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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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">V</font> being   <a href="msaterm.html#M4" title="MSATERM:mode.4">Variables</a> of <font color="Olive" title="b1">A</font><br/> for <font color="Olive" title="b3">t</font> being  <a href="msaterm.html#NM3" title="MSATERM:NM.3">c-Term</a> of <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">V</font><br/> for <font color="Olive" title="b4">f</font> being   <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Olive" title="b2">V</font>,the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Olive" title="b1">A</font><br/> for <font color="Olive" title="b5">e1</font>, <font color="Olive" title="b6">e2</font> being  <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>  st <font color="Olive" title="b5">e1</font> <a href="msaterm.html#R1" title="MSATERM:pred.1">is_an_evaluation_of</a> <font color="Olive" title="b3">t</font>,<font color="Olive" title="b4">f</font> &amp; <font color="Olive" title="b6">e2</font> <a href="msaterm.html#R1" title="MSATERM:pred.1">is_an_evaluation_of</a> <font color="Olive" title="b3">t</font>,<font color="Olive" title="b4">f</font> holds <br/><font color="Olive" title="b5">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">e2</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="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">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="Olive" title="b1">V</font><br/> for <font color="Olive" title="b3">f</font> being   <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Olive" title="b1">V</font>,the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font><br/> for <font color="Olive" title="b4">e1</font>, <font color="Olive" title="b5">e2</font> being  <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>  st <font color="Olive" title="b4">e1</font> <a href="msaterm.html#R1" title="MSATERM:pred.1">is_an_evaluation_of</a> <font color="Olive" title="b2">t</font>,<font color="Olive" title="b3">f</font> &amp; <font color="Olive" title="b5">e2</font> <a href="msaterm.html#R1" title="MSATERM:pred.1">is_an_evaluation_of</a> <font color="Olive" title="b2">t</font>,<font color="Olive" title="b3">f</font> holds <br/><font color="Olive" title="b4">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">e2</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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><br/> for <font color="Olive" title="b2">f</font> being   <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Maroon" title="c3">V</font>,the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font><br/> for <font color="Olive" title="b3">e1</font>, <font color="Olive" title="b4">e2</font> being  <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>  st <font color="Olive" title="b3">e1</font> <a href="msaterm.html#R1" title="MSATERM:pred.1">is_an_evaluation_of</a> <font color="Olive" title="b1">t</font>,<font color="Olive" title="b2">f</font> &amp; <font color="Olive" title="b4">e2</font> <a href="msaterm.html#R1" title="MSATERM:pred.1">is_an_evaluation_of</a> <font color="Olive" title="b1">t</font>,<font color="Olive" title="b2">f</font> holds <br/><font color="Olive" title="b3">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">e2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">t</font> be   <a href="msaterm.html#NM3" title="MSATERM:NM.3">c-Term</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being   <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Maroon" title="c3">V</font>,the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font><br/> for <font color="Olive" title="b2">e1</font>, <font color="Olive" title="b3">e2</font> being  <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>  st <font color="Olive" title="b2">e1</font> <a href="msaterm.html#R1" title="MSATERM:pred.1">is_an_evaluation_of</a> <font color="Maroon" title="c4">t</font>,<font color="Olive" title="b1">f</font> &amp; <font color="Olive" title="b3">e2</font> <a href="msaterm.html#R1" title="MSATERM:pred.1">is_an_evaluation_of</a> <font color="Maroon" title="c4">t</font>,<font color="Olive" title="b1">f</font> holds <br/><font color="Olive" title="b2">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">e2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">f</font> be    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Maroon" title="c3">V</font>,the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">e1</font>, <font color="Olive" title="b2">e2</font> being  <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>  st <font color="Olive" title="b1">e1</font> <a href="msaterm.html#R1" title="MSATERM:pred.1">is_an_evaluation_of</a> <font color="Maroon" title="c4">t</font>,<font color="Maroon" title="c5">f</font> &amp; <font color="Olive" title="b2">e2</font> <a href="msaterm.html#R1" title="MSATERM:pred.1">is_an_evaluation_of</a> <font color="Maroon" title="c4">t</font>,<font color="Maroon" title="c5">f</font> holds <br/><font color="Olive" title="b1">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">e2</font></span><br/>









<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <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">means </span>for <font color="Olive" title="b1">e1</font>, <font color="Olive" title="b2">e2</font> being  <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>  st <font color="Olive" title="b1">e1</font> <a href="msaterm.html#R1" title="MSATERM:pred.1">is_an_evaluation_of</a> $1,<font color="Maroon" title="c5">f</font> &amp; <font color="Olive" title="b2">e2</font> <a href="msaterm.html#R1" title="MSATERM:pred.1">is_an_evaluation_of</a> $1,<font color="Maroon" title="c5">f</font> holds <br/><font color="Olive" title="b1">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">e2</font>;<br/>
<div><span class="lab"><font color="Green" title="E41">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">s</font> be   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="c6">s</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">v</font> <a href="msaterm.html#K5" title="MSATERM:func.5">-term</a> <font color="Maroon" title="c2">A</font>]</span><br/><span class="kw">let </span><font color="Maroon" title="c7">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="c6">s</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">v</font> <a href="msaterm.html#K5" title="MSATERM:func.5">-term</a> <font color="Maroon" title="c2">A</font>]</span><br/><span class="kw">thus </span><a NAME="E1:64_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">v</font> <a href="msaterm.html#K5" title="MSATERM:func.5">-term</a> <font color="Maroon" title="c2">A</font>]
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">e1</font>, <font color="Maroon" title="c9">e2</font> be   <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">e1</font> <a href="msaterm.html#R1" title="MSATERM:pred.1">is_an_evaluation_of</a> <font color="Maroon" title="c7">v</font> <a href="msaterm.html#K5" title="MSATERM:func.5">-term</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c5">f</font> &amp; <font color="Maroon" title="c9">e2</font> <a href="msaterm.html#R1" title="MSATERM:pred.1">is_an_evaluation_of</a> <font color="Maroon" title="c7">v</font> <a href="msaterm.html#K5" title="MSATERM:func.5">-term</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c5">f</font> implies <font color="Maroon" title="c8">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e2</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c10">t</font> = <font color="Maroon" title="c7">v</font> <a href="msaterm.html#K5" title="MSATERM:func.5">-term</a> <font color="Maroon" title="c2">A</font>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:64_1_1"/><span class="lab"><font color="Green" title="E42">A2</font></span>: 
<font color="Maroon" title="c8">e1</font> <a href="msaterm.html#R1" title="MSATERM:pred.1">is_an_evaluation_of</a> <font color="Maroon" title="c7">v</font> <a href="msaterm.html#K5" title="MSATERM:func.5">-term</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c5">f</font>
 <span class="kw">and </span><br/><a NAME="E2:64_1_1"/><span class="lab"><font color="Green" title="E43">A3</font></span>: 
<font color="Maroon" title="c9">e2</font> <a href="msaterm.html#R1" title="MSATERM:pred.1">is_an_evaluation_of</a> <font color="Maroon" title="c7">v</font> <a href="msaterm.html#K5" title="MSATERM:func.5">-term</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c5">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e2</font></span><br/>

<a NAME="E3:64_1_1"/><span class="lab"><font color="Green" title="E44">A4</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="msaterm.html#K5" title="MSATERM:func.5">-term</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_1"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="ref" href="msaterm.html#D9" target="_self" title="MSATERM:def.9">Def9</a></span>;<br/>
<a NAME="E4:64_1_1"/><span class="lab"><font color="Green" title="E45">A5</font></span>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  is   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c8">e1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_1.html#T47" title="TREES_1:th.47">TREES_1:47</a></span>;<br/>
<a NAME="E5:64_1_1"/><span class="lab"><font color="Green" title="E46">A6</font></span>: 
<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="c7">v</font>,<font color="Maroon" title="c6">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></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>  <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">v</font>,<font color="Maroon" title="c6">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#T3" title="TREES_4:th.3">TREES_4:3</a></span>;<br/>
<a NAME="E6:64_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">e1</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">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c6">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_1"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E4:64_1_1"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="ref" href="msaterm.html#D9" target="_self" title="MSATERM:def.9">Def9</a></span>;<br/>
<a NAME="E7:64_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A7</font></span>: 
<font color="Maroon" title="c8">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c6">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_1_1"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="ref" href="trees_4.html#T3" title="TREES_4:th.3">TREES_4:3</a>, <a class="ref" href="trees_4.html#T5" title="TREES_4:th.5">TREES_4:5</a></span>;<br/>
<a NAME="E8:64_1_1"/><span class="lab"><font color="Green" title="E48">A8</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="msaterm.html#K5" title="MSATERM:func.5">-term</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_1_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="ref" href="msaterm.html#D9" target="_self" title="MSATERM:def.9">Def9</a></span>;<br/>
<a NAME="E9:64_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">e2</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">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c6">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_1_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E3:64_1_1"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="txt" href="#E5:64_1_1"><span class="lab"><font color="Green" title="E46">A6</font></span></a>, <a class="txt" href="#E4:64_1_1"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="ref" href="msaterm.html#D9" target="_self" title="MSATERM:def.9">Def9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:64_1_1"/>
<font color="Maroon" title="c8">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:64_1_1"><span class="lab"><font color="Green" title="E48">A8</font></span></a>, <a class="txt" href="#E7:64_1_1"><span class="lab"><font color="Green" title="E47">A7</font></span></a>, <a class="ref" href="trees_4.html#T3" title="TREES_4:th.3">TREES_4:3</a>, <a class="ref" href="trees_4.html#T5" title="TREES_4:th.5">TREES_4:5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E42">A9</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">o</font> be   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">p</font> being  <a href="msaterm.html#NM4" title="MSATERM:NM.4">ArgumentSeq</a> of <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">V</font>  st ( 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#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">p</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">t</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<span class="p1">(<span class="default"><a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <font color="Maroon" title="c6">o</font>,<span class="p2">(<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="Olive" title="b1">p</font>]</span><br/><span class="kw">let </span><font color="Maroon" title="c7">p</font> be   <a href="msaterm.html#NM4" title="MSATERM:NM.4">ArgumentSeq</a> of <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">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">(<span class="default"><a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <font color="Maroon" title="c6">o</font>,<span class="p2">(<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="c7">p</font>] )</span><br/><span class="kw">assume </span><a NAME="E1:64_2"/><span class="lab"><font color="Green" title="E43">A10</font></span>: 
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#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">p</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">t</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<span class="p1">(<span class="default"><a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <font color="Maroon" title="c6">o</font>,<span class="p2">(<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="c7">p</font>]</span><br/><span class="kw">thus </span><a NAME="E2:64_2"/>
<font color="Maroon">S<sub>1</sub></font>[<span class="p1">(<span class="default"><a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <font color="Maroon" title="c6">o</font>,<span class="p2">(<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="c7">p</font>]
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c8">t</font> = <span class="p1">(<span class="default"><a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <font color="Maroon" title="c6">o</font>,<span class="p2">(<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="c7">p</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c9">e1</font>, <font color="Maroon" title="c10">e2</font> be   <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">e1</font> <a href="msaterm.html#R1" title="MSATERM:pred.1">is_an_evaluation_of</a> <span class="p1">(<span class="default"><a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <font color="Maroon" title="c6">o</font>,<span class="p2">(<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="c7">p</font>,<font color="Maroon" title="c5">f</font> &amp; <font color="Maroon" title="c10">e2</font> <a href="msaterm.html#R1" title="MSATERM:pred.1">is_an_evaluation_of</a> <span class="p1">(<span class="default"><a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <font color="Maroon" title="c6">o</font>,<span class="p2">(<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="c7">p</font>,<font color="Maroon" title="c5">f</font> implies <font color="Maroon" title="c9">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">e2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:64_2_1"/><span class="lab"><font color="Green" title="E44">A11</font></span>: 
<font color="Maroon" title="c9">e1</font> <a href="msaterm.html#R1" title="MSATERM:pred.1">is_an_evaluation_of</a> <span class="p1">(<span class="default"><a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <font color="Maroon" title="c6">o</font>,<span class="p2">(<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="c7">p</font>,<font color="Maroon" title="c5">f</font>
 <span class="kw">and </span><br/><a NAME="E2:64_2_1"/><span class="lab"><font color="Green" title="E45">A12</font></span>: 
<font color="Maroon" title="c10">e2</font> <a href="msaterm.html#R1" title="MSATERM:pred.1">is_an_evaluation_of</a> <span class="p1">(<span class="default"><a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> <font color="Maroon" title="c6">o</font>,<span class="p2">(<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="c7">p</font>,<font color="Maroon" title="c5">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">e2</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c11">q1</font> being   <a href="trees_3.html#V6" title="TREES_3:attr.6">DTree-yielding</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E4:64_2_1"/><span class="lab"><font color="Green" title="E46">A13</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">q1</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="c7">p</font>
 <span class="kw">and </span><br/><a NAME="E5:64_2_1"/><span class="lab"><font color="Green" title="E47">A14</font></span>: 
<font color="Maroon" title="c9">e1</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="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <font color="Maroon" title="c11">q1</font></span>)</span></span>)</span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c11">q1</font>
 <span class="kw">and </span><br/><a NAME="E6:64_2_1"/><span class="lab"><font color="Green" title="E48">A15</font></span>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><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="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">p</font> &amp; <font color="Olive" title="b2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> holds <br/> ex <font color="Olive" title="b3">vt</font> being  <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> st <br/>( <font color="Olive" title="b3">vt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">vt</font> <a href="msaterm.html#R1" title="MSATERM:pred.1">is_an_evaluation_of</a> <font color="Olive" title="b2">t</font>,<font color="Maroon" title="c5">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_2_1"><span class="lab"><font color="Green" title="E44">A11</font></span></a>, <a class="ref" href="msaterm.html#T35" target="_self" title="MSATERM:th.35">Th35</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">q2</font> being   <a href="trees_3.html#V6" title="TREES_3:attr.6">DTree-yielding</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E8:64_2_1"/><span class="lab"><font color="Green" title="E49">A16</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">q2</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="c7">p</font>
 <span class="kw">and </span><br/><a NAME="E9:64_2_1"/><span class="lab"><font color="Green" title="E50">A17</font></span>: 
<font color="Maroon" title="c10">e2</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="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c6">o</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <font color="Maroon" title="c12">q2</font></span>)</span></span>)</span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c12">q2</font>
 <span class="kw">and </span><br/><a NAME="E10:64_2_1"/><span class="lab"><font color="Green" title="E51">A18</font></span>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><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="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">p</font> &amp; <font color="Olive" title="b2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> holds <br/> ex <font color="Olive" title="b3">vt</font> being  <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> st <br/>( <font color="Olive" title="b3">vt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">vt</font> <a href="msaterm.html#R1" title="MSATERM:pred.1">is_an_evaluation_of</a> <font color="Olive" title="b2">t</font>,<font color="Maroon" title="c5">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_2_1"><span class="lab"><font color="Green" title="E45">A12</font></span></a>, <a class="ref" href="msaterm.html#T35" target="_self" title="MSATERM:th.35">Th35</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E52">A19</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c13">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p</font> implies <font color="Maroon" title="c11">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:64_2_1_1"/><span class="lab"><font color="Green" title="E53">A20</font></span>: 
<font color="Maroon" title="c13">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">t</font> = <font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</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="txt" href="msaterm.html#E2"><span class="lab"><font color="Green" title="E2">Lm2</font></span></a></span>;<br/><a NAME="E3:64_2_1_1"/><span class="lab"><font color="Green" title="E54">A21</font></span>: 
 ex <font color="Olive" title="b1">vt2</font> being  <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> st <br/>( <font color="Olive" title="b1">vt2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b1">vt2</font> <a href="msaterm.html#R1" title="MSATERM:pred.1">is_an_evaluation_of</a> <font color="Maroon" title="c14">t</font>,<font color="Maroon" title="c5">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:64_2_1"><span class="lab"><font color="Green" title="E51">A18</font></span></a>, <a class="txt" href="#E1:64_2_1_1"><span class="lab"><font color="Green" title="E53">A20</font></span></a>, <a class="txt" href="msaterm.html#E44"><span class="lab"><font color="Green" title="E29">Lm9</font></span></a></span>;<br/><a NAME="E4:64_2_1_1"/>
 ex <font color="Olive" title="b1">vt1</font> being  <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> st <br/>( <font color="Olive" title="b1">vt1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b1">vt1</font> <a href="msaterm.html#R1" title="MSATERM:pred.1">is_an_evaluation_of</a> <font color="Maroon" title="c14">t</font>,<font color="Maroon" title="c5">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:64_2_1"><span class="lab"><font color="Green" title="E48">A15</font></span></a>, <a class="txt" href="#E1:64_2_1_1"><span class="lab"><font color="Green" title="E53">A20</font></span></a>, <a class="txt" href="msaterm.html#E44"><span class="lab"><font color="Green" title="E29">Lm9</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:64_2_1_1"/>
<font color="Maroon" title="c11">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_2"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="txt" href="#E1:64_2_1_1"><span class="lab"><font color="Green" title="E53">A20</font></span></a>, <a class="txt" href="#E3:64_2_1_1"><span class="lab"><font color="Green" title="E54">A21</font></span></a>, <a class="txt" href="msaterm.html#E44"><span class="lab"><font color="Green" title="E29">Lm9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E53">A22</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c13">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">q1</font> implies <font color="Maroon" title="c11">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> )</span><br/><span class="kw">assume </span><a NAME="E1:64_2_1_2"/>
<font color="Maroon" title="c13">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">q1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font></span><br/><a NAME="E2:64_2_1_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">q1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="msaterm.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:64_2_1_2"/>
<font color="Maroon" title="c11">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_2_1"><span class="lab"><font color="Green" title="E46">A13</font></span></a>, <a class="txt" href="#E11:64_2_1"><span class="lab"><font color="Green" title="E52">A19</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E13:64_2_1"/><span class="lab"><font color="Green" title="E54">A23</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">q2</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="c12">q2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E14:64_2_1"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">q1</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="c11">q1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E15:64_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">q2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_2_1"><span class="lab"><font color="Green" title="E46">A13</font></span></a>, <a class="txt" href="#E8:64_2_1"><span class="lab"><font color="Green" title="E49">A16</font></span></a>, <a class="txt" href="#E13:64_2_1"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="txt" href="#E12:64_2_1"><span class="lab"><font color="Green" title="E53">A22</font></span></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:64_2_1"/>
<font color="Maroon" title="c9">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:64_2_1"><span class="lab"><font color="Green" title="E47">A14</font></span></a>, <a class="txt" href="#E9:64_2_1"><span class="lab"><font color="Green" title="E50">A17</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E43">A24</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="64_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">s</font> be   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font> <a href="msaterm.html#K4" title="MSATERM:func.4">-term</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">V</font>]</span><br/><span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">x</font> <a href="msaterm.html#K4" title="MSATERM:func.4">-term</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">V</font>]</span><br/><span class="kw">thus </span><a NAME="E1:64_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">x</font> <a href="msaterm.html#K4" title="MSATERM:func.4">-term</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">V</font>]
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">e1</font>, <font color="Maroon" title="c9">e2</font> be   <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">e1</font> <a href="msaterm.html#R1" title="MSATERM:pred.1">is_an_evaluation_of</a> <font color="Maroon" title="c7">x</font> <a href="msaterm.html#K4" title="MSATERM:func.4">-term</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c5">f</font> &amp; <font color="Maroon" title="c9">e2</font> <a href="msaterm.html#R1" title="MSATERM:pred.1">is_an_evaluation_of</a> <font color="Maroon" title="c7">x</font> <a href="msaterm.html#K4" title="MSATERM:func.4">-term</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c5">f</font> implies <font color="Maroon" title="c8">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e2</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c10">t</font> = <font color="Maroon" title="c7">x</font> <a href="msaterm.html#K4" title="MSATERM:func.4">-term</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">V</font>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:64_3_1"/><span class="lab"><font color="Green" title="E44">A25</font></span>: 
<font color="Maroon" title="c8">e1</font> <a href="msaterm.html#R1" title="MSATERM:pred.1">is_an_evaluation_of</a> <font color="Maroon" title="c7">x</font> <a href="msaterm.html#K4" title="MSATERM:func.4">-term</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c5">f</font>
 <span class="kw">and </span><br/><a NAME="E2:64_3_1"/><span class="lab"><font color="Green" title="E45">A26</font></span>: 
<font color="Maroon" title="c9">e2</font> <a href="msaterm.html#R1" title="MSATERM:pred.1">is_an_evaluation_of</a> <font color="Maroon" title="c7">x</font> <a href="msaterm.html#K4" title="MSATERM:func.4">-term</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c5">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e2</font></span><br/>

<a NAME="E3:64_3_1"/><span class="lab"><font color="Green" title="E46">A27</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="msaterm.html#K4" title="MSATERM:func.4">-term</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">V</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_3_1"><span class="lab"><font color="Green" title="E44">A25</font></span></a>, <a class="ref" href="msaterm.html#D9" target="_self" title="MSATERM:def.9">Def9</a></span>;<br/>
<a NAME="E4:64_3_1"/><span class="lab"><font color="Green" title="E47">A28</font></span>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  is   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c8">e1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_1.html#T47" title="TREES_1:th.47">TREES_1:47</a></span>;<br/>
<a NAME="E5:64_3_1"/><span class="lab"><font color="Green" title="E48">A29</font></span>: 
<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="c7">x</font>,<font color="Maroon" title="c6">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></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>  <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">x</font>,<font color="Maroon" title="c6">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#T3" title="TREES_4:th.3">TREES_4:3</a></span>;<br/>
<a NAME="E6:64_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">e1</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> <font color="Maroon" title="c7">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_3_1"><span class="lab"><font color="Green" title="E44">A25</font></span></a>, <a class="txt" href="#E4:64_3_1"><span class="lab"><font color="Green" title="E47">A28</font></span></a>, <a class="ref" href="msaterm.html#D9" target="_self" title="MSATERM:def.9">Def9</a></span>;<br/>
<a NAME="E7:64_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A30</font></span>: 
<font color="Maroon" title="c8">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c7">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_3_1"><span class="lab"><font color="Green" title="E46">A27</font></span></a>, <a class="ref" href="trees_4.html#T3" title="TREES_4:th.3">TREES_4:3</a>, <a class="ref" href="trees_4.html#T5" title="TREES_4:th.5">TREES_4:5</a></span>;<br/>
<a NAME="E8:64_3_1"/><span class="lab"><font color="Green" title="E50">A31</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="msaterm.html#K4" title="MSATERM:func.4">-term</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">V</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_3_1"><span class="lab"><font color="Green" title="E45">A26</font></span></a>, <a class="ref" href="msaterm.html#D9" target="_self" title="MSATERM:def.9">Def9</a></span>;<br/>
<a NAME="E9:64_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">e2</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> <font color="Maroon" title="c7">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_3_1"><span class="lab"><font color="Green" title="E45">A26</font></span></a>, <a class="txt" href="#E3:64_3_1"><span class="lab"><font color="Green" title="E46">A27</font></span></a>, <a class="txt" href="#E5:64_3_1"><span class="lab"><font color="Green" title="E48">A29</font></span></a>, <a class="txt" href="#E4:64_3_1"><span class="lab"><font color="Green" title="E47">A28</font></span></a>, <a class="ref" href="msaterm.html#D9" target="_self" title="MSATERM:def.9">Def9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:64_3_1"/>
<font color="Maroon" title="c8">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:64_3_1"><span class="lab"><font color="Green" title="E50">A31</font></span></a>, <a class="txt" href="#E7:64_3_1"><span class="lab"><font color="Green" title="E49">A30</font></span></a>, <a class="ref" href="trees_4.html#T3" title="TREES_4:th.3">TREES_4:3</a>, <a class="ref" href="trees_4.html#T5" title="TREES_4:th.5">TREES_4:5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E4:64"/>
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> 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#S3" target="_self" title="MSATERM:sch.3">MSATERM:sch 3</a>(<a class="txt" href="#E3:64"><span class="lab"><font color="Green" title="E43">A24</font></span></a>, <a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E2:64"><span class="lab"><font color="Green" title="E42">A9</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E5:64"/>
for <font color="Olive" title="b1">e1</font>, <font color="Olive" title="b2">e2</font> being  <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>  st <font color="Olive" title="b1">e1</font> <a href="msaterm.html#R1" title="MSATERM:pred.1">is_an_evaluation_of</a> <font color="Maroon" title="c4">t</font>,<font color="Maroon" title="c5">f</font> &amp; <font color="Olive" title="b2">e2</font> <a href="msaterm.html#R1" title="MSATERM:pred.1">is_an_evaluation_of</a> <font color="Maroon" title="c4">t</font>,<font color="Maroon" title="c5">f</font> holds <br/><font color="Olive" title="b1">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">e2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
