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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="Red">::  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>, <font color="Olive" title="b3">n1</font>, <font color="Olive" title="b4">n2</font> being  <a href="lang1.html#NM2" title="LANG1:NM.2">String</a> of <font color="Maroon" title="c1">G</font>  st <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> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b3">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="Olive" title="b1">s</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="Olive" title="b4">n2</font> <a href="lang1.html#R2" title="LANG1:pred.2">==&gt;</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">n1</font> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</a> <font color="Olive" title="b4">n2</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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">n1</font>, <font color="Olive" title="b3">n2</font> being  <a href="lang1.html#NM2" title="LANG1:NM.2">String</a> of <font color="Maroon" title="c1">G</font>  st <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> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b2">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="c2">s</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="Olive" title="b3">n2</font> <a href="lang1.html#R2" title="LANG1:pred.2">==&gt;</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">n1</font> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</a> <font color="Olive" title="b3">n2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">n</font>, <font color="Maroon" title="c4">n1</font>, <font color="Maroon" title="c5">n2</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="Red">::  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> implies <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="c2">s</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> <a href="lang1.html#R2" title="LANG1:pred.2">==&gt;</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="c3">n</font></span>)</span> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</a> <font color="Maroon" title="c5">n2</font> )</span><br/>









<span class="kw">assume </span><a NAME="E1:10"/><span class="lab"><font color="Green" title="E2">A1</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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="c2">s</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> <a href="lang1.html#R2" title="LANG1:pred.2">==&gt;</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="c3">n</font></span>)</span> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</a> <font color="Maroon" title="c5">n2</font></span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c4">n1</font>
; <span class="comment"><font color="Red">:: 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="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n2</font>, <font color="Olive" title="b2">n3</font> being  <a href="lang1.html#NM2" title="LANG1:NM.2">String</a> of <font color="Maroon" title="c1">G</font> ex <font color="Olive" title="b3">s</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c1">G</font> st <br/>( <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="c2">s</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> <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="Olive" title="b3">s</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="Olive" title="b1">n2</font> &amp; <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="c3">n</font></span>)</span> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</a> <font color="Maroon" title="c5">n2</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="Olive" title="b2">n3</font></span>)</span> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</a> <font color="Olive" title="b1">n2</font> &amp; <font color="Olive" title="b3">s</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <font color="Olive" title="b2">n3</font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c5">n2</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n3</font> being  <a href="lang1.html#NM2" title="LANG1:NM.2">String</a> of <font color="Maroon" title="c1">G</font> ex <font color="Olive" title="b2">s</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c1">G</font> st <br/>( <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="c2">s</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> <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="Olive" title="b2">s</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> &amp; <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="c3">n</font></span>)</span> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</a> <font color="Maroon" title="c5">n2</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="Olive" title="b1">n3</font></span>)</span> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</a> <font color="Maroon" title="c5">n2</font> &amp; <font color="Olive" title="b2">s</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <font color="Olive" title="b1">n3</font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c3">n</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <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> st <br/>( <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="c2">s</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> <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="Olive" title="b1">s</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> &amp; <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="c3">n</font></span>)</span> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</a> <font color="Maroon" title="c5">n2</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="c3">n</font></span>)</span> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</a> <font color="Maroon" title="c5">n2</font> &amp; <font color="Olive" title="b1">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">take </span>
<font color="Maroon" title="c2">s</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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="c2">s</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> <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="c2">s</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> &amp; <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="c3">n</font></span>)</span> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</a> <font color="Maroon" title="c5">n2</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="c3">n</font></span>)</span> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</a> <font color="Maroon" title="c5">n2</font> &amp; <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">thus </span><a NAME="E2:10"/>
( <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="c2">s</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> <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="c2">s</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> &amp; <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="c3">n</font></span>)</span> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</a> <font color="Maroon" title="c5">n2</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="c3">n</font></span>)</span> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</a> <font color="Maroon" title="c5">n2</font> &amp; <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="#E1:10"><span class="lab"><font color="Green" title="E2">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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