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<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">X</font> be  <a href="relat_1.html#V2">V2</a>  <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>, <font color="Maroon">vk</font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">o</font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">t</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>;<br/><b>let </b><font color="Maroon">a</font> be    <a href="msaterm.html#M1">ArgumentSeq</a> of  <a href="msaterm.html#K2">Sym</a> <font color="Maroon">o</font>,<font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">ak</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">vk</font>;<br/>















<b>assume </b><a NAME="E1:51"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">a</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</font> &amp; <font color="Maroon">ak</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> )
 ;<br/>

<b>consider </b><font color="Maroon">dt</font> being  <a href="finset_1.html#V1">finite</a> <a href="trees_2.html#NM4">DecoratedTree</a>, <font color="Maroon">tt</font> being  <a href="finset_1.html#V1">finite</a> <a href="trees_1.html#NM1">Tree</a><b> such that </b><br/><a NAME="E3:51"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">dt</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> &amp; <font color="Maroon">tt</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">dt</font> &amp;  <a href="msafree2.html#K9">depth</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="trees_1.html#K6">height</a> <font color="Maroon">tt</font> )
 <b>by </b><i><a class="ref" href="msafree2.html#D14">MSAFREE2:def 14</a></i>;<br/>
<b>consider </b><font color="Maroon">dtk</font> being  <a href="finset_1.html#V1">finite</a> <a href="trees_2.html#NM4">DecoratedTree</a>, <font color="Maroon">ttk</font> being  <a href="finset_1.html#V1">finite</a> <a href="trees_1.html#NM1">Tree</a><b> such that </b><br/><a NAME="E5:51"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">dtk</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ak</font> &amp; <font color="Maroon">ttk</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">dtk</font> &amp;  <a href="msafree2.html#K9">depth</a> <font color="Maroon">ak</font> <a href="hidden.html#R1">=</a>  <a href="trees_1.html#K6">height</a> <font color="Maroon">ttk</font> )
 <b>by </b><i><a class="ref" href="msafree2.html#D14">MSAFREE2:def 14</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> as  <a href="trees_3.html#V6">DTree-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a> ;<br/>
<b>consider </b><font color="Maroon">q</font> being  <a href="trees_3.html#V6">DTree-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E8:51"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="trees_3.html#K13">tree</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">q</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="trees_4.html#D4">TREES_4:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">da</font> =  <a href="trees_4.html#K11">doms</a> <font color="Maroon">a</font> as  <a href="trees_3.html#V5">FinTree-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a> ;<br/>
<a NAME="E10:51"/><i><font color="Green">E48</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">a'</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a'</font>
 
<b>by </b><i><a class="ref" href="trees_3.html#T39">TREES_3:39</a></i>;<br/>
<a NAME="E11:51"/>
<font color="Maroon">ttk</font> <a href="hidden.html#R1">=</a> <font color="Maroon">da</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 
<b>by </b><i>, , <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>;<br/>
<a NAME="E12:51"/><b>then </b>
<font color="Maroon">ttk</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">da</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E13:51"/>
 <a href="msafree2.html#K9">depth</a> <font color="Maroon">ak</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="msafree2.html#K9">depth</a> <font color="Maroon">t</font>
 <b>by </b><i>, , , <a class="ref" href="trees_3.html#T81">TREES_3:81</a></i>;<br/>


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