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<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="lang1.html#L1" title="LANG1:struct.1">DTConstrStr</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">s</font> being   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c1">G</font><br/>  for <font color="Olive" title="b2">n</font> being   <a href="lang1.html#NM2" title="LANG1:NM.2">String</a> of <font color="Maroon" title="c1">G</font>  st <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Olive" title="b1">s</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span> <a href="lang1.html#R2" title="LANG1:pred.2">==&gt;</a> <font color="Olive" title="b2">n</font> holds <br/><font color="Olive" title="b1">s</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <font color="Olive" title="b2">n</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">s</font> be   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c1">G</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="lang1.html#NM2" title="LANG1:NM.2">String</a> of <font color="Maroon" title="c1">G</font>  st <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span> <a href="lang1.html#R2" title="LANG1:pred.2">==&gt;</a> <font color="Olive" title="b1">n</font> holds <br/><font color="Maroon" title="c2">s</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <font color="Olive" title="b1">n</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">n</font> be   <a href="lang1.html#NM2" title="LANG1:NM.2">String</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span> <a href="lang1.html#R2" title="LANG1:pred.2">==&gt;</a> <font color="Maroon" title="c3">n</font> implies <font color="Maroon" title="c2">s</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <font color="Maroon" title="c3">n</font> )</span><br/>





<span class="kw">given </span><font color="Maroon" title="c4">n1</font>, <font color="Maroon" title="c5">n2</font>, <font color="Maroon" title="c6">n3</font> being   <a href="lang1.html#NM2" title="LANG1:NM.2">String</a> of <font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c7">t</font> being   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><a NAME="E2:12"/><span class="lab"><font color="Green" title="E4">A1</font></span>: 
<span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n1</font> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</a> <span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c7">t</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</a> <font color="Maroon" title="c5">n2</font>
 <span class="kw">and </span><br/><a NAME="E3:12"/><span class="lab"><font color="Green" title="E5">A2</font></span>: 
<font color="Maroon" title="c3">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n1</font> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</a> <font color="Maroon" title="c6">n3</font></span>)</span> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</a> <font color="Maroon" title="c5">n2</font>
 <span class="kw">and </span><br/><a NAME="E4:12"/><span class="lab"><font color="Green" title="E6">A3</font></span>: 
<font color="Maroon" title="c7">t</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <font color="Maroon" title="c6">n3</font>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="lang1.html#D4" target="_self" title="LANG1:def.4">LANG1:def 4</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" title="c2">s</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <font color="Maroon" title="c3">n</font></span><br/>

<a NAME="E5:12"/><span class="lab"><font color="Green" title="E7">A4</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c7">t</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>;<br/>
<a NAME="E6:12"/><span class="lab"><font color="Green" title="E8">A5</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n1</font> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</a> <span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c7">t</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c7">t</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" title="FINSEQ_1:th.22">FINSEQ_1:22</a></span>;<br/>
<a NAME="E7:12"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n1</font> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</a> <span class="p3"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c7">t</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">n2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="ref" href="finseq_1.html#T22" title="FINSEQ_1:th.22">FINSEQ_1:22</a></span>;<br/>
<a NAME="E8:12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E9">A6</font></span>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">n2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:12"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="txt" href="#E6:12"><span class="lab"><font color="Green" title="E8">A5</font></span></a>, <a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>;<br/>
<a NAME="E9:12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A7</font></span>: 
<font color="Maroon" title="c5">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T7" title="NAT_1:th.7">NAT_1:7</a></span>;<br/>
<a NAME="E10:12"/><span class="lab"><font color="Green" title="E11">A8</font></span>: 
<font color="Maroon" title="c6">n3</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T34" title="FINSEQ_1:th.34">FINSEQ_1:34</a></span>;<br/>
<a NAME="E11:12"/><span class="lab"><font color="Green" title="E12">A9</font></span>: 
<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c6">n3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T34" title="FINSEQ_1:th.34">FINSEQ_1:34</a></span>;<br/>
<a NAME="E12:12"/><span class="lab"><font color="Green" title="E13">A10</font></span>: 
<span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">s</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>;<br/>
<a NAME="E13:12"/><span class="lab"><font color="Green" title="E14">A11</font></span>: 
<span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c7">t</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c7">t</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T34" title="FINSEQ_1:th.34">FINSEQ_1:34</a></span>;<br/>
<a NAME="E14:12"/><span class="lab"><font color="Green" title="E15">A12</font></span>: 
<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c7">t</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c7">t</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T34" title="FINSEQ_1:th.34">FINSEQ_1:34</a></span>;<br/>
<a NAME="E15:12"/>
<font color="Maroon" title="c4">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:12"><span class="lab"><font color="Green" title="E9">A6</font></span></a>, <a class="ref" href="nat_1.html#T7" title="NAT_1:th.7">NAT_1:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:12"/>
<font color="Maroon" title="c2">s</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="txt" href="#E4:12"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="txt" href="#E9:12"><span class="lab"><font color="Green" title="E10">A7</font></span></a>, <a class="txt" href="#E14:12"><span class="lab"><font color="Green" title="E15">A12</font></span></a>, <a class="txt" href="#E13:12"><span class="lab"><font color="Green" title="E14">A11</font></span></a>, <a class="txt" href="#E11:12"><span class="lab"><font color="Green" title="E12">A9</font></span></a>, <a class="txt" href="#E10:12"><span class="lab"><font color="Green" title="E11">A8</font></span></a>, <a class="txt" href="#E12:12"><span class="lab"><font color="Green" title="E13">A10</font></span></a>, <a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</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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