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<b>let </b><font color="Maroon" title="c12">it1</font>, <font color="Maroon" title="c13">it2</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <font color="Maroon" title="c8">G</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">G</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  st <br/>( <font color="Maroon" title="c12">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">tsg</font> &amp; ( for <font color="Olive" title="b2">t</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c8">G</font>  st <font color="Olive" title="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lang1.html#K5" title="LANG1:func.5">Terminals</a> <font color="Maroon" title="c8">G</font> holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Olive" title="b2">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Olive" title="b2">t</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> ) &amp; ( for <font color="Olive" title="b2">nt</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c8">G</font><br/> for <font color="Olive" title="b3">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" title="c8">G</font><br/> for <font color="Olive" title="b4">rts</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st <font color="Olive" title="b4">rts</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b3">ts</font> &amp; <font color="Olive" title="b2">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <font color="Olive" title="b4">rts</font> holds <br/>for <font color="Olive" title="b5">x</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">G</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>   st <font color="Olive" title="b5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Olive" title="b3">ts</font> holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Olive" title="b3">ts</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Olive" title="b2">nt</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Olive" title="b5">x</font></span>)</span> ) ) &amp;  ex <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <font color="Maroon" title="c8">G</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">G</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  st <br/>( <font color="Maroon" title="c13">it2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">tsg</font> &amp; ( for <font color="Olive" title="b2">t</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c8">G</font>  st <font color="Olive" title="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lang1.html#K5" title="LANG1:func.5">Terminals</a> <font color="Maroon" title="c8">G</font> holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Olive" title="b2">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Olive" title="b2">t</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> ) &amp; ( for <font color="Olive" title="b2">nt</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c8">G</font><br/> for <font color="Olive" title="b3">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" title="c8">G</font><br/> for <font color="Olive" title="b4">rts</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st <font color="Olive" title="b4">rts</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b3">ts</font> &amp; <font color="Olive" title="b2">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <font color="Olive" title="b4">rts</font> holds <br/>for <font color="Olive" title="b5">x</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">G</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>   st <font color="Olive" title="b5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Olive" title="b3">ts</font> holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Olive" title="b3">ts</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Olive" title="b2">nt</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Olive" title="b5">x</font></span>)</span> ) ) implies <font color="Maroon" title="c12">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">it2</font> )</span><br/>

<b>given </b><font color="Maroon" title="c14">f1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <font color="Maroon" title="c8">G</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">G</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> <b> such that </b><a NAME="E2:55_4_2"/><i><font color="Green" title="E42">A17</font></i>: 
( <font color="Maroon" title="c12">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">tsg</font> &amp; ( for <font color="Olive" title="b1">t</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c8">G</font>  st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lang1.html#K5" title="LANG1:func.5">Terminals</a> <font color="Maroon" title="c8">G</font> holds <br/><font color="Maroon" title="c14">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Olive" title="b1">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Olive" title="b1">t</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> ) &amp; ( for <font color="Olive" title="b1">nt</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c8">G</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" title="c8">G</font><br/> for <font color="Olive" title="b3">rts</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st <font color="Olive" title="b3">rts</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b2">ts</font> &amp; <font color="Olive" title="b1">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <font color="Olive" title="b3">rts</font> holds <br/>for <font color="Olive" title="b4">x</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">G</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>   st <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f1</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Olive" title="b2">ts</font> holds <br/><font color="Maroon" title="c14">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Olive" title="b2">ts</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Olive" title="b1">nt</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Olive" title="b4">x</font></span>)</span> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( for <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <font color="Maroon" title="c8">G</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">G</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>   holds <br/>(  not <font color="Maroon" title="c13">it2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">tsg</font> or  ex <font color="Olive" title="b2">t</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c8">G</font> st <br/>( <font color="Olive" title="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lang1.html#K5" title="LANG1:func.5">Terminals</a> <font color="Maroon" title="c8">G</font> &amp; not <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Olive" title="b2">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Olive" title="b2">t</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> ) or  ex <font color="Olive" title="b2">nt</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c8">G</font> ex <font color="Olive" title="b3">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" title="c8">G</font> ex <font color="Olive" title="b4">rts</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Olive" title="b4">rts</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b3">ts</font> &amp; <font color="Olive" title="b2">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <font color="Olive" title="b4">rts</font> &amp;  ex <font color="Olive" title="b5">x</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">G</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  st <br/>( <font color="Olive" title="b5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Olive" title="b3">ts</font> &amp; not <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Olive" title="b3">ts</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Olive" title="b2">nt</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Olive" title="b5">x</font></span>)</span> ) ) ) or <font color="Maroon" title="c12">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">it2</font> )</span><br/>

<b>given </b><font color="Maroon" title="c15">f2</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <font color="Maroon" title="c8">G</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">G</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> <b> such that </b><a NAME="E4:55_4_2"/><i><font color="Green" title="E43">A18</font></i>: 
( <font color="Maroon" title="c13">it2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">tsg</font> &amp; ( for <font color="Olive" title="b1">t</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c8">G</font>  st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lang1.html#K5" title="LANG1:func.5">Terminals</a> <font color="Maroon" title="c8">G</font> holds <br/><font color="Maroon" title="c15">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Olive" title="b1">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Olive" title="b1">t</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> ) &amp; ( for <font color="Olive" title="b1">nt</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c8">G</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" title="c8">G</font><br/> for <font color="Olive" title="b3">rts</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st <font color="Olive" title="b3">rts</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b2">ts</font> &amp; <font color="Olive" title="b1">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <font color="Olive" title="b3">rts</font> holds <br/>for <font color="Olive" title="b4">x</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">G</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>   st <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">f2</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Olive" title="b2">ts</font> holds <br/><font color="Maroon" title="c15">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Olive" title="b2">ts</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Olive" title="b1">nt</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Olive" title="b4">x</font></span>)</span> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">it2</font></span><br/>

<div><i><font color="Green" title="E44">A19</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="55_4_2_1"><b>now </b></a><div class="add"><div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">nt</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c8">G</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" title="c8">G</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> holds <br/><font color="Maroon" title="c14">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Olive" title="b2">ts</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>11</sub></font>(<font color="Olive" title="b1">nt</font>, <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b2">ts</font>,<font color="Maroon" title="c14">f1</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Olive" title="b2">ts</font>)</span>
<div class="add"><b>let </b><font color="Maroon" title="c16">t</font> be   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c8">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c16">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lang1.html#K5" title="LANG1:func.5">Terminals</a> <font color="Maroon" title="c8">G</font> implies <font color="Maroon" title="c14">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c16">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>10</sub></font>(<font color="Maroon" title="c16">t</font>) )</span><br/><b>assume </b><a NAME="E1:55_4_2_1_1"/>
<font color="Maroon" title="c16">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lang1.html#K5" title="LANG1:func.5">Terminals</a> <font color="Maroon" title="c8">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c16">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>10</sub></font>(<font color="Maroon" title="c16">t</font>)</span><br/><b>hence </b><font color="Maroon" title="c14">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c16">t</font></span>)</span> = 
<span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c16">t</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span>
<b>by </b><i><a class="txt" href="dtconstr.html#E2:55_4_2"><i><font color="Green" title="E42">A17</font></i></a></i>
<br/>.= 
<font color="Maroon">H<sub>10</sub></font>(<font color="Maroon" title="c16">t</font>)
<b>by </b><i><a class="txt" href="dtconstr.html#E7:55"><i><font color="Green" title="E41">A14</font></i></a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div><b>let </b><font color="Maroon" title="c16">nt</font> be   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c8">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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" title="c8">G</font>  st <font color="Maroon" title="c16">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> holds <br/><font color="Maroon" title="c14">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Olive" title="b1">ts</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>11</sub></font>(<font color="Maroon" title="c16">nt</font>, <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b1">ts</font>,<font color="Maroon" title="c14">f1</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Olive" title="b1">ts</font>)</span><br/><b>let </b><font color="Maroon" title="c17">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" title="c8">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c16">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="c17">ts</font> implies <font color="Maroon" title="c14">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c17">ts</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>11</sub></font>(<font color="Maroon" title="c16">nt</font>, <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c17">ts</font>,<font color="Maroon" title="c14">f1</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c17">ts</font>) )</span><br/><b>assume </b><a NAME="E2:55_4_2_1"/>
<font color="Maroon" title="c16">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="c17">ts</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c17">ts</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>11</sub></font>(<font color="Maroon" title="c16">nt</font>, <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c17">ts</font>,<font color="Maroon" title="c14">f1</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c17">ts</font>)</span><br/><b>hence </b><font color="Maroon" title="c14">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c17">ts</font></span>)</span> = 
<span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c16">nt</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">f1</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c17">ts</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="dtconstr.html#E2:55_4_2"><i><font color="Green" title="E42">A17</font></i></a></i>
<br/>.= 
<font color="Maroon">H<sub>11</sub></font>(<font color="Maroon" title="c16">nt</font>, <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c17">ts</font>,<font color="Maroon" title="c14">f1</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c17">ts</font>)
<b>by </b><i><a class="txt" href="dtconstr.html#E7:55"><i><font color="Green" title="E41">A14</font></i></a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E45">A20</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="55_4_2_2"><b>now </b></a><div class="add"><div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">nt</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c8">G</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" title="c8">G</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> holds <br/><font color="Maroon" title="c15">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Olive" title="b2">ts</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>11</sub></font>(<font color="Olive" title="b1">nt</font>, <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b2">ts</font>,<font color="Maroon" title="c15">f2</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Olive" title="b2">ts</font>)</span>
<div class="add"><b>let </b><font color="Maroon" title="c16">t</font> be   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c8">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c16">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lang1.html#K5" title="LANG1:func.5">Terminals</a> <font color="Maroon" title="c8">G</font> implies <font color="Maroon" title="c15">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c16">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>10</sub></font>(<font color="Maroon" title="c16">t</font>) )</span><br/><b>assume </b><a NAME="E1:55_4_2_2_1"/>
<font color="Maroon" title="c16">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lang1.html#K5" title="LANG1:func.5">Terminals</a> <font color="Maroon" title="c8">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c15">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c16">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>10</sub></font>(<font color="Maroon" title="c16">t</font>)</span><br/><b>hence </b><font color="Maroon" title="c15">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c16">t</font></span>)</span> = 
<span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c16">t</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span>
<b>by </b><i><a class="txt" href="dtconstr.html#E4:55_4_2"><i><font color="Green" title="E43">A18</font></i></a></i>
<br/>.= 
<font color="Maroon">H<sub>10</sub></font>(<font color="Maroon" title="c16">t</font>)
<b>by </b><i><a class="txt" href="dtconstr.html#E7:55"><i><font color="Green" title="E41">A14</font></i></a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div><b>let </b><font color="Maroon" title="c16">nt</font> be   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c8">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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" title="c8">G</font>  st <font color="Maroon" title="c16">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> holds <br/><font color="Maroon" title="c15">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Olive" title="b1">ts</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>11</sub></font>(<font color="Maroon" title="c16">nt</font>, <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b1">ts</font>,<font color="Maroon" title="c15">f2</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Olive" title="b1">ts</font>)</span><br/><b>let </b><font color="Maroon" title="c17">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" title="c8">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c16">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="c17">ts</font> implies <font color="Maroon" title="c15">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c17">ts</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>11</sub></font>(<font color="Maroon" title="c16">nt</font>, <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c17">ts</font>,<font color="Maroon" title="c15">f2</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c17">ts</font>) )</span><br/><b>assume </b><a NAME="E2:55_4_2_2"/>
<font color="Maroon" title="c16">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="c17">ts</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c15">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c17">ts</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>11</sub></font>(<font color="Maroon" title="c16">nt</font>, <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c17">ts</font>,<font color="Maroon" title="c15">f2</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c17">ts</font>)</span><br/><b>hence </b><font color="Maroon" title="c15">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c17">ts</font></span>)</span> = 
<span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c16">nt</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">f2</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c17">ts</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="dtconstr.html#E4:55_4_2"><i><font color="Green" title="E43">A18</font></i></a></i>
<br/>.= 
<font color="Maroon">H<sub>11</sub></font>(<font color="Maroon" title="c16">nt</font>, <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c17">ts</font>,<font color="Maroon" title="c15">f2</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c17">ts</font>)
<b>by </b><i><a class="txt" href="dtconstr.html#E7:55"><i><font color="Green" title="E41">A14</font></i></a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E7:55_4_2"/>
<font color="Maroon" title="c14">f1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c15">f2</font>
 
<b>from </b><i><a class="ref" href="dtconstr.html#S9" target="_self" title="DTCONSTR:sch.9">DTCONSTR:sch 9</a>(<a class="txt" href="dtconstr.html#E5:55_4_2"><i><font color="Green" title="E44">A19</font></i></a>, <a class="txt" href="dtconstr.html#E6:55_4_2"><i><font color="Green" title="E45">A20</font></i></a>);<br/></i>
<b>hence </b><a NAME="E8:55_4_2"/>
<font color="Maroon" title="c12">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">it2</font>
 <b>by </b><i><a class="txt" href="dtconstr.html#E2:55_4_2"><i><font color="Green" title="E42">A17</font></i></a>, <a class="txt" href="dtconstr.html#E4:55_4_2"><i><font color="Green" title="E43">A18</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
