:71"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A4</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:71"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="ref" href="facirc_1.html#T11" title="FACIRC_1:th.11">FACIRC_1:11</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E52">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">n</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:71_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><a NAME="E2:71_1"/><span class="kw">then </span>
 <a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c9">n</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <span class="p2">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c7">s2</font>,<font color="Maroon" title="c9">n</font>)</span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="txt" href="#E2:71"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="ref" href="circtrm1.html#T46" target="_self" title="CIRCTRM1:th.46">Th47</a></span>;<br/><span class="kw">then </span> <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c8">s1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) = 
<span class="p1">(<span class="default"><a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <span class="p2">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c7">s2</font>,<font color="Maroon" title="c9">n</font>)</span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="facirc_1.html#T12" title="FACIRC_1:th.12">FACIRC_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c7">s2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="facirc_1.html#T12" title="FACIRC_1:th.12">FACIRC_1:12</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:71_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E7:71"/>
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E5:71"><span class="lab"><font color="Green" title="E51">A4</font></span></a>, <a class="txt" href="#E6:71"><span class="lab"><font color="Green" title="E52">A5</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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