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<b>let </b><font color="Maroon" title="c1">S</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="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ; <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">A</font> being <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a> <a href="msafree2.html#V4" title="MSAFREE2:attr.4">locally-finite</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">V</font> being   <a href="msaterm.html#M4" title="MSATERM:mode.4">Variables</a> of <font color="Olive" title="b1">A</font><br/> for <font color="Olive" title="b3">X</font> being   <a href="msaterm.html#M3" title="MSATERM:mode.3">SetWithCompoundTerm</a> of <font color="Maroon" title="c1">S</font>,<font color="Olive" title="b2">V</font><br/> for <font color="Olive" title="b4">s</font> being  <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <span class="p1">(<span class="default"><font color="Olive" title="b3">X</font> <a href="circtrm1.html#K6" title="CIRCTRM1:func.6">-Circuit</a> <font color="Olive" title="b1">A</font></span>)</span><br/> for <font color="Olive" title="b5">f</font> being   <a href="circtrm1.html#M1" title="CIRCTRM1:mode.1">CompatibleValuation</a> of <font color="Olive" title="b4">s</font><br/> for <font color="Olive" title="b6">t</font> being  <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S</font>,<font color="Olive" title="b2">V</font>  st <font color="Olive" title="b6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_9.html#K9" title="TREES_9:func.9">Subtrees</a> <font color="Olive" title="b3">X</font> holds <br/>(  <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Olive" title="b4">s</font>,<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b6">t</font></span>)</span></span>)</span></span>)</span> <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> <font color="Olive" title="b6">t</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Olive" title="b4">s</font>,<span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b6">t</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">t</font> <a href="circtrm1.html#K7" title="CIRCTRM1:func.7">@</a> <font color="Olive" title="b5">f</font>,<font color="Olive" title="b1">A</font> )</span><br/><b>let </b><font color="Maroon" title="c2">A</font> be  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a> <a href="msafree2.html#V4" title="MSAFREE2:attr.4">locally-finite</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</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">V</font> being   <a href="msaterm.html#M4" title="MSATERM:mode.4">Variables</a> of <font color="Maroon" title="c2">A</font><br/> for <font color="Olive" title="b2">X</font> being   <a href="msaterm.html#M3" title="MSATERM:mode.3">SetWithCompoundTerm</a> of <font color="Maroon" title="c1">S</font>,<font color="Olive" title="b1">V</font><br/> for <font color="Olive" title="b3">s</font> being  <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <span class="p1">(<span class="default"><font color="Olive" title="b2">X</font> <a href="circtrm1.html#K6" title="CIRCTRM1:func.6">-Circuit</a> <font color="Maroon" title="c2">A</font></span>)</span><br/> for <font color="Olive" title="b4">f</font> being   <a href="circtrm1.html#M1" title="CIRCTRM1:mode.1">CompatibleValuation</a> of <font color="Olive" title="b3">s</font><br/> for <font color="Olive" title="b5">t</font> being  <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S</font>,<font color="Olive" title="b1">V</font>  st <font color="Olive" title="b5">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_9.html#K9" title="TREES_9:func.9">Subtrees</a> <font color="Olive" title="b2">X</font> holds <br/>(  <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Olive" title="b3">s</font>,<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b5">t</font></span>)</span></span>)</span></span>)</span> <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> <font color="Olive" title="b5">t</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Olive" title="b3">s</font>,<span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b5">t</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">t</font> <a href="circtrm1.html#K7" title="CIRCTRM1:func.7">@</a> <font color="Olive" title="b4">f</font>,<font color="Maroon" title="c2">A</font> )</span><br/><b>let </b><font color="Maroon" title="c3">V</font> be    <a href="msaterm.html#M4" title="MSATERM:mode.4">Variables</a> of <font color="Maroon" title="c2">A</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">X</font> being   <a href="msaterm.html#M3" title="MSATERM:mode.3">SetWithCompoundTerm</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c3">V</font><br/> for <font color="Olive" title="b2">s</font> being  <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <span class="p1">(<span class="default"><font color="Olive" title="b1">X</font> <a href="circtrm1.html#K6" title="CIRCTRM1:func.6">-Circuit</a> <font color="Maroon" title="c2">A</font></span>)</span><br/> for <font color="Olive" title="b3">f</font> being   <a href="circtrm1.html#M1" title="CIRCTRM1:mode.1">CompatibleValuation</a> of <font color="Olive" title="b2">s</font><br/> for <font color="Olive" title="b4">t</font> being  <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c3">V</font>  st <font color="Olive" title="b4">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_9.html#K9" title="TREES_9:func.9">Subtrees</a> <font color="Olive" title="b1">X</font> holds <br/>(  <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Olive" title="b2">s</font>,<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b4">t</font></span>)</span></span>)</span></span>)</span> <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> <font color="Olive" title="b4">t</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Olive" title="b2">s</font>,<span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b4">t</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">t</font> <a href="circtrm1.html#K7" title="CIRCTRM1:func.7">@</a> <font color="Olive" title="b3">f</font>,<font color="Maroon" title="c2">A</font> )</span><br/><b>let </b><font color="Maroon" title="c4">X</font> be    <a href="msaterm.html#M3" title="MSATERM:mode.3">SetWithCompoundTerm</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c3">V</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">s</font> being  <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="circtrm1.html#K6" title="CIRCTRM1:func.6">-Circuit</a> <font color="Maroon" title="c2">A</font></span>)</span><br/> for <font color="Olive" title="b2">f</font> being   <a href="circtrm1.html#M1" title="CIRCTRM1:mode.1">CompatibleValuation</a> of <font color="Olive" title="b1">s</font><br/> for <font color="Olive" title="b3">t</font> being  <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c3">V</font>  st <font color="Olive" title="b3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_9.html#K9" title="TREES_9:func.9">Subtrees</a> <font color="Maroon" title="c4">X</font> holds <br/>(  <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b3">t</font></span>)</span></span>)</span></span>)</span> <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> <font color="Olive" title="b3">t</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Olive" title="b1">s</font>,<span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b3">t</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">t</font> <a href="circtrm1.html#K7" title="CIRCTRM1:func.7">@</a> <font color="Olive" title="b2">f</font>,<font color="Maroon" title="c2">A</font> )</span><br/><b>let </b><font color="Maroon" title="c5">q</font> be   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="circtrm1.html#K6" title="CIRCTRM1:func.6">-Circuit</a> <font color="Maroon" title="c2">A</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="circtrm1.html#M1" title="CIRCTRM1:mode.1">CompatibleValuation</a> of <font color="Maroon" title="c5">q</font><br/> for <font color="Olive" title="b2">t</font> being  <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c3">V</font>  st <font color="Olive" title="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_9.html#K9" title="TREES_9:func.9">Subtrees</a> <font color="Maroon" title="c4">X</font> holds <br/>(  <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">q</font>,<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b2">t</font></span>)</span></span>)</span></span>)</span> <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> <font color="Olive" title="b2">t</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">q</font>,<span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b2">t</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">t</font> <a href="circtrm1.html#K7" title="CIRCTRM1:func.7">@</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c2">A</font> )</span><br/><b>let </b><font color="Maroon" title="c6">f</font> be    <a href="circtrm1.html#M1" title="CIRCTRM1:mode.1">CompatibleValuation</a> of <font color="Maroon" title="c5">q</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">t</font> being  <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c3">V</font>  st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_9.html#K9" title="TREES_9:func.9">Subtrees</a> <font color="Maroon" title="c4">X</font> holds <br/>(  <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">q</font>,<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">t</font></span>)</span></span>)</span></span>)</span> <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> <font color="Olive" title="b1">t</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">q</font>,<span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">t</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> <a href="circtrm1.html#K7" title="CIRCTRM1:func.7">@</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c2">A</font> )</span><br/>











<a NAME="E1:39"/><i><font color="Green" title="E24">A1</font></i>: 
<font color="Maroon" title="c4">X</font> <a href="circtrm1.html#K1" title="CIRCTRM1:func.1">-CircuitStr</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#G1" title="MSUALG_1:aggr.1">ManySortedSign</a>(# <span class="p1">(<span class="default"><a href="trees_9.html#K9" title="TREES_9:func.9">Subtrees</a> <font color="Maroon" title="c4">X</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">S</font>,<span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="trees_9.html#K12" title="TREES_9:func.12">-Subtrees</a> <font color="Maroon" title="c4">X</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">S</font>,<span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <font color="Maroon" title="c4">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="funct_3.html#K6" title="FUNCT_3:func.6">incl</a> <span class="p2">(<span class="default"><span class="p3"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">S</font>,<span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="trees_9.html#K12" title="TREES_9:func.12">-Subtrees</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span> #)
 
;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="trees_2.html#NM4" title="TREES_2:NM.4">DecoratedTree</a>] <b>means </b>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_9.html#K9" title="TREES_9:func.9">Subtrees</a> <font color="Maroon" title="c4">X</font> implies (  <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">q</font>,<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> $1</span>)</span></span>)</span></span>)</span> <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> $1 &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">q</font>,<span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> $1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 <a href="circtrm1.html#K7" title="CIRCTRM1:func.7">@</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c2">A</font> ) );<br/>
<a NAME="E2:39"/><i><font color="Green" title="E25">A2</font></i>: 
for <font color="Olive" title="b1">s</font> being  <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> holds  <font color="Maroon">S<sub>1</sub></font>[ <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">v</font>,<font color="Olive" title="b1">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">s</font> be   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S</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">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">s</font> holds  <font color="Maroon">S<sub>1</sub></font>[ <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Maroon" title="c7">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>]</span><br/><b>let </b><font color="Maroon" title="c8">v</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">s</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[ <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c7">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>]</span><br/>



<b>assume </b><a NAME="E1:39_1"/><i><font color="Green" title="E26">A3</font></i>: 
 <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c7">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_9.html#K9" title="TREES_9:func.9">Subtrees</a> <font color="Maroon" title="c4">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">q</font>,<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p4">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p5"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c7">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c7">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">q</font>,<span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p5">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p0"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c7">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c7">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c7">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span> <a href="circtrm1.html#K7" title="CIRCTRM1:func.7">@</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c2">A</font> )</span><br/>

<a NAME="E2:39_1"/><b>then </b><i><font color="Green" title="E27">A4</font></i>: 
 <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c7">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="circtrm1.html#K1" title="CIRCTRM1:func.1">-CircuitStr</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="circtrm1.html#T12" target="_self" title="CIRCTRM1:th.12">Th12</a></i>;<br/>
<b>hence </b><a NAME="E3:39_1"/>
 <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">q</font>,<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p4">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p5"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c7">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c7">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="facirc_1.html#T18" title="FACIRC_1:th.18">FACIRC_1:18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">q</font>,<span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p5">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p0"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c7">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c7">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c7">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span> <a href="circtrm1.html#K7" title="CIRCTRM1:func.7">@</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c2">A</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c9">t</font> =  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c7">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="msaterm.html#NM3" title="MSATERM:NM.3">c-Term</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">V</font> <b>by </b><i><a class="ref" href="msaterm.html#T8" title="MSATERM:th.8">MSATERM:8</a></i>;<br/>
<a NAME="E5:39_1"/><i><font color="Green" title="E28">A5</font></i>: 
<font color="Maroon" title="c9">t</font> is   <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c3">V</font>
 
<b>by </b><i><a class="ref" href="msaterm.html#T4" title="MSATERM:th.4">MSATERM:4</a></i>;<br/>
<a NAME="E6:39_1"/>
<font color="Maroon" title="c5">q</font> <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c7">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="#E2:39_1"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="facirc_1.html#T18" title="FACIRC_1:th.18">FACIRC_1:18</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">q</font>,<span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p5">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p0"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c7">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c7">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span> = 
<font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c7">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="facirc_1.html#D9" title="FACIRC_1:def.9">FACIRC_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">v</font>
<b>by </b><i><a class="txt" href="#E1:39_1"><i><font color="Green" title="E26">A3</font></i></a>, <a class="ref" href="circtrm1.html#D8" target="_self" title="CIRCTRM1:def.8">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="msaterm.html#K5" title="MSATERM:func.5">-term</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="msaterm.html#K9" title="MSATERM:func.9">@</a> <font color="Maroon" title="c6">f</font>
<b>by </b><i><a class="ref" href="msaterm.html#T42" title="MSATERM:th.42">MSATERM:42</a></i>
<br/>.= 
<font color="Maroon" title="c9">t</font> <a href="msaterm.html#K9" title="MSATERM:func.9">@</a> <font color="Maroon" title="c6">f</font>
<b>by </b><i><a class="ref" href="msaterm.html#D4" title="MSATERM:def.4">MSATERM:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c7">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span> <a href="circtrm1.html#K7" title="CIRCTRM1:func.7">@</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c2">A</font>
<b>by </b><i><a class="txt" href="#E5:39_1"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="circtrm1.html#D7" target="_self" title="CIRCTRM1:def.7">Def7</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="E3:39"/><i><font color="Green" title="E26">A6</font></i>: 
for <font color="Olive" title="b1">o</font> being  <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">p</font> being   <a href="msaterm.html#M1" title="MSATERM:mode.1">ArgumentSeq</a> of  <a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <font color="Olive" title="b1">o</font>,<font color="Maroon" title="c3">V</font>  st ( for <font color="Olive" title="b3">t</font> being  <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c3">V</font>  st <font color="Olive" title="b3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b2">p</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b3">t</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Olive" title="b2">p</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">o</font> be   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</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">p</font> being   <a href="msaterm.html#M1" title="MSATERM:mode.1">ArgumentSeq</a> of  <a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <font color="Maroon" title="c7">o</font>,<font color="Maroon" title="c3">V</font>  st ( for <font color="Olive" title="b2">t</font> being  <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c3">V</font>  st <font color="Olive" title="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">p</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">t</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Olive" title="b1">p</font>]</span><br/><b>let </b><font color="Maroon" title="c8">p</font> be    <a href="msaterm.html#M1" title="MSATERM:mode.1">ArgumentSeq</a> of  <a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <font color="Maroon" title="c7">o</font>,<font color="Maroon" title="c3">V</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">t</font> being  <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c3">V</font>  st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">p</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">t</font>] ) implies <font color="Maroon">S<sub>1</sub></font>[<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c8">p</font>] )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:39_2"/><i><font color="Green" title="E27">A7</font></i>: 
for <font color="Olive" title="b1">t</font> being  <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c3">V</font>  st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">p</font> &amp; <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_9.html#K9" title="TREES_9:func.9">Subtrees</a> <font color="Maroon" title="c4">X</font> holds <br/>(  <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">q</font>,<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">t</font></span>)</span></span>)</span></span>)</span> <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> <font color="Olive" title="b1">t</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">q</font>,<span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">t</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> <a href="circtrm1.html#K7" title="CIRCTRM1:func.7">@</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c2">A</font> )
 <b>and </b><br/><a NAME="E2:39_2"/><i><font color="Green" title="E28">A8</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_9.html#K9" title="TREES_9:func.9">Subtrees</a> <font color="Maroon" title="c4">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">q</font>,<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p4">(<span class="default"><span class="p5"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c8">p</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">q</font>,<span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p5">(<span class="default"><span class="p0"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="circtrm1.html#K7" title="CIRCTRM1:func.7">@</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c2">A</font> )</span><br/>

<b>consider </b><font color="Maroon" title="c9">tt</font> being    <a href="trees_3.html#M3" title="TREES_3:mode.3">Element</a> of <font color="Maroon" title="c4">X</font>, <font color="Maroon" title="c10">n</font> being   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c9">tt</font><b> such that </b><br/><a NAME="E4:39_2"/><i><font color="Green" title="E29">A9</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">tt</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Maroon" title="c10">n</font>
 <b>by </b><i><a class="txt" href="#E2:39_2"><i><font color="Green" title="E28">A8</font></i></a>, <a class="ref" href="trees_9.html#T20" title="TREES_9:th.20">TREES_9:20</a></i>;<br/>
<a NAME="E5:39_2"/>
(  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">tt</font></span>)</span> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c10">n</font> &amp; <font color="Maroon" title="c10">n</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="c10">n</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a>, <a class="ref" href="trees_1.html#T47" title="TREES_1:th.47">TREES_1:47</a></i>;<br/>
<b>then </b><font color="Maroon" title="c9">tt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">tt</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="ref" href="trees_2.html#D11" title="TREES_2:def.11">TREES_2:def 11</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="#E4:39_2"><i><font color="Green" title="E29">A9</font></i></a>, <a class="ref" href="trees_4.html#D4" title="TREES_4:def.4">TREES_4:def 4</a></i>
;<br/>
<a NAME="E7:39_2"/><b>then </b>
<font color="Maroon" title="c9">tt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">S</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T129" title="ZFMISC_1:th.129">ZFMISC_1:129</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">g</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c8">p</font> as   <a href="circcomb.html#NM1" title="CIRCCOMB:NM.1">Gate</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="circtrm1.html#K1" title="CIRCTRM1:func.1">-CircuitStr</a> </span>)</span> <b>by </b><i><a class="txt" href="#E4:39_2"><i><font color="Green" title="E29">A9</font></i></a>, <a class="ref" href="trees_9.html#T25" title="TREES_9:th.25">TREES_9:25</a></i>;<br/>
<a NAME="E9:39_2"/><i><font color="Green" title="E30">A10</font></i>: 
 <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T34" title="FUNCT_1:th.34">FUNCT_1:34</a></i>;<br/>
<a NAME="E10:39_2"/><i><font color="Green" title="E31">A11</font></i>: 
<font color="Maroon" title="c11">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="ref" href="trees_4.html#D4" title="TREES_4:def.4">TREES_4:def 4</a></i>;<br/>
<a NAME="E11:39_2"/>
<font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c11">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="circtrm1.html#T13" target="_self" title="CIRCTRM1:th.13">Th13</a></i>;<br/>
<a NAME="E12:39_2"/><b>then </b><i><font color="Green" title="E32">A12</font></i>: 
 <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font>
 
<b>by </b><i><a class="ref" href="trees_4.html#T15" title="TREES_4:th.15">TREES_4:15</a></i>;<br/>
<a NAME="E13:39_2"/><i><font color="Green" title="E33">A13</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c11">g</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="trees_9.html#K9" title="TREES_9:func.9">Subtrees</a> <font color="Maroon" title="c4">X</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E14:39_2"/><i><font color="Green" title="E34">A14</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> <span class="p1">(<span class="default"><a href="trees_4.html#K11" title="TREES_4:func.11">doms</a> <font color="Maroon" title="c8">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="trees_4.html#T10" title="TREES_4:th.10">TREES_4:10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c11">g</font></span>)</span> implies  <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">q</font>,<span class="p1">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span></span>)</span> <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> <font color="Maroon" title="c12">x</font> )</span><br/><b>assume </b><a NAME="E1:39_2_2"/><i><font color="Green" title="E35">A15</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c11">g</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">q</font>,<span class="p1">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span></span>)</span> <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> <font color="Maroon" title="c12">x</font></span><br/><b>then reconsider </b><font color="Maroon" title="c13">t</font> = <font color="Maroon" title="c12">x</font> as    <a href="trees_3.html#M3" title="TREES_3:mode.3">Element</a> of  <a href="trees_9.html#K9" title="TREES_9:func.9">Subtrees</a> <font color="Maroon" title="c4">X</font> <b>by </b><i><a class="txt" href="#E13:39_2"><i><font color="Green" title="E33">A13</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">t</font> = <font color="Maroon" title="c13">t</font> as   <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c3">V</font> <b>by </b><i><a class="txt" href="#E1:39"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="circtrm1.html#T4" target="_self" title="CIRCTRM1:th.4">Th4</a></i>;<br/><a NAME="E4:39_2_2"/><i><font color="Green" title="E36">A16</font></i>: 
(  <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">q</font>,<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c14">t</font></span>)</span></span>)</span></span>)</span> <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> <font color="Maroon" title="c14">t</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">q</font>,<span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c14">t</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">t</font> <a href="circtrm1.html#K7" title="CIRCTRM1:func.7">@</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c2">A</font> )
 <b>by </b><i><a class="txt" href="#E1:39_2"><i><font color="Green" title="E27">A7</font></i></a>, <a class="txt" href="#E12:39_2"><i><font color="Green" title="E32">A12</font></i></a>, <a class="txt" href="#E1:39_2_2"><i><font color="Green" title="E35">A15</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c15">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:39_2_2"/><i><font color="Green" title="E37">A17</font></i>: 
( <font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">p</font> &amp; <font color="Maroon" title="c14">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">z</font> )
 <b>by </b><i><a class="txt" href="#E12:39_2"><i><font color="Green" title="E32">A12</font></i></a>, <a class="txt" href="#E1:39_2_2"><i><font color="Green" title="E35">A15</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E7:39_2_2"/>
( <font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="trees_4.html#K11" title="TREES_4:func.11">doms</a> <font color="Maroon" title="c8">p</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="trees_4.html#K11" title="TREES_4:func.11">doms</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c14">t</font> )
 <b>by </b><i><a class="txt" href="#E6:39_2_2"><i><font color="Green" title="E37">A17</font></i></a>, <a class="ref" href="funct_6.html#T31" title="FUNCT_6:th.31">FUNCT_6:31</a></i>;<br/><a NAME="E8:39_2_2"/><b>then </b>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c14">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="trees_4.html#K11" title="TREES_4:func.11">doms</a> <font color="Maroon" title="c8">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E9:39_2_2"/><b>then </b>
 <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c14">t</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p1">(<span class="default"><a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> <span class="p2">(<span class="default"><a href="trees_4.html#K11" title="TREES_4:func.11">doms</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="trees_3.html#T81" title="TREES_3:th.81">TREES_3:81</a></i>;<br/><a NAME="E10:39_2_2"/><b>then </b>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c14">t</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p1">(<span class="default"><a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> <span class="p2">(<span class="default"><a href="trees_4.html#K11" title="TREES_4:func.11">doms</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c16">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E12:39_2_2"/><i><font color="Green" title="E38">A18</font></i>: 
 <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p1">(<span class="default"><a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> <span class="p2">(<span class="default"><a href="trees_4.html#K11" title="TREES_4:func.11">doms</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c14">t</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">i</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c17">i</font> = <font color="Maroon" title="c16">i</font> as    <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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E14:39_2_2"/>
 <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">q</font>,<span class="p1">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">q</font>,<span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c14">t</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c17">i</font>
 <b>by </b><i><a class="txt" href="#E14:39_2"><i><font color="Green" title="E34">A14</font></i></a>, <a class="txt" href="#E12:39_2_2"><i><font color="Green" title="E38">A18</font></i></a>, <a class="ref" href="facirc_1.html#T13" title="FACIRC_1:th.13">FACIRC_1:13</a></i>;<br/><b>hence </b><a NAME="E15:39_2_2"/>
 <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">q</font>,<span class="p1">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span></span>)</span> <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> <font color="Maroon" title="c12">x</font>
 <b>by </b><i><a class="txt" href="#E4:39_2_2"><i><font color="Green" title="E36">A16</font></i></a>, <a class="ref" href="facirc_1.html#T17" title="FACIRC_1:th.17">FACIRC_1:17</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><b>end;</b></div>
<a NAME="E16:39_2"/><b>then </b>
 <a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">q</font>,<span class="p2">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p4">(<span class="default"><span class="p5"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c8">p</font>
 
<b>by </b><i><a class="txt" href="#E9:39_2"><i><font color="Green" title="E30">A10</font></i></a>, <a class="ref" href="facirc_1.html#T19" title="FACIRC_1:th.19">FACIRC_1:19</a></i>;<br/>
<b>hence </b><a NAME="E17:39_2"/>
 <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">q</font>,<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p4">(<span class="default"><span class="p5"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c8">p</font>
 <b>by </b><i><a class="ref" href="facirc_1.html#T12" title="FACIRC_1:th.12">FACIRC_1:12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">q</font>,<span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p5">(<span class="default"><span class="p0"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="circtrm1.html#K7" title="CIRCTRM1:func.7">@</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c2">A</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c12">t</font> = <span class="p1">(<span class="default"><a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <font color="Maroon" title="c7">o</font>,<font color="Maroon" title="c3">V</font></span>)</span> <a href="msaterm.html#K6" title="MSATERM:func.6">-tree</a> <font color="Maroon" title="c8">p</font> as   <a href="msaterm.html#NM3" title="MSATERM:NM.3">c-Term</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">V</font> <b>by </b><i><a class="ref" href="msaterm.html#T27" title="MSATERM:th.27">MSATERM:27</a></i>;<br/>
<a NAME="E19:39_2"/><i><font color="Green" title="E35">A19</font></i>: 
(  <a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <font color="Maroon" title="c7">o</font>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c3">V</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <font color="Maroon" title="c7">o</font>,<font color="Maroon" title="c3">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<b>by </b><i><a class="ref" href="msafree.html#D11" title="MSAFREE:def.11">MSAFREE:def 11</a></i>;<br/>
<a NAME="E20:39_2"/><i><font color="Green" title="E36">A20</font></i>: 
<font color="Maroon" title="c12">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c8">p</font>
 
<b>by </b><i><a class="ref" href="msafree.html#D11" title="MSAFREE:def.11">MSAFREE:def 11</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">q</font>,<span class="p2">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c13">vp</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E22:39_2"/><i><font color="Green" title="E37">A21</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">vp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">p</font>
 <b>and </b><br/><a NAME="E23:39_2"/><i><font color="Green" title="E38">A22</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">vp</font> holds <br/><font color="Maroon" title="c13">vp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>)
 <b>from </b><i><a class="ref" href="finseq_1.html#S2" title="FINSEQ_1:sch.2">FINSEQ_1:sch 2</a>();<br/></i>
<a NAME="E24:39_2"/><i><font color="Green" title="E39">A23</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">vp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">p</font></span>)</span> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">p</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E22:39_2"><i><font color="Green" title="E37">A21</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E25:39_2"/><i><font color="Green" title="E40">A24</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c7">o</font></span>)</span>
 
<b>by </b><i><a class="ref" href="msaterm.html#T22" title="MSATERM:th.22">MSATERM:22</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">p</font> implies  ex <font color="Olive" title="b1">t'</font> being  <a href="msaterm.html#NM3" title="MSATERM:NM.3">c-Term</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">V</font> st <br/>( <font color="Olive" title="b1">t'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> &amp;  <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Olive" title="b1">t'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c7">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> ) )</span><br/><b>assume </b><a NAME="E1:39_2_3"/><i><font color="Green" title="E41">A25</font></i>: 
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">p</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">t'</font> being  <a href="msaterm.html#NM3" title="MSATERM:NM.3">c-Term</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">V</font> st <br/>( <font color="Olive" title="b1">t'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> &amp;  <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Olive" title="b1">t'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c7">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> )</span><br/><b>then reconsider </b><font color="Maroon" title="c15">t</font> = <font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> as   <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c3">V</font> <b>by </b><i><a class="ref" href="msaterm.html#T22" title="MSATERM:th.22">MSATERM:22</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">t'</font> = <font color="Maroon" title="c15">t</font> as   <a href="msaterm.html#NM3" title="MSATERM:NM.3">c-Term</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">V</font> <b>by </b><i><a class="ref" href="msaterm.html#T27" title="MSATERM:th.27">MSATERM:27</a></i>;<br/><b>take </b><font color="Maroon" title="c17">t'</font> = <font color="Maroon" title="c16">t'</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c17">t'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> &amp;  <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c17">t'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c7">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> )</span><br/><a NAME="E4:39_2_3"/>
 <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c15">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c17">t'</font>
 <b>by </b><i><a class="ref" href="circtrm1.html#T1" target="_self" title="CIRCTRM1:th.1">Th1</a></i>;<br/><b>hence </b><a NAME="E5:39_2_3"/>
( <font color="Maroon" title="c17">t'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> &amp;  <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c17">t'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c7">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> )
 <b>by </b><i><a class="txt" href="#E1:39_2_3"><i><font color="Green" title="E41">A25</font></i></a>, <a class="ref" href="msaterm.html#T23" title="MSATERM:th.23">MSATERM:23</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><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c14">p'</font> = <font color="Maroon" title="c8">p</font> as   <a href="msaterm.html#NM4" title="MSATERM:NM.4">ArgumentSeq</a> of <font color="Maroon" title="c7">o</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">V</font> <b>by </b><i><a class="txt" href="#E25:39_2"><i><font color="Green" title="E40">A24</font></i></a>, <a class="ref" href="msaterm.html#T24" title="MSATERM:th.24">MSATERM:24</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">p</font> implies for <font color="Olive" title="b1">t</font> being  <a href="msaterm.html#NM3" title="MSATERM:NM.3">c-Term</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">V</font>  st <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> holds <br/><font color="Maroon" title="c13">vp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> <a href="msaterm.html#K9" title="MSATERM:func.9">@</a> <font color="Maroon" title="c6">f</font> )</span><br/><b>assume </b><a NAME="E1:39_2_4"/><i><font color="Green" title="E41">A26</font></i>: 
<font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">p</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">t</font> being  <a href="msaterm.html#NM3" title="MSATERM:NM.3">c-Term</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">V</font>  st <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> holds <br/><font color="Maroon" title="c13">vp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> <a href="msaterm.html#K9" title="MSATERM:func.9">@</a> <font color="Maroon" title="c6">f</font></span><br/><b>let </b><font color="Maroon" title="c16">t</font> be   <a href="msaterm.html#NM3" title="MSATERM:NM.3">c-Term</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">V</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#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> implies <font color="Maroon" title="c13">vp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">t</font> <a href="msaterm.html#K9" title="MSATERM:func.9">@</a> <font color="Maroon" title="c6">f</font> )</span><br/><b>assume </b><a NAME="E2:39_2_4"/><i><font color="Green" title="E42">A27</font></i>: 
<font color="Maroon" title="c16">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">vp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">t</font> <a href="msaterm.html#K9" title="MSATERM:func.9">@</a> <font color="Maroon" title="c6">f</font></span><br/><a NAME="E3:39_2_4"/><b>then </b><i><font color="Green" title="E43">A28</font></i>: 
<font color="Maroon" title="c16">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">p</font>
 <b>by </b><i><a class="txt" href="#E1:39_2_4"><i><font color="Green" title="E41">A26</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c17">tt</font> = <font color="Maroon" title="c16">t</font> as    <a href="trees_3.html#M3" title="TREES_3:mode.3">Element</a> of  <a href="trees_9.html#K9" title="TREES_9:func.9">Subtrees</a> <font color="Maroon" title="c4">X</font> <b>by </b><i><a class="txt" href="#E12:39_2"><i><font color="Green" title="E32">A12</font></i></a>, <a class="txt" href="#E13:39_2"><i><font color="Green" title="E33">A13</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c18">tt</font> = <font color="Maroon" title="c17">tt</font> as   <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c3">V</font> <b>by </b><i><a class="txt" href="#E1:39"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="circtrm1.html#T4" target="_self" title="CIRCTRM1:th.4">Th4</a></i>;<br/><a NAME="E6:39_2_4"/><i><font color="Green" title="E44">A29</font></i>: 
(  <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">q</font>,<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c16">t</font></span>)</span></span>)</span></span>)</span> <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> <font color="Maroon" title="c18">tt</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">q</font>,<span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c16">t</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">tt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">tt</font> <a href="circtrm1.html#K7" title="CIRCTRM1:func.7">@</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c2">A</font> )
 <b>by </b><i><a class="txt" href="#E1:39_2"><i><font color="Green" title="E27">A7</font></i></a>, <a class="txt" href="#E3:39_2_4"><i><font color="Green" title="E43">A28</font></i></a></i>;<br/><a NAME="E7:39_2_4"/><i><font color="Green" title="E45">A30</font></i>: 
<font color="Maroon" title="c13">vp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">q</font>,<span class="p2">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p4">(<span class="default"><span class="p5"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">t</font>
 <b>by </b><i><a class="txt" href="#E20:39_2"><i><font color="Green" title="E36">A20</font></i></a>, <a class="txt" href="#E23:39_2"><i><font color="Green" title="E38">A22</font></i></a>, <a class="txt" href="#E24:39_2"><i><font color="Green" title="E39">A23</font></i></a>, <a class="txt" href="#E1:39_2_4"><i><font color="Green" title="E41">A26</font></i></a>, <a class="txt" href="#E2:39_2_4"><i><font color="Green" title="E42">A27</font></i></a></i>;<br/><a NAME="E8:39_2_4"/>
( <font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="trees_4.html#K11" title="TREES_4:func.11">doms</a> <font color="Maroon" title="c8">p</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="trees_4.html#K11" title="TREES_4:func.11">doms</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c16">t</font> )
 <b>by </b><i><a class="txt" href="#E1:39_2_4"><i><font color="Green" title="E41">A26</font></i></a>, <a class="txt" href="#E2:39_2_4"><i><font color="Green" title="E42">A27</font></i></a>, <a class="ref" href="funct_6.html#T31" title="FUNCT_6:th.31">FUNCT_6:31</a></i>;<br/><a NAME="E9:39_2_4"/><b>then </b>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c16">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="trees_4.html#K11" title="TREES_4:func.11">doms</a> <font color="Maroon" title="c8">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E10:39_2_4"/><b>then </b>
 <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c16">t</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p1">(<span class="default"><a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> <span class="p2">(<span class="default"><a href="trees_4.html#K11" title="TREES_4:func.11">doms</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="trees_3.html#T81" title="TREES_3:th.81">TREES_3:81</a></i>;<br/><a NAME="E11:39_2_4"/><b>then </b>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c16">t</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p1">(<span class="default"><a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> <span class="p2">(<span class="default"><a href="trees_4.html#K11" title="TREES_4:func.11">doms</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c19">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E13:39_2_4"/><i><font color="Green" title="E46">A31</font></i>: 
 <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p1">(<span class="default"><a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> <span class="p2">(<span class="default"><a href="trees_4.html#K11" title="TREES_4:func.11">doms</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c16">t</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c19">j</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c20">j</font> = <font color="Maroon" title="c19">j</font> as    <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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E15:39_2_4"/>
 <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">q</font>,<span class="p1">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">q</font>,<span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c16">t</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c20">j</font>
 <b>by </b><i><a class="txt" href="#E14:39_2"><i><font color="Green" title="E34">A14</font></i></a>, <a class="txt" href="#E13:39_2_4"><i><font color="Green" title="E46">A31</font></i></a>, <a class="ref" href="facirc_1.html#T13" title="FACIRC_1:th.13">FACIRC_1:13</a></i>;<br/><a NAME="E16:39_2_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">q</font>,<span class="p2">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p4">(<span class="default"><span class="p5"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">q</font>,<span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c16">t</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">t</font>
 <b>by </b><i><a class="txt" href="#E6:39_2_4"><i><font color="Green" title="E44">A29</font></i></a>, <a class="ref" href="facirc_1.html#D9" title="FACIRC_1:def.9">FACIRC_1:def 9</a></i>;<br/><b>hence </b><a NAME="E17:39_2_4"/>
<font color="Maroon" title="c13">vp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">t</font> <a href="msaterm.html#K9" title="MSATERM:func.9">@</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="txt" href="#E6:39_2_4"><i><font color="Green" title="E44">A29</font></i></a>, <a class="txt" href="#E7:39_2_4"><i><font color="Green" title="E45">A30</font></i></a>, <a class="ref" href="circtrm1.html#D7" target="_self" title="CIRCTRM1:def.7">Def7</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><b>end;</b></div>
<a NAME="E29:39_2"/><b>then </b><i><font color="Green" title="E41">A32</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <font color="Maroon" title="c7">o</font>,<span class="p3">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c3">V</font></span>)</span></span>)</span> <a href="msaterm.html#K6" title="MSATERM:func.6">-tree</a> <font color="Maroon" title="c14">p'</font></span>)</span> <a href="msaterm.html#K9" title="MSATERM:func.9">@</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c7">o</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">vp</font>
 
<b>by </b><i><a class="txt" href="#E22:39_2"><i><font color="Green" title="E37">A21</font></i></a>, <a class="ref" href="msaterm.html#T43" title="MSATERM:th.43">MSATERM:43</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2_5"><b>now </b></a><div class="add"><a NAME="E1:39_2_5"/>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="circtrm1.html#K1" title="CIRCTRM1:func.1">-CircuitStr</a> </span>)</span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">q</font>,<span class="p2">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">t</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="circtrm1.html#K1" title="CIRCTRM1:func.1">-CircuitStr</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="#E12:39_2"><i><font color="Green" title="E32">A12</font></i></a>, <a class="ref" href="circuit1.html#T4" title="CIRCUIT1:th.4">CIRCUIT1:4</a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/><b>hence </b><a NAME="E2:39_2_5"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">q</font>,<span class="p3">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">p</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>; <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">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">p</font></span>)</span> holds <br/><font color="Maroon" title="c13">vp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">q</font>,<span class="p3">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font></span><br/><b>let </b><font color="Maroon" title="c15">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">p</font></span>)</span> implies <font color="Maroon" title="c13">vp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">q</font>,<span class="p3">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">z</font> )</span><br/><b>assume </b><a NAME="E3:39_2_5"/><i><font color="Green" title="E42">A33</font></i>: 
<font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">p</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="c13">vp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">q</font>,<span class="p3">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">z</font></span><br/><b>then reconsider </b><font color="Maroon" title="c16">i</font> = <font color="Maroon" title="c15">z</font> as    <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>  ;<br/><a NAME="E5:39_2_5"/>
<font color="Maroon" title="c13">vp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">q</font>,<span class="p2">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E23:39_2"><i><font color="Green" title="E38">A22</font></i></a>, <a class="txt" href="#E3:39_2_5"><i><font color="Green" title="E42">A33</font></i></a>, <a class="txt" href="#E24:39_2"><i><font color="Green" title="E39">A23</font></i></a></i>;<br/><b>hence </b><a NAME="E6:39_2_5"/>
<font color="Maroon" title="c13">vp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">q</font>,<span class="p3">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">z</font>
 <b>by </b><i><a class="txt" href="#E24:39_2"><i><font color="Green" title="E39">A23</font></i></a>, <a class="txt" href="#E3:39_2_5"><i><font color="Green" title="E42">A33</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</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><b>end;</b></div>
<a NAME="E31:39_2"/><b>then </b><i><font color="Green" title="E42">A34</font></i>: 
<font color="Maroon" title="c13">vp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">q</font>,<span class="p2">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c11">g</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E12:39_2"><i><font color="Green" title="E32">A12</font></i></a>, <a class="txt" href="#E24:39_2"><i><font color="Green" title="E39">A23</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<a NAME="E32:39_2"/>
 <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c11">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="circtrm1.html#K6" title="CIRCTRM1:func.6">-Circuit</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c7">o</font>,<font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="#E10:39_2"><i><font color="Green" title="E31">A11</font></i></a>, <a class="ref" href="circtrm1.html#T17" target="_self" title="CIRCTRM1:th.17">Th17</a></i>;<br/>
<a NAME="E33:39_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c7">o</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">vp</font> <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#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">q</font>,<span class="p3">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">t</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">t</font>
 
<b>by </b><i><a class="txt" href="#E9:39_2"><i><font color="Green" title="E30">A10</font></i></a>, <a class="txt" href="#E20:39_2"><i><font color="Green" title="E36">A20</font></i></a>, <a class="txt" href="#E31:39_2"><i><font color="Green" title="E42">A34</font></i></a>, <a class="ref" href="facirc_1.html#T10" title="FACIRC_1:th.10">FACIRC_1:10</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">q</font>,<span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p5">(<span class="default"><span class="p0"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c8">p</font></span>)</span> = 
<font color="Maroon" title="c12">t</font> <a href="msaterm.html#K9" title="MSATERM:func.9">@</a> <font color="Maroon" title="c6">f</font>
<b>by </b><i><a class="txt" href="#E19:39_2"><i><font color="Green" title="E35">A19</font></i></a>, <a class="txt" href="#E29:39_2"><i><font color="Green" title="E41">A32</font></i></a>, <a class="ref" href="facirc_1.html#T12" title="FACIRC_1:th.12">FACIRC_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="circtrm1.html#K7" title="CIRCTRM1:func.7">@</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c2">A</font>
<b>by </b><i><a class="txt" href="#E19:39_2"><i><font color="Green" title="E35">A19</font></i></a>, <a class="ref" href="circtrm1.html#D7" target="_self" title="CIRCTRM1:def.7">Def7</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>thus </b><a NAME="E4:39"/>
for <font color="Olive" title="b1">t</font> being  <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c3">V</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">t</font>]
 <b>from </b><i><a class="ref" href="msaterm.html#S1" title="MSATERM:sch.1">MSATERM:sch 1</a>(<a class="txt" href="#E2:39"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="#E3:39"><i><font color="Green" title="E26">A6</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>
