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<span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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">finite-yielding</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> over <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>  st (  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="Olive" title="b2">V</font><br/>  for <font color="Olive" title="b5">o</font> being   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font>  holds <br/>(  not <font color="Olive" title="b4">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="trees_9.html#K9" title="TREES_9:func.9">Subtrees</a> <font color="Olive" title="b3">X</font> or  not <font color="Olive" title="b4">t</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="Olive" title="b5">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> or  not  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Olive" title="b5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) ) holds <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="tarski.html#R2" title="TARSKI: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#K5" title="FACIRC_1:func.5">Following</a> (<font color="Olive" title="b4">s</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#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b6">t</font></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#K5" title="FACIRC_1:func.5">Following</a> (<font color="Olive" title="b4">s</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#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b6">t</font></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/><span class="kw">let </span><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">finite-yielding</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> over <font color="Maroon" title="c1">S</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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>  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="Olive" title="b1">V</font><br/>  for <font color="Olive" title="b4">o</font> being   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font>  holds <br/>(  not <font color="Olive" title="b3">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="trees_9.html#K9" title="TREES_9:func.9">Subtrees</a> <font color="Olive" title="b2">X</font> or  not <font color="Olive" title="b3">t</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="Olive" title="b4">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> or  not  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Olive" title="b4">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) ) holds <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="tarski.html#R2" title="TARSKI: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#K5" title="FACIRC_1:func.5">Following</a> (<font color="Olive" title="b3">s</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#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b5">t</font></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#K5" title="FACIRC_1:func.5">Following</a> (<font color="Olive" title="b3">s</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#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b5">t</font></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/><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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>  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><br/>  for <font color="Olive" title="b3">o</font> being   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font>  holds <br/>(  not <font color="Olive" title="b2">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="trees_9.html#K9" title="TREES_9:func.9">Subtrees</a> <font color="Olive" title="b1">X</font> or  not <font color="Olive" title="b2">t</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="Olive" title="b3">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> or  not  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Olive" title="b3">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) ) holds <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="tarski.html#R2" title="TARSKI: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#K5" title="FACIRC_1:func.5">Following</a> (<font color="Olive" title="b2">s</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#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b4">t</font></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#K5" title="FACIRC_1:func.5">Following</a> (<font color="Olive" title="b2">s</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#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b4">t</font></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/><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">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><br/>  for <font color="Olive" title="b2">o</font> being   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font>  holds <br/>(  not <font color="Olive" title="b1">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="trees_9.html#K9" title="TREES_9:func.9">Subtrees</a> <font color="Maroon" title="c4">X</font> or  not <font color="Olive" title="b1">t</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="Olive" title="b2">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> or  not  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Olive" title="b2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) ) implies  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="tarski.html#R2" title="TARSKI: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#K5" title="FACIRC_1:func.5">Following</a> (<font color="Olive" title="b1">s</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#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b3">t</font></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#K5" title="FACIRC_1:func.5">Following</a> (<font color="Olive" title="b1">s</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#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b3">t</font></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/>







<span class="kw">assume </span><a NAME="E1:40"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
 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><br/>  for <font color="Olive" title="b2">o</font> being   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font>  holds <br/>(  not <font color="Olive" title="b1">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="trees_9.html#K9" title="TREES_9:func.9">Subtrees</a> <font color="Maroon" title="c4">X</font> or  not <font color="Olive" title="b1">t</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="Olive" title="b2">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> or  not  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Olive" title="b2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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="tarski.html#R2" title="TARSKI: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#K5" title="FACIRC_1:func.5">Following</a> (<font color="Olive" title="b1">s</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#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b3">t</font></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#K5" title="FACIRC_1:func.5">Following</a> (<font color="Olive" title="b1">s</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#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b3">t</font></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/>

<span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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="tarski.html#R2" title="TARSKI: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#K5" title="FACIRC_1:func.5">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#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b2">t</font></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#K5" title="FACIRC_1:func.5">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#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b2">t</font></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/><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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="tarski.html#R2" title="TARSKI: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#K5" title="FACIRC_1:func.5">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#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b1">t</font></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#K5" title="FACIRC_1:func.5">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#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b1">t</font></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="E2:40"/><span class="lab"><font color="Green" title="E24">A2</font></span>: 
<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/>
<span class="kw">defpred </span><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>] <span class="kw">means </span>( $1 <a href="tarski.html#R2" title="TARSKI: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#K5" title="FACIRC_1:func.5">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#NK1" title="RELAT_1:NK.1">dom</a> $1</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#K5" title="FACIRC_1:func.5">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#NK1" title="RELAT_1:NK.1">dom</a> $1</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="E3:40"/><span class="lab"><font color="Green" title="E25">A3</font></span>: 
 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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="40_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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/><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">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/>



<span class="kw">assume </span><a NAME="E1:40_1"/><span class="lab"><font color="Green" title="E26">A4</font></span>: 
 <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="tarski.html#R2" title="TARSKI: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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="facirc_1.html#K5" title="FACIRC_1:func.5">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#NK1" title="RELAT_1:NK.1">dom</a> <span class="p3">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p4"><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>) <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#K5" title="FACIRC_1:func.5">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#NK1" title="RELAT_1:NK.1">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="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="E2:40_1"/><span class="kw">then </span>
 <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="tarski.html#R2" title="TARSKI: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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="circtrm1.html#T11" target="_self" onmouseover="rs('circtrm1/T11')" onmouseout="rh()">Th11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:40_1"/>
 <a href="facirc_1.html#K5" title="FACIRC_1:func.5">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#NK1" title="RELAT_1:NK.1">dom</a> <span class="p3">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p4"><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>) <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="facirc_1.html#T18" onmouseover="rs('facirc_1/T18')" onmouseout="rh()">FACIRC_1:18</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">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#NK1" title="RELAT_1:NK.1">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="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/>

<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="msaterm.html#T8" onmouseover="rs('msaterm/T8')" onmouseout="rh()">MSATERM:8</a></span>;<br/>
<a NAME="E5:40_1"/><span class="lab"><font color="Green" title="E27">A5</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msaterm.html#T4" onmouseover="rs('msaterm/T4')" onmouseout="rh()">MSATERM:4</a></span>;<br/>
<a NAME="E6:40_1"/>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</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>  <a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_4.html#T3" onmouseover="rs('trees_4/T3')" onmouseout="rh()">TREES_4:3</a></span>;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">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#NK1" title="RELAT_1:NK.1">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="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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="facirc_1.html#T11" onmouseover="rs('facirc_1/T11')" onmouseout="rh()">FACIRC_1:11</a>, <a class="ref" href="trees_1.html#T42" onmouseover="rs('trees_1/T42')" onmouseout="rh()">TREES_1:42</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="ref" href="circtrm1.html#D8" target="_self" onmouseover="rs('circtrm1/D8')" onmouseout="rh()">Def8</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="msaterm.html#T42" onmouseover="rs('msaterm/T42')" onmouseout="rh()">MSATERM:42</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="msaterm.html#D4" onmouseover="rs('msaterm/D4')" onmouseout="rh()">MSATERM:def 4</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40_1"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="ref" href="circtrm1.html#D7" target="_self" onmouseover="rs('circtrm1/D7')" onmouseout="rh()">Def7</a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:40"/><span class="lab"><font color="Green" title="E26">A6</font></span>: 
 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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="40_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">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/><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">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/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:40_2"/><span class="lab"><font color="Green" title="E27">A7</font></span>: 
 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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c8">p</font> &amp; <font color="Olive" title="b1">t</font> <a href="tarski.html#R2" title="TARSKI: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#K5" title="FACIRC_1:func.5">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#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b1">t</font></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#K5" title="FACIRC_1:func.5">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#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b1">t</font></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 class="kw">and </span><br/><a NAME="E2:40_2"/><span class="lab"><font color="Green" title="E28">A8</font></span>: 
<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="tarski.html#R2" title="TARSKI: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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="facirc_1.html#K5" title="FACIRC_1:func.5">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#NK1" title="RELAT_1:NK.1">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> <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#K5" title="FACIRC_1:func.5">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#NK1" title="RELAT_1:NK.1">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> <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/>

<span class="kw">consider </span><font color="Maroon" title="c9">tt</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">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><span class="kw"> such that </span><br/><a NAME="E4:40_2"/><span class="lab"><font color="Green" title="E29">A9</font></span>: 
<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="msaterm.html#K8" title="MSATERM:func.8">|</a> <font color="Maroon" title="c10">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_2"><span class="lab"><font color="Green" title="E28">A8</font></span></a>, <a class="ref" href="trees_9.html#T19" onmouseover="rs('trees_9/T19')" onmouseout="rh()">TREES_9:19</a></span>;<br/>
<a NAME="E5:40_2"/><span class="lab"><font color="Green" title="E30">A10</font></span>: 
 <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_1.html#T22" onmouseover="rs('trees_1/T22')" onmouseout="rh()">TREES_1:22</a></span>;<br/>
<a NAME="E6:40_2"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T34" onmouseover="rs('finseq_1/T34')" onmouseout="rh()">FINSEQ_1:34</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c9">tt</font> <a href="msaterm.html#K3" title="MSATERM:func.3">.</a> <font color="Maroon" title="c10">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">tt</font> <a href="msaterm.html#K8" title="MSATERM:func.8">|</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40_2"><span class="lab"><font color="Green" title="E30">A10</font></span></a>, <a class="ref" href="trees_2.html#D10" onmouseover="rs('trees_2/D10')" onmouseout="rh()">TREES_2:def 10</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_2"><span class="lab"><font color="Green" title="E29">A9</font></span></a>, <a class="ref" href="trees_4.html#D4" onmouseover="rs('trees_4/D4')" onmouseout="rh()">TREES_4:def 4</a></span>
;<br/>
<a NAME="E8:40_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">tt</font> <a href="msaterm.html#K3" title="MSATERM:func.3">.</a> <font color="Maroon" title="c10">n</font> <a href="tarski.html#R2" title="TARSKI: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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T106" onmouseover="rs('zfmisc_1/T106')" onmouseout="rh()">ZFMISC_1:106</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_2"><span class="lab"><font color="Green" title="E29">A9</font></span></a>, <a class="ref" href="trees_9.html#T24" onmouseover="rs('trees_9/T24')" onmouseout="rh()">TREES_9:24</a></span>;<br/>
<a NAME="E10:40_2"/><span class="lab"><font color="Green" title="E31">A11</font></span>: 
 <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>
 
;<br/>
<a NAME="E11:40_2"/><span class="lab"><font color="Green" title="E32">A12</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_4.html#D4" onmouseover="rs('trees_4/D4')" onmouseout="rh()">TREES_4:def 4</a></span>;<br/>
<a NAME="E12:40_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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="circtrm1.html#T12" target="_self" onmouseover="rs('circtrm1/T12')" onmouseout="rh()">Th12</a></span>;<br/>
<a NAME="E13:40_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A13</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_4.html#T15" onmouseover="rs('trees_4/T15')" onmouseout="rh()">TREES_4:15</a></span>;<br/>
<a NAME="E14:40_2"/><span class="lab"><font color="Green" title="E34">A14</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" onmouseover="rs('finseq_1/D4')" onmouseout="rh()">FINSEQ_1:def 4</a></span>;<br/>
<a NAME="E15:40_2"/><span class="lab"><font color="Green" title="E35">A15</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">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#K11" title="TREES_3:func.11">tree</a> <span class="p1">(<span class="default"><a href="trees_4.html#K7" title="TREES_4:func.7">doms</a> <font color="Maroon" title="c8">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_4.html#T10" onmouseover="rs('trees_4/T10')" onmouseout="rh()">TREES_4:10</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="40_2_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not  <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p2">(<span class="default"><span class="p3"><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_2_2"/>
 <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p2">(<span class="default"><span class="p3"><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:40_2_2"/><span class="kw">then </span>
<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>  <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="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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:40_2"><span class="lab"><font color="Green" title="E32">A12</font></span></a>, <a class="ref" href="trees_1.html#T43" onmouseover="rs('trees_1/T43')" onmouseout="rh()">TREES_1:43</a>, <a class="ref" href="trees_4.html#T5" onmouseover="rs('trees_4/T5')" onmouseout="rh()">TREES_4:5</a></span>;<br/><a NAME="E3:40_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_4.html#T17" onmouseover="rs('trees_4/T17')" onmouseout="rh()">TREES_4:17</a></span>;<br/><a NAME="E4:40_2_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ;<br/><a NAME="E5:40_2_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="msaterm.html#T22" onmouseover="rs('msaterm/T22')" onmouseout="rh()">MSATERM:22</a></span>;<br/><a NAME="E6:40_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A16</font></span>: 
 <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c7">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E7:40_2_2"/>
<font color="Maroon" title="c11">g</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="circtrm1.html#T4" target="_self" onmouseover="rs('circtrm1/T4')" onmouseout="rh()">Th4</a></span>;<br/><span class="kw">hence </span><a NAME="E8:40_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E2:40_2"><span class="lab"><font color="Green" title="E28">A8</font></span></a>, <a class="txt" href="#E11:40_2"><span class="lab"><font color="Green" title="E32">A12</font></span></a>, <a class="txt" href="#E6:40_2_2"><span class="lab"><font color="Green" title="E36">A16</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">h</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E18:40_2"/><span class="lab"><font color="Green" title="E36">A17</font></span>: 
 <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p2">(<span class="default"><span class="p3"><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">h</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T6" onmouseover="rs('nat_1/T6')" onmouseout="rh()">NAT_1:6</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">h</font> = <font color="Maroon" title="c12">h</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="40_2_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">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> holds <br/> <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c13">h</font>) <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> <font color="Olive" title="b1">x</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">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#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c13">h</font>) <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> <font color="Maroon" title="c14">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:40_2_3"/><span class="lab"><font color="Green" title="E37">A18</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c13">h</font>) <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> <font color="Maroon" title="c14">x</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">t</font> = <font color="Maroon" title="c14">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="trees_9.html#K9" title="TREES_9:func.9">Subtrees</a> <font color="Maroon" title="c4">X</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:40_2"><span class="lab"><font color="Green" title="E34">A14</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c16">t</font> = <font color="Maroon" title="c15">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="circtrm1.html#T4" target="_self" onmouseover="rs('circtrm1/T4')" onmouseout="rh()">Th4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c17">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E5:40_2_3"/><span class="lab"><font color="Green" title="E38">A19</font></span>: 
<font color="Maroon" title="c17">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>
 <span class="kw">and </span><br/><a NAME="E6:40_2_3"/><span class="lab"><font color="Green" title="E39">A20</font></span>: 
<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="c17">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:40_2"><span class="lab"><font color="Green" title="E33">A13</font></span></a>, <a class="txt" href="#E1:40_2_3"><span class="lab"><font color="Green" title="E37">A18</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/><a NAME="E7:40_2_3"/><span class="lab"><font color="Green" title="E40">A21</font></span>: 
<font color="Maroon" title="c17">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#K7" title="TREES_4:func.7">doms</a> <font color="Maroon" title="c8">p</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40_2_3"><span class="lab"><font color="Green" title="E38">A19</font></span></a>, <a class="txt" href="#E6:40_2_3"><span class="lab"><font color="Green" title="E39">A20</font></span></a>, <a class="ref" href="funct_6.html#T22" onmouseover="rs('funct_6/T22')" onmouseout="rh()">FUNCT_6:22</a></span>;<br/><a NAME="E8:40_2_3"/>
<span class="p1">(<span class="default"><a href="trees_4.html#K7" title="TREES_4:func.7">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="c17">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c16">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40_2_3"><span class="lab"><font color="Green" title="E38">A19</font></span></a>, <a class="txt" href="#E6:40_2_3"><span class="lab"><font color="Green" title="E39">A20</font></span></a>, <a class="ref" href="funct_6.html#T22" onmouseover="rs('funct_6/T22')" onmouseout="rh()">FUNCT_6:22</a></span>;<br/><a NAME="E9:40_2_3"/><span class="kw">then </span>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c16">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><a href="trees_4.html#K7" title="TREES_4:func.7">doms</a> <font color="Maroon" title="c8">p</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:40_2_3"><span class="lab"><font color="Green" title="E40">A21</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/><a NAME="E10:40_2_3"/><span class="kw">then </span>
 <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">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#K11" title="TREES_3:func.11">tree</a> <span class="p2">(<span class="default"><a href="trees_4.html#K7" title="TREES_4:func.7">doms</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_3.html#T78" onmouseover="rs('trees_3/T78')" onmouseout="rh()">TREES_3:78</a></span>;<br/><a NAME="E11:40_2_3"/><span class="kw">then </span>
 <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c16">t</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">h</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:40_2"><span class="lab"><font color="Green" title="E35">A15</font></span></a>, <a class="txt" href="#E18:40_2"><span class="lab"><font color="Green" title="E36">A17</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E13:40_2_3"/><span class="lab"><font color="Green" title="E41">A22</font></span>: 
<font color="Maroon" title="c13">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c16">t</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c18">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T10" onmouseover="rs('nat_1/T10')" onmouseout="rh()">NAT_1:10</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c19">i</font> = <font color="Maroon" title="c18">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/><a NAME="E15:40_2_3"/>
 <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c13">h</font>) <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a>  <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">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#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c16">t</font></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c19">i</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:40_2_3"><span class="lab"><font color="Green" title="E41">A22</font></span></a>, <a class="ref" href="facirc_1.html#T13" onmouseover="rs('facirc_1/T13')" onmouseout="rh()">FACIRC_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E16:40_2_3"/>
 <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c13">h</font>) <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> <font color="Maroon" title="c14">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_2"><span class="lab"><font color="Green" title="E27">A7</font></span></a>, <a class="txt" href="#E13:40_2"><span class="lab"><font color="Green" title="E33">A13</font></span></a>, <a class="txt" href="#E1:40_2_3"><span class="lab"><font color="Green" title="E37">A18</font></span></a>, <a class="ref" href="facirc_1.html#T17" onmouseover="rs('facirc_1/T17')" onmouseout="rh()">FACIRC_1:17</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E21:40_2"/><span class="kw">then </span>
 <a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c13">h</font>)</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:40_2"><span class="lab"><font color="Green" title="E31">A11</font></span></a>, <a class="ref" href="facirc_1.html#T19" onmouseover="rs('facirc_1/T19')" onmouseout="rh()">FACIRC_1:19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E22:40_2"/>
 <a href="facirc_1.html#K5" title="FACIRC_1:func.5">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#NK1" title="RELAT_1:NK.1">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> <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E18:40_2"><span class="lab"><font color="Green" title="E36">A17</font></span></a>, <a class="ref" href="facirc_1.html#T12" onmouseover="rs('facirc_1/T12')" onmouseout="rh()">FACIRC_1:12</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">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#NK1" title="RELAT_1:NK.1">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> <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/>

<span class="kw">reconsider </span><font color="Maroon" title="c14">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="msaterm.html#T27" onmouseover="rs('msaterm/T27')" onmouseout="rh()">MSATERM:27</a></span>;<br/>
<a NAME="E24:40_2"/><span class="lab"><font color="Green" title="E37">A23</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msafree.html#D9" onmouseover="rs('msafree/D9')" onmouseout="rh()">MSAFREE:def 9</a></span>;<br/>
<a NAME="E25:40_2"/><span class="lab"><font color="Green" title="E38">A24</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msafree.html#D9" onmouseover="rs('msafree/D9')" onmouseout="rh()">MSAFREE:def 9</a></span>;<br/>
<a NAME="E26:40_2"/><span class="lab"><font color="Green" title="E39">A25</font></span>: 
<font color="Maroon" title="c14">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msafree.html#D9" onmouseover="rs('msafree/D9')" onmouseout="rh()">MSAFREE:def 9</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ) <span class="kw">-&gt; </span>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  = <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c13">h</font>)</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/>
<span class="kw">consider </span><font color="Maroon" title="c15">vp</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E28:40_2"/><span class="lab"><font color="Green" title="E40">A26</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">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>
 <span class="kw">and </span><br/><a NAME="E29:40_2"/><span class="lab"><font color="Green" title="E41">A27</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">vp</font> holds <br/><font color="Maroon" title="c15">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>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_1.html#S2" onmouseover="rs('finseq_1/S2')" onmouseout="rh()">FINSEQ_1:sch 2</a>();<br/></span>
<a NAME="E30:40_2"/><span class="lab"><font color="Green" title="E42">A28</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:40_2"><span class="lab"><font color="Green" title="E40">A26</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E31:40_2"/><span class="lab"><font color="Green" title="E43">A29</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E32:40_2"/><span class="lab"><font color="Green" title="E44">A30</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msaterm.html#T22" onmouseover="rs('msaterm/T22')" onmouseout="rh()">MSATERM:22</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="40_2_4"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">p</font> holds <br/> ex <font color="Olive" title="b2">t9</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="b2">t9</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="Olive" title="b1">i</font> &amp;  <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Olive" title="b2">t9</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="Olive" title="b1">i</font> )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c16">i</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">i</font> <a href="tarski.html#R2" title="TARSKI: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">t9</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">t9</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="c16">i</font> &amp;  <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Olive" title="b1">t9</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="c16">i</font> ) )</span><br/><span class="kw">assume </span><a NAME="E1:40_2_4"/><span class="lab"><font color="Green" title="E45">A31</font></span>: 
<font color="Maroon" title="c16">i</font> <a href="tarski.html#R2" title="TARSKI: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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">t9</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">t9</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="c16">i</font> &amp;  <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Olive" title="b1">t9</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="c16">i</font> )</span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">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="c16">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="msaterm.html#T22" onmouseover="rs('msaterm/T22')" onmouseout="rh()">MSATERM:22</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c18">t9</font> = <font color="Maroon" title="c17">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="msaterm.html#T27" onmouseover="rs('msaterm/T27')" onmouseout="rh()">MSATERM:27</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c19">t9</font> = <font color="Maroon" title="c18">t9</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c19">t9</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="c16">i</font> &amp;  <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c19">t9</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="c16">i</font> )</span><br/><a NAME="E4:40_2_4"/>
 <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>  <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c19">t9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="circtrm1.html#T1" target="_self" onmouseover="rs('circtrm1/T1')" onmouseout="rh()">Th1</a></span>;<br/><span class="kw">hence </span><a NAME="E5:40_2_4"/>
( <font color="Maroon" title="c19">t9</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="c16">i</font> &amp;  <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c19">t9</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="c16">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_2_4"><span class="lab"><font color="Green" title="E45">A31</font></span></a>, <a class="ref" href="msaterm.html#T23" onmouseover="rs('msaterm/T23')" onmouseout="rh()">MSATERM:23</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">p9</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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:40_2"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="ref" href="msaterm.html#T24" onmouseover="rs('msaterm/T24')" onmouseout="rh()">MSATERM:24</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="40_2_5"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">p</font> holds <br/> for <font color="Olive" title="b2">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="b2">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="Olive" title="b1">i</font> holds <br/><font color="Maroon" title="c15">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="Olive" title="b2">t</font> <a href="msaterm.html#K9" title="MSATERM:func.9">@</a> <font color="Maroon" title="c6">f</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c17">i</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">i</font> <a href="tarski.html#R2" title="TARSKI: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="c17">i</font> holds <br/><font color="Maroon" title="c15">vp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">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/><span class="kw">assume </span><a NAME="E1:40_2_5"/><span class="lab"><font color="Green" title="E45">A32</font></span>: 
<font color="Maroon" title="c17">i</font> <a href="tarski.html#R2" title="TARSKI: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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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="c17">i</font> holds <br/><font color="Maroon" title="c15">vp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">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/><span class="kw">let </span><font color="Maroon" title="c18">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">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="c17">i</font> implies <font color="Maroon" title="c15">vp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">t</font> <a href="msaterm.html#K9" title="MSATERM:func.9">@</a> <font color="Maroon" title="c6">f</font> )</span><br/><span class="kw">assume </span><a NAME="E2:40_2_5"/><span class="lab"><font color="Green" title="E46">A33</font></span>: 
<font color="Maroon" title="c18">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="c17">i</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">vp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">t</font> <a href="msaterm.html#K9" title="MSATERM:func.9">@</a> <font color="Maroon" title="c6">f</font></span><br/><a NAME="E3:40_2_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A34</font></span>: 
<font color="Maroon" title="c18">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c8">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_2_5"><span class="lab"><font color="Green" title="E45">A32</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">tt</font> = <font color="Maroon" title="c18">t</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="trees_9.html#K9" title="TREES_9:func.9">Subtrees</a> <font color="Maroon" title="c4">X</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:40_2"><span class="lab"><font color="Green" title="E33">A13</font></span></a>, <a class="txt" href="#E14:40_2"><span class="lab"><font color="Green" title="E34">A14</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">tt</font> = <font color="Maroon" title="c19">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="circtrm1.html#T4" target="_self" onmouseover="rs('circtrm1/T4')" onmouseout="rh()">Th4</a></span>;<br/><a NAME="E6:40_2_5"/><span class="lab"><font color="Green" title="E48">A35</font></span>: 
 <a href="facirc_1.html#K5" title="FACIRC_1:func.5">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#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c18">t</font></span>)</span></span>)</span>) <a href="facirc_1.html#R1" title="FACIRC_1:pred.1">is_stable_at</a> <font color="Maroon" title="c20">tt</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_2"><span class="lab"><font color="Green" title="E27">A7</font></span></a>, <a class="txt" href="#E3:40_2_5"><span class="lab"><font color="Green" title="E47">A34</font></span></a></span>;<br/><a NAME="E7:40_2_5"/><span class="lab"><font color="Green" title="E49">A36</font></span>: 
<span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">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#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c18">t</font></span>)</span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">tt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_2"><span class="lab"><font color="Green" title="E27">A7</font></span></a>, <a class="txt" href="#E3:40_2_5"><span class="lab"><font color="Green" title="E47">A34</font></span></a></span>;<br/><a NAME="E8:40_2_5"/><span class="lab"><font color="Green" title="E50">A37</font></span>: 
<font color="Maroon" title="c15">vp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c13">h</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:40_2"><span class="lab"><font color="Green" title="E41">A27</font></span></a>, <a class="txt" href="#E30:40_2"><span class="lab"><font color="Green" title="E42">A28</font></span></a>, <a class="txt" href="#E31:40_2"><span class="lab"><font color="Green" title="E43">A29</font></span></a>, <a class="txt" href="#E1:40_2_5"><span class="lab"><font color="Green" title="E45">A32</font></span></a>, <a class="txt" href="#E2:40_2_5"><span class="lab"><font color="Green" title="E46">A33</font></span></a></span>;<br/><a NAME="E9:40_2_5"/><span class="lab"><font color="Green" title="E51">A38</font></span>: 
<font color="Maroon" title="c17">i</font> <a href="tarski.html#R2" title="TARSKI: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#K7" title="TREES_4:func.7">doms</a> <font color="Maroon" title="c8">p</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_2_5"><span class="lab"><font color="Green" title="E45">A32</font></span></a>, <a class="txt" href="#E2:40_2_5"><span class="lab"><font color="Green" title="E46">A33</font></span></a>, <a class="ref" href="funct_6.html#T22" onmouseover="rs('funct_6/T22')" onmouseout="rh()">FUNCT_6:22</a></span>;<br/><a NAME="E10:40_2_5"/>
<span class="p1">(<span class="default"><a href="trees_4.html#K7" title="TREES_4:func.7">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="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c18">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_2_5"><span class="lab"><font color="Green" title="E45">A32</font></span></a>, <a class="txt" href="#E2:40_2_5"><span class="lab"><font color="Green" title="E46">A33</font></span></a>, <a class="ref" href="funct_6.html#T22" onmouseover="rs('funct_6/T22')" onmouseout="rh()">FUNCT_6:22</a></span>;<br/><a NAME="E11:40_2_5"/><span class="kw">then </span>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c18">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><a href="trees_4.html#K7" title="TREES_4:func.7">doms</a> <font color="Maroon" title="c8">p</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:40_2_5"><span class="lab"><font color="Green" title="E51">A38</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/><a NAME="E12:40_2_5"/><span class="kw">then </span>
 <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c18">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#K11" title="TREES_3:func.11">tree</a> <span class="p2">(<span class="default"><a href="trees_4.html#K7" title="TREES_4:func.7">doms</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_3.html#T78" onmouseover="rs('trees_3/T78')" onmouseout="rh()">TREES_3:78</a></span>;<br/><a NAME="E13:40_2_5"/><span class="kw">then </span>
 <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c18">t</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">h</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:40_2"><span class="lab"><font color="Green" title="E35">A15</font></span></a>, <a class="txt" href="#E18:40_2"><span class="lab"><font color="Green" title="E36">A17</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c21">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E15:40_2_5"/><span class="lab"><font color="Green" title="E52">A39</font></span>: 
<font color="Maroon" title="c13">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c18">t</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c21">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T10" onmouseover="rs('nat_1/T10')" onmouseout="rh()">NAT_1:10</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c22">j</font> = <font color="Maroon" title="c21">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/><a NAME="E17:40_2_5"/>
 <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c13">h</font>) <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a>  <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">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#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c18">t</font></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c22">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:40_2_5"><span class="lab"><font color="Green" title="E52">A39</font></span></a>, <a class="ref" href="facirc_1.html#T13" onmouseover="rs('facirc_1/T13')" onmouseout="rh()">FACIRC_1:13</a></span>;<br/><a NAME="E18:40_2_5"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c13">h</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">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#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c18">t</font></span>)</span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:40_2_5"><span class="lab"><font color="Green" title="E48">A35</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E19:40_2_5"/>
<font color="Maroon" title="c15">vp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">t</font> <a href="msaterm.html#K9" title="MSATERM:func.9">@</a> <font color="Maroon" title="c6">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:40_2_5"><span class="lab"><font color="Green" title="E49">A36</font></span></a>, <a class="txt" href="#E8:40_2_5"><span class="lab"><font color="Green" title="E50">A37</font></span></a>, <a class="ref" href="circtrm1.html#D7" target="_self" onmouseover="rs('circtrm1/D7')" onmouseout="rh()">Def7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E36:40_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A40</font></span>: 
<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="c16">p9</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="c15">vp</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:40_2"><span class="lab"><font color="Green" title="E40">A26</font></span></a>, <a class="ref" href="msaterm.html#T43" onmouseover="rs('msaterm/T43')" onmouseout="rh()">MSATERM:43</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="40_2_6"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c13">h</font>)</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> &amp; (  for <font color="Olive" title="b1">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c15">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#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c13">h</font>)</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></a><div class="add"><a NAME="E1:40_2_6"/><span class="lab"><font color="Green" title="E46">A41</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:40_2"><span class="lab"><font color="Green" title="E33">A13</font></span></a>, <a class="ref" href="finseq_1.html#D4" onmouseover="rs('finseq_1/D4')" onmouseout="rh()">FINSEQ_1:def 4</a></span>;<br/><a NAME="E2:40_2_6"/>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c13">h</font>)</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="circuit1.html#T3" onmouseover="rs('circuit1/T3')" onmouseout="rh()">CIRCUIT1:3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:40_2_6"/>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c13">h</font>)</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_2_6"><span class="lab"><font color="Green" title="E46">A41</font></span></a>, <a class="ref" href="relat_1.html#T27" onmouseover="rs('relat_1/T27')" onmouseout="rh()">RELAT_1:27</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c15">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#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c13">h</font>)</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/><span class="kw">let </span><font color="Maroon" title="c17">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">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="c15">vp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">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#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c13">h</font>)</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="c17">z</font> )</span><br/><span class="kw">assume </span><a NAME="E4:40_2_6"/><span class="lab"><font color="Green" title="E47">A42</font></span>: 
<font color="Maroon" title="c17">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">vp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">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#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c13">h</font>)</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="c17">z</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">i</font> = <font color="Maroon" title="c17">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  ;<br/><a NAME="E6:40_2_6"/>
<font color="Maroon" title="c15">vp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c13">h</font>)</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="c18">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:40_2"><span class="lab"><font color="Green" title="E41">A27</font></span></a>, <a class="txt" href="#E30:40_2"><span class="lab"><font color="Green" title="E42">A28</font></span></a>, <a class="txt" href="#E4:40_2_6"><span class="lab"><font color="Green" title="E47">A42</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:40_2_6"/>
<font color="Maroon" title="c15">vp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">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#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c13">h</font>)</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="c17">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:40_2"><span class="lab"><font color="Green" title="E43">A29</font></span></a>, <a class="txt" href="#E4:40_2_6"><span class="lab"><font color="Green" title="E47">A42</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E38:40_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A43</font></span>: 
<font color="Maroon" title="c15">vp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c13">h</font>)</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:40_2"><span class="lab"><font color="Green" title="E33">A13</font></span></a>, <a class="txt" href="#E30:40_2"><span class="lab"><font color="Green" title="E42">A28</font></span></a>, <a class="txt" href="#E31:40_2"><span class="lab"><font color="Green" title="E43">A29</font></span></a>, <a class="ref" href="funct_1.html#T2" onmouseover="rs('funct_1/T2')" onmouseout="rh()">FUNCT_1:2</a></span>;<br/>
<a NAME="E39:40_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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:40_2"><span class="lab"><font color="Green" title="E32">A12</font></span></a>, <a class="ref" href="circtrm1.html#T16" target="_self" onmouseover="rs('circtrm1/T16')" onmouseout="rh()">Th16</a></span>;<br/>
<a NAME="E40:40_2"/><span class="kw">then </span>
<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="c15">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#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c13">h</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:40_2"><span class="lab"><font color="Green" title="E31">A11</font></span></a>, <a class="txt" href="#E26:40_2"><span class="lab"><font color="Green" title="E39">A25</font></span></a>, <a class="txt" href="#E38:40_2"><span class="lab"><font color="Green" title="E46">A43</font></span></a>, <a class="ref" href="facirc_1.html#T10" onmouseover="rs('facirc_1/T10')" onmouseout="rh()">FACIRC_1:10</a></span>;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">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#NK1" title="RELAT_1:NK.1">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> <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="c14">t</font> <a href="msaterm.html#K9" title="MSATERM:func.9">@</a> <font color="Maroon" title="c6">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:40_2"><span class="lab"><font color="Green" title="E36">A17</font></span></a>, <a class="txt" href="#E24:40_2"><span class="lab"><font color="Green" title="E37">A23</font></span></a>, <a class="txt" href="#E25:40_2"><span class="lab"><font color="Green" title="E38">A24</font></span></a>, <a class="txt" href="#E36:40_2"><span class="lab"><font color="Green" title="E45">A40</font></span></a>, <a class="ref" href="facirc_1.html#T12" onmouseover="rs('facirc_1/T12')" onmouseout="rh()">FACIRC_1:12</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:40_2"><span class="lab"><font color="Green" title="E38">A24</font></span></a>, <a class="ref" href="circtrm1.html#D7" target="_self" onmouseover="rs('circtrm1/D7')" onmouseout="rh()">Def7</a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E5:40"/>
 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>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="msaterm.html#S1" onmouseover="rs('msaterm/S1')" onmouseout="rh()">MSATERM:sch 1</a>(<a class="txt" href="#E3:40"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E4:40"><span class="lab"><font color="Green" title="E26">A6</font></span></a>);</span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
