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<a NAME="E1:19_1"/><span class="lab"><font color="Green" title="E10">A3</font></span>: 
 for <font color="Olive" title="b1">nt</font> being   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>  for <font color="Olive" title="b2">ts</font> being    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b1">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b2">ts</font> &amp; (  for <font color="Olive" title="b3">t</font> being   <a href="trees_2.html#NM5" title="TREES_2:NM.5">DecoratedTree</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b3">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b2">ts</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b3">t</font>] ) holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">nt</font> <a href="trees_4.html#K8" title="TREES_4:func.8">-tree</a> <font color="Olive" title="b2">ts</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="19_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">nt</font> be   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon">F<sub>1</sub></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">ts</font> being    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon" title="c1">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b1">ts</font> &amp; (  for <font color="Olive" title="b2">t</font> being   <a href="trees_2.html#NM5" title="TREES_2:NM.5">DecoratedTree</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b2">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b1">ts</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">t</font>] ) holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">nt</font> <a href="trees_4.html#K8" title="TREES_4:func.8">-tree</a> <font color="Olive" title="b1">ts</font>]</span><br/><span class="kw">let </span><font color="Maroon" title="c2">ts</font> be    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <font color="Maroon">F<sub>1</sub></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"> ( <font color="Maroon" title="c1">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c2">ts</font> &amp; (  for <font color="Olive" title="b1">t</font> being   <a href="trees_2.html#NM5" title="TREES_2:NM.5">DecoratedTree</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b1">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">ts</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">t</font>] ) implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">nt</font> <a href="trees_4.html#K8" title="TREES_4:func.8">-tree</a> <font color="Maroon" title="c2">ts</font>] )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:19_1_1"/><span class="lab"><font color="Green" title="E11">A4</font></span>: 
<font color="Maroon" title="c1">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c2">ts</font>
 <span class="kw">and </span><br/><a NAME="E2:19_1_1"/><span class="lab"><font color="Green" title="E12">A5</font></span>: 
 for <font color="Olive" title="b1">t</font> being   <a href="trees_2.html#NM5" title="TREES_2:NM.5">DecoratedTree</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b1">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">ts</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">t</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"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">nt</font> <a href="trees_4.html#K8" title="TREES_4:func.8">-tree</a> <font color="Maroon" title="c2">ts</font>]</span><br/>

<a NAME="E3:19_1_1"/><span class="lab"><font color="Green" title="E13">A6</font></span>: 
<font color="Maroon" title="c1">nt</font> is   <a href="dtconstr.html#NM3" title="DTCONSTR:NM.3">NonTerminal</a> of <font color="Maroon">F<sub>1</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1_1"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="ref" href="bintree1.html#T10" target="_self" onmouseover="rs('bintree1/T10')" onmouseout="rh()">Th10</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c3">tl</font>, <font color="Maroon" title="c4">tr</font> being    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <font color="Maroon">F<sub>1</sub></font>()<span class="kw"> such that </span><br/><a NAME="E5:19_1_1"/><span class="lab"><font color="Green" title="E14">A7</font></span>: 
 <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c2">ts</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="bintree1.html#K1" title="BINTREE1:func.1">root-label</a> <font color="Maroon" title="c3">tl</font></span>)</span>,<span class="p2">(<span class="default"><a href="bintree1.html#K1" title="BINTREE1:func.1">root-label</a> <font color="Maroon" title="c4">tr</font></span>)</span></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E6:19_1_1"/>
<font color="Maroon" title="c3">tl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">ts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <span class="kw">and </span><br/><a NAME="E7:19_1_1"/>
<font color="Maroon" title="c4">tr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">ts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
 <span class="kw">and </span><br/><a NAME="E8:19_1_1"/><span class="lab"><font color="Green" title="E15">A8</font></span>: 
<font color="Maroon" title="c1">nt</font> <a href="trees_4.html#K8" title="TREES_4:func.8">-tree</a> <font color="Maroon" title="c2">ts</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">nt</font> <a href="trees_4.html#K6" title="TREES_4:func.6">-tree</a> (<font color="Maroon" title="c3">tl</font>,<font color="Maroon" title="c4">tr</font>)
 <span class="kw">and </span><br/><a NAME="E9:19_1_1"/><span class="lab"><font color="Green" title="E16">A9</font></span>: 
( <font color="Maroon" title="c3">tl</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">ts</font> &amp; <font color="Maroon" title="c4">tr</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">ts</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1_1"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="ref" href="bintree1.html#T10" target="_self" onmouseover="rs('bintree1/T10')" onmouseout="rh()">Th10</a></span>;<br/>
<a NAME="E10:19_1_1"/>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">tl</font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">tr</font>] )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19_1_1"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="txt" href="#E9:19_1_1"><span class="lab"><font color="Green" title="E16">A9</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:19_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">nt</font> <a href="trees_4.html#K8" title="TREES_4:func.8">-tree</a> <font color="Maroon" title="c2">ts</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="txt" href="#E1:19_1_1"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="txt" href="#E3:19_1_1"><span class="lab"><font color="Green" title="E13">A6</font></span></a>, <a class="txt" href="#E5:19_1_1"><span class="lab"><font color="Green" title="E14">A7</font></span></a>, <a class="txt" href="#E8:19_1_1"><span class="lab"><font color="Green" title="E15">A8</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><span class="kw">end;</span></div>
<a NAME="E2:19_1"/><span class="lab"><font color="Green" title="E11">A10</font></span>: 
 for <font color="Olive" title="b1">s</font> being   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b1">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="dtconstr.html#K6" title="DTCONSTR:func.6">Terminals</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon">P<sub>1</sub></font>[ <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Olive" title="b1">s</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E8">A1</font></span></a></span>;<br/>
<a NAME="E3:19_1"/>
 for <font color="Olive" title="b1">t</font> being   <a href="trees_2.html#NM5" title="TREES_2:NM.5">DecoratedTree</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b1">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">t</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="dtconstr.html#S7" onmouseover="rs('dtconstr/S7')" onmouseout="rh()">DTCONSTR:sch 7</a>(<a class="txt" href="#E2:19_1"><span class="lab"><font color="Green" title="E11">A10</font></span></a>, <a class="txt" href="#E1:19_1"><span class="lab"><font color="Green" title="E10">A3</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E4:19_1"/>
 for <font color="Olive" title="b1">t</font> being    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">t</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"> verum</span><br/>


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