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<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="msaterm.html#M4" target="_self">Variables</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="msaterm.html#NM4">ArgumentSeq</a> of <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>











<a NAME="E1:61"/><i><font color="Green">E38</font></i>: 
 <a href="msaterm.html#K2" target="_self">Sym</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="ref" href="msafree.html#D11">MSAFREE:def 11</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>7</sub></font> be  <a href="trees_3.html#V6">DTree-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:61"/><i><font color="Green">E39</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E3:61"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="msaterm.html#NM3">c-Term</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>3</sub></font> being <a href="finset_1.html#V1">finite</a> <a href="trees_2.html#NM4">DecoratedTree</a> st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="msaterm.html#R1" target="_self">is_an_evaluation_of</a> <font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<a NAME="E4:61"/><i><font color="Green">E41</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:61_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <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">c<sub>7</sub></font></span>)</span>
 ;<br/><a NAME="E2:61_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="trees_3.html#T39">TREES_3:39</a></i>;<br/><a NAME="E3:61_1"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="msaterm.html#K1" target="_self">-Terms</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="msaterm.html#E2:61"><i><font color="Green">E39</font></i></a>, <a class="txt" href="msaterm.html#E4:61"><i><font color="Green">E41</font></i></a>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> as   <a href="msaterm.html#NM3">c-Term</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> ;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="msaterm.html#E1:61_1"><i><font color="Green">E42</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>11</sub></font> being  <a href="finset_1.html#V1">finite</a> <a href="trees_2.html#NM4">DecoratedTree</a><b> such that </b><br/><a NAME="E7:61_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="msaterm.html#R1" target="_self">is_an_evaluation_of</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="msaterm.html#E2:61"><i><font color="Green">E39</font></i></a>, <a class="txt" href="msaterm.html#E3:61"><i><font color="Green">E40</font></i></a>, <a class="txt" href="msaterm.html#E4:61"><i><font color="Green">E41</font></i></a>, <a class="txt" href="msaterm.html#E2:61_1"><i><font color="Green">E43</font></i></a></i>;<br/><a NAME="E8:61_1"/>
<span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="msaterm.html#E2:61_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="msaterm.html#E7:61_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>;<br/><b>hence </b><a NAME="E9:61_1"/>
<span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> is  <a href="finset_1.html#V1">finite</a> <a href="trees_1.html#NM1">Tree</a>
 ;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> =  <a href="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>7</sub></font> as  <a href="trees_3.html#V5">FinTree-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i><a class="ref" href="trees_3.html#T25">TREES_3:25</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="trees_3.html#K16">roots</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">c<sub>7</sub></font>;<br/>
<a NAME="E7:61"/><i><font color="Green">E42</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="trees_3.html#K16">roots</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="trees_3.html#K13">tree</a> <font color="Maroon">c<sub>8</sub></font> &amp;  <a href="trees_3.html#K13">tree</a> <span class="p1">(<span class="default"><a href="trees_4.html#K11">doms</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="trees_4.html#T10">TREES_4:10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="trees_3.html#K16">roots</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">c<sub>7</sub></font> as  <a href="finset_1.html#V1">finite</a> <a href="trees_2.html#NM4">DecoratedTree</a> <b>by </b><i><a class="ref" href="finset_1.html#T29">FINSET_1:29</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>10</sub></font>
;<br/>

<b>thus </b><a NAME="E9:61"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="trees_3.html#K16">roots</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<a NAME="E10:61"/><i><font color="Green">E43</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="trees_4.html#K11">doms</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="trees_4.html#K11">doms</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <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">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="ref" href="trees_3.html#T39">TREES_3:39</a>, <a class="ref" href="trees_3.html#T40">TREES_3:40</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:61_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:61_2"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="msaterm.html#K1" target="_self">-Terms</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> as   <a href="msaterm.html#NM3">c-Term</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> ;<br/><b>consider </b><font color="Maroon">c<sub>13</sub></font> being  <a href="finset_1.html#V1">finite</a> <a href="trees_2.html#NM4">DecoratedTree</a><b> such that </b><br/><a NAME="E5:61_2"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="msaterm.html#R1" target="_self">is_an_evaluation_of</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="msaterm.html#E3:61"><i><font color="Green">E40</font></i></a>, <a class="txt" href="msaterm.html#E1:61_2"><i><font color="Green">E44</font></i></a></i>;<br/><b>thus </b><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> = 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="msaterm.html#E2:61"><i><font color="Green">E39</font></i></a>, <a class="txt" href="msaterm.html#E4:61"><i><font color="Green">E41</font></i></a>, <a class="txt" href="msaterm.html#E1:61_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="msaterm.html#E5:61_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="msaterm.html#E5:61_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="msaterm.html#D9" target="_self">Def9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="trees_4.html#K11">doms</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="msaterm.html#E1:61_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:61"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msaterm.html#K2" target="_self">Sym</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p3">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="msaterm.html#K6" target="_self">-tree</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="msaterm.html#E1:61"><i><font color="Green">E38</font></i></a>, <a class="txt" href="msaterm.html#E2:61"><i><font color="Green">E39</font></i></a>, <a class="txt" href="msaterm.html#E4:61"><i><font color="Green">E41</font></i></a>, <a class="txt" href="msaterm.html#E7:61"><i><font color="Green">E42</font></i></a>, <a class="txt" href="msaterm.html#E10:61"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="msaterm.html#D9" target="_self">MSATERM:def 9</a><br/>

<b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">c<sub>10</sub></font>;<br/>

<a NAME="E13:61"/><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="ref" href="trees_4.html#D4">TREES_4:def 4</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>;<br/><b>assume </b><a NAME="E1:61_3"/><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msaterm.html#K2" target="_self">Sym</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p3">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="msaterm.html#K6" target="_self">-tree</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:61_3_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:61_3_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="msaterm.html#E1:61"><i><font color="Green">E38</font></i></a>, <a class="txt" href="msaterm.html#E13:61"><i><font color="Green">E44</font></i></a>, <a class="txt" href="msaterm.html#E1:61_3"><i><font color="Green">E45</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E3:61_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="msaterm.html#E24"><i><font color="Green">Lemma17</font></i></a></i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">c<sub>15</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:61_3"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="modal_1.html#T4">MODAL_1:4</a></i>;<br/><a NAME="E5:61_3"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="msaterm.html#E2:61"><i><font color="Green">E39</font></i></a>, <a class="txt" href="msaterm.html#E7:61"><i><font color="Green">E42</font></i></a>, <a class="txt" href="msaterm.html#E10:61"><i><font color="Green">E43</font></i></a>, <a class="txt" href="msaterm.html#E4:61_3"><i><font color="Green">E46</font></i></a>, <a class="ref" href="trees_3.html#T51">TREES_3:51</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="msaterm.html#NM3">c-Term</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="msaterm.html#E2"><i><font color="Green">Lemma2</font></i></a></i>;<br/><a NAME="E7:61_3"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="msaterm.html#E5:61_3"><i><font color="Green">E47</font></i></a>, <a class="txt" href="msaterm.html#E44"><i><font color="Green">Lemma30</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>17</sub></font> being  <a href="finset_1.html#V1">finite</a> <a href="trees_2.html#NM4">DecoratedTree</a><b> such that </b><br/><a NAME="E9:61_3"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="msaterm.html#R1" target="_self">is_an_evaluation_of</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="msaterm.html#E3:61"><i><font color="Green">E40</font></i></a></i>;<br/><a NAME="E10:61_3"/><i><font color="Green">E50</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>16</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="msaterm.html#E2:61"><i><font color="Green">E39</font></i></a>, <a class="txt" href="msaterm.html#E4:61"><i><font color="Green">E41</font></i></a>, <a class="txt" href="msaterm.html#E7:61_3"><i><font color="Green">E48</font></i></a>, <a class="txt" href="msaterm.html#E9:61_3"><i><font color="Green">E49</font></i></a>, <a class="ref" href="msaterm.html#D9" target="_self">Def9</a>, <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">c<sub>17</sub></font> <b>by </b><i><a class="txt" href="msaterm.html#E2:61"><i><font color="Green">E39</font></i></a>, <a class="txt" href="msaterm.html#E4:61"><i><font color="Green">E41</font></i></a>, <a class="txt" href="msaterm.html#E5:61_3"><i><font color="Green">E47</font></i></a>, <a class="txt" href="msaterm.html#E7:61_3"><i><font color="Green">E48</font></i></a>, <a class="txt" href="msaterm.html#E9:61_3"><i><font color="Green">E49</font></i></a>, <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>;<br/><a NAME="E12:61_3"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="msaterm.html#E1:61_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="msaterm.html#E4:61_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="msaterm.html#E5:61_3"><i><font color="Green">E47</font></i></a>, <a class="txt" href="msaterm.html#E10:61_3"><i><font color="Green">E50</font></i></a>, <a class="ref" href="trees_4.html#T12">TREES_4:12</a></i>;<br/><b>thus </b><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> = 
<font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="txt" href="msaterm.html#E2:61"><i><font color="Green">E39</font></i></a>, <a class="txt" href="msaterm.html#E4:61_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="msaterm.html#E5:61_3"><i><font color="Green">E47</font></i></a>, <a class="txt" href="msaterm.html#E9:61_3"><i><font color="Green">E49</font></i></a>, <a class="ref" href="trees_4.html#T12">TREES_4:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="msaterm.html#E9:61_3"><i><font color="Green">E49</font></i></a>, <a class="txt" href="msaterm.html#E12:61_3"><i><font color="Green">E51</font></i></a>, <a class="ref" href="msaterm.html#D9" target="_self">Def9</a></i>
;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>;<br/><b>assume </b><a NAME="E1:61_4"/><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msaterm.html#K2" target="_self">Sym</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p3">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="msaterm.html#K6" target="_self">-tree</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:61_4_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:61_4_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="msaterm.html#E1:61"><i><font color="Green">E38</font></i></a>, <a class="txt" href="msaterm.html#E13:61"><i><font color="Green">E44</font></i></a>, <a class="txt" href="msaterm.html#E1:61_4"><i><font color="Green">E45</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E3:61_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="msaterm.html#E24"><i><font color="Green">Lemma17</font></i></a></i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">c<sub>15</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:61_4"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="modal_1.html#T4">MODAL_1:4</a></i>;<br/><a NAME="E5:61_4"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="msaterm.html#E2:61"><i><font color="Green">E39</font></i></a>, <a class="txt" href="msaterm.html#E7:61"><i><font color="Green">E42</font></i></a>, <a class="txt" href="msaterm.html#E10:61"><i><font color="Green">E43</font></i></a>, <a class="txt" href="msaterm.html#E4:61_4"><i><font color="Green">E46</font></i></a>, <a class="ref" href="trees_3.html#T51">TREES_3:51</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="msaterm.html#NM3">c-Term</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="msaterm.html#E2"><i><font color="Green">Lemma2</font></i></a></i>;<br/><a NAME="E7:61_4"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="msaterm.html#E5:61_4"><i><font color="Green">E47</font></i></a>, <a class="txt" href="msaterm.html#E44"><i><font color="Green">Lemma30</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>17</sub></font> being  <a href="finset_1.html#V1">finite</a> <a href="trees_2.html#NM4">DecoratedTree</a><b> such that </b><br/><a NAME="E9:61_4"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="msaterm.html#R1" target="_self">is_an_evaluation_of</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="msaterm.html#E3:61"><i><font color="Green">E40</font></i></a></i>;<br/><a NAME="E10:61_4"/><i><font color="Green">E50</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>16</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="msaterm.html#E2:61"><i><font color="Green">E39</font></i></a>, <a class="txt" href="msaterm.html#E4:61"><i><font color="Green">E41</font></i></a>, <a class="txt" href="msaterm.html#E7:61_4"><i><font color="Green">E48</font></i></a>, <a class="txt" href="msaterm.html#E9:61_4"><i><font color="Green">E49</font></i></a>, <a class="ref" href="msaterm.html#D9" target="_self">Def9</a>, <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">c<sub>17</sub></font> <b>by </b><i><a class="txt" href="msaterm.html#E2:61"><i><font color="Green">E39</font></i></a>, <a class="txt" href="msaterm.html#E4:61"><i><font color="Green">E41</font></i></a>, <a class="txt" href="msaterm.html#E5:61_4"><i><font color="Green">E47</font></i></a>, <a class="txt" href="msaterm.html#E7:61_4"><i><font color="Green">E48</font></i></a>, <a class="txt" href="msaterm.html#E9:61_4"><i><font color="Green">E49</font></i></a>, <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>;<br/><a NAME="E12:61_4"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="msaterm.html#E1:61_4"><i><font color="Green">E45</font></i></a>, <a class="txt" href="msaterm.html#E4:61_4"><i><font color="Green">E46</font></i></a>, <a class="txt" href="msaterm.html#E5:61_4"><i><font color="Green">E47</font></i></a>, <a class="txt" href="msaterm.html#E10:61_4"><i><font color="Green">E50</font></i></a>, <a class="ref" href="trees_4.html#T12">TREES_4:12</a></i>;<br/><b>thus </b><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> = 
<font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="txt" href="msaterm.html#E2:61"><i><font color="Green">E39</font></i></a>, <a class="txt" href="msaterm.html#E4:61_4"><i><font color="Green">E46</font></i></a>, <a class="txt" href="msaterm.html#E5:61_4"><i><font color="Green">E47</font></i></a>, <a class="txt" href="msaterm.html#E9:61_4"><i><font color="Green">E49</font></i></a>, <a class="ref" href="trees_4.html#T12">TREES_4:12</a></i>
<br/>.= 
<font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="msaterm.html#E9:61_4"><i><font color="Green">E49</font></i></a>, <a class="txt" href="msaterm.html#E12:61_4"><i><font color="Green">E51</font></i></a>, <a class="ref" href="msaterm.html#D9" target="_self">Def9</a></i>
;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E16:61"/><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msaterm.html#K2" target="_self">Sym</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p3">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="msaterm.html#K6" target="_self">-tree</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_5"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_5"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_5_1"><b>suppose </b></a><a NAME="E1:61_5_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msaterm.html#K2" target="_self">Sym</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p3">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="msaterm.html#K6" target="_self">-tree</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> = 
 <a href="msaterm.html#K2" target="_self">Sym</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="trees_4.html#D4">TREES_4:def 4</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="msafree.html#D11">MSAFREE:def 11</a></i>
;<br/><a NAME="E3:61_5_1"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp;  <a href="trees_9.html#K2">succ</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="trees_3.html#K16">roots</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="msaterm.html#E16:61"><i><font color="Green">E45</font></i></a>, <a class="txt" href="msaterm.html#E1:61_5_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="trees_9.html#T31">TREES_9:31</a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E4:61_5_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="trees_9.html#K2">succ</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="msaterm.html#E1:61_5_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="trees_4.html#D4">TREES_4:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_5_2"><b>suppose </b></a><a NAME="E1:61_5_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">c<sub>14</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:61_5_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="modal_1.html#T4">MODAL_1:4</a></i>;<br/><a NAME="E4:61_5_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="msaterm.html#E2:61"><i><font color="Green">E39</font></i></a>, <a class="txt" href="msaterm.html#E7:61"><i><font color="Green">E42</font></i></a>, <a class="txt" href="msaterm.html#E10:61"><i><font color="Green">E43</font></i></a>, <a class="txt" href="msaterm.html#E3:61_5_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="trees_3.html#T51">TREES_3:51</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="msaterm.html#NM3">c-Term</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="msaterm.html#E2"><i><font color="Green">Lemma2</font></i></a></i>;<br/><a NAME="E6:61_5_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="msaterm.html#E4:61_5_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="msaterm.html#E44"><i><font color="Green">Lemma30</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>16</sub></font> being  <a href="finset_1.html#V1">finite</a> <a href="trees_2.html#NM4">DecoratedTree</a><b> such that </b><br/><a NAME="E8:61_5_2"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="msaterm.html#R1" target="_self">is_an_evaluation_of</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="msaterm.html#E3:61"><i><font color="Green">E40</font></i></a></i>;<br/><a NAME="E9:61_5_2"/><i><font color="Green">E50</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>15</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="msaterm.html#E2:61"><i><font color="Green">E39</font></i></a>, <a class="txt" href="msaterm.html#E4:61"><i><font color="Green">E41</font></i></a>, <a class="txt" href="msaterm.html#E6:61_5_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="msaterm.html#E8:61_5_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="msaterm.html#D9" target="_self">Def9</a>, <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">c<sub>16</sub></font> <b>by </b><i><a class="txt" href="msaterm.html#E2:61"><i><font color="Green">E39</font></i></a>, <a class="txt" href="msaterm.html#E4:61"><i><font color="Green">E41</font></i></a>, <a class="txt" href="msaterm.html#E4:61_5_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="msaterm.html#E6:61_5_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="msaterm.html#E8:61_5_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="lang1.html#K3">*&gt;</a></span> as   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">c<sub>10</sub></font> <b>by </b><i><a class="txt" href="msaterm.html#E3:61_5_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="trees_1.html#T46">TREES_1:46</a></i>;<br/><a NAME="E12:61_5_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="trees_2.html#K5">|</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="msaterm.html#E2:61"><i><font color="Green">E39</font></i></a>, <a class="txt" href="msaterm.html#E4:61_5_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="msaterm.html#E8:61_5_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="trees_4.html#D4">TREES_4:def 4</a></i>;<br/><a NAME="E13:61_5_2"/>
<font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="msaterm.html#E16:61"><i><font color="Green">E45</font></i></a>, <a class="txt" href="msaterm.html#E3:61_5_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="msaterm.html#E4:61_5_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="msaterm.html#E9:61_5_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="trees_4.html#T12">TREES_4:12</a></i>;<br/><b>then </b><font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> = 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="trees_9.html#K2">succ</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="msaterm.html#E8:61_5_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="msaterm.html#D9" target="_self">Def9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="trees_9.html#K2">succ</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="msaterm.html#E3:61_5_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="msaterm.html#E12:61_5_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="trees_9.html#T32">TREES_9:32</a></i>
;<br/><b>hence </b><a NAME="E15:61_5_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="trees_9.html#K2">succ</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="msaterm.html#E2:61"><i><font color="Green">E39</font></i></a>, <a class="txt" href="msaterm.html#E3:61_5_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="msaterm.html#E4:61_5_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="msaterm.html#E8:61_5_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="trees_4.html#T12">TREES_4:12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
