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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="msafree2.html#NM1">Vertex</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K12">FreeGen</a> <font color="Olive">b<sub>1</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p1">(<span class="default"><a href="trees_4.html#K1">root-tree</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span> ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K12">FreeGen</a> <font color="Olive">b<sub>1</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p1">(<span class="default"><a href="trees_4.html#K1">root-tree</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,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></span>)</span> ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K6">id</a> <span class="p1">(<span class="default"><a href="msafree.html#K12">FreeGen</a> <font color="Olive">b<sub>1</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> ) );<br/>
<div><i><font color="Green">E3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:3_1_1_1"/><i><font color="Green">E4</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="msafree2.html#NM1">Vertex</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><a NAME="E3:3_1_1_1"/>
 <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="msafree2.html#T5">MSAFREE2:5</a></i>;<br/><a NAME="E4:3_1_1_1"/><b>then </b><i><font color="Green">E5</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>;<br/><a NAME="E5:3_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="circuit2.html#E1:3_1_1_1"><i><font color="Green">E4</font></i></a>, <a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>;<br/><a NAME="E6:3_1_1_1"/><b>then </b><i><font color="Green">E6</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>thus </b><a NAME="E7:3_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:3_1_1_1_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 </span><b>by </b><i><a class="txt" href="circuit2.html#E4:3_1_1_1"><i><font color="Green">E5</font></i></a>, <a class="txt" href="circuit2.html#E6:3_1_1_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:3_1_1_1_1_1_1"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><a href="msafree.html#K12">FreeGen</a> <font color="Maroon">c<sub>5</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p1">(<span class="default"><a href="trees_4.html#K1">root-tree</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span> as    <a href="hidden.html#M1">set</a>  ;<br/><b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/><b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/><b>thus </b><a NAME="E3:3_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>thus </b><a NAME="E4:3_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K12">FreeGen</a> <font color="Maroon">c<sub>5</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p1">(<span class="default"><a href="trees_4.html#K1">root-tree</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span> )
 ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:3_1_1_1_1_1_1_1"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:3_1_1_1_1_1_1_1"/>
 <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="msafree2.html#T6">MSAFREE2:6</a></i>;<br/><b>hence </b><a NAME="E3:3_1_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K12">FreeGen</a> <font color="Maroon">c<sub>5</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p1">(<span class="default"><a href="trees_4.html#K1">root-tree</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,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></span>)</span>
 <b>by </b><i><a class="txt" href="circuit2.html#E1:3_1_1_1_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="circuit2.html#E1:3_1_1_1_1_1_1_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E6:3_1_1_1_1_1_1"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><a NAME="E7:3_1_1_1_1_1_1"/><i><font color="Green">E9</font></i>: 
<span class="p1">(<span class="default"><a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/><a NAME="E8:3_1_1_1_1_1_1"/>
 <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/><b>hence </b><a NAME="E9:3_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K6">id</a> <span class="p1">(<span class="default"><a href="msafree.html#K12">FreeGen</a> <font color="Maroon">c<sub>5</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="circuit2.html#E1:3_1_1_1_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="circuit2.html#E6:3_1_1_1_1_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="circuit2.html#E7:3_1_1_1_1_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:3_1_1_1_1_1_2"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><a href="msafree.html#K12">FreeGen</a> <font color="Maroon">c<sub>5</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p1">(<span class="default"><a href="trees_4.html#K1">root-tree</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,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></span>)</span> as    <a href="hidden.html#M1">set</a>  ;<br/><b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/><b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/><b>thus </b><a NAME="E3:3_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:3_1_1_1_1_1_2_1"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:3_1_1_1_1_1_2_1"/>
 <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="msafree2.html#T6">MSAFREE2:6</a></i>;<br/><b>hence </b><a NAME="E3:3_1_1_1_1_1_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K12">FreeGen</a> <font color="Maroon">c<sub>5</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p1">(<span class="default"><a href="trees_4.html#K1">root-tree</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="circuit2.html#E1:3_1_1_1_1_1_2"><i><font color="Green">E7</font></i></a>, <a class="txt" href="circuit2.html#E1:3_1_1_1_1_1_2_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div>
<b>end;</b></div><b>thus </b><a NAME="E5:3_1_1_1_1_1_2"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K12">FreeGen</a> <font color="Maroon">c<sub>5</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p1">(<span class="default"><a href="trees_4.html#K1">root-tree</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,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></span>)</span> )
 ;<br/><b>assume </b><a NAME="E6:3_1_1_1_1_1_2"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><a NAME="E7:3_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/><b>hence </b><a NAME="E8:3_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K6">id</a> <span class="p1">(<span class="default"><a href="msafree.html#K12">FreeGen</a> <font color="Maroon">c<sub>5</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="circuit2.html#E1:3_1_1_1_1_1_2"><i><font color="Green">E7</font></i></a>, <a class="txt" href="circuit2.html#E6:3_1_1_1_1_1_2"><i><font color="Green">E8</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:3_1_1_1_1_1_3"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="relat_1.html#K6">id</a> <span class="p1">(<span class="default"><a href="msafree.html#K12">FreeGen</a> <font color="Maroon">c<sub>5</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> as    <a href="hidden.html#M1">set</a>  ;<br/><b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/><b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/><b>thus </b><a NAME="E3:3_1_1_1_1_1_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:3_1_1_1_1_1_3_1"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:3_1_1_1_1_1_3_1"/><i><font color="Green">E9</font></i>: 
<span class="p1">(<span class="default"><a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/><a NAME="E3:3_1_1_1_1_1_3_1"/>
 <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/><b>hence </b><a NAME="E4:3_1_1_1_1_1_3_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K12">FreeGen</a> <font color="Maroon">c<sub>5</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p1">(<span class="default"><a href="trees_4.html#K1">root-tree</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="circuit2.html#E1:3_1_1_1_1_1_3"><i><font color="Green">E7</font></i></a>, <a class="txt" href="circuit2.html#E1:3_1_1_1_1_1_3_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="circuit2.html#E2:3_1_1_1_1_1_3_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div>
<b>end;</b></div><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:3_1_1_1_1_1_3_2"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:3_1_1_1_1_1_3_2"/>
<span class="p1">(<span class="default"><a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/><b>hence </b><a NAME="E3:3_1_1_1_1_1_3_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K12">FreeGen</a> <font color="Maroon">c<sub>5</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p1">(<span class="default"><a href="trees_4.html#K1">root-tree</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,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></span>)</span>
 <b>by </b><i><a class="txt" href="circuit2.html#E1:3_1_1_1_1_1_3"><i><font color="Green">E7</font></i></a>, <a class="txt" href="circuit2.html#E1:3_1_1_1_1_1_3_2"><i><font color="Green">E8</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E6:3_1_1_1_1_1_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><b>thus </b><a NAME="E7:3_1_1_1_1_1_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K6">id</a> <span class="p1">(<span class="default"><a href="msafree.html#K12">FreeGen</a> <font color="Maroon">c<sub>5</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:3_1_1"/><i><font color="Green">E4</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="pboole.html#S3">PBOOLE:sch 3</a>(<a class="txt" href="circuit2.html#E1:3_1_1"><i><font color="Green">E3</font></i></a>);<br/></i>
<div><i><font color="Green">E5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:3_1_1_2"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="msafree2.html#NM1">Vertex</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><a NAME="E3:3_1_1_2"/>
 <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="msafree2.html#T5">MSAFREE2:5</a></i>;<br/><a NAME="E4:3_1_1_2"/><b>then </b><i><font color="Green">E7</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>;<br/><a NAME="E5:3_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="circuit2.html#E1:3_1_1_2"><i><font color="Green">E6</font></i></a>, <a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>;<br/><a NAME="E6:3_1_1_2"/><b>then </b><i><font color="Green">E8</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E7:3_1_1_2"/><i><font color="Green">E9</font></i>: 
 <a href="msafree.html#K12">FreeGen</a> <font color="Maroon">c<sub>6</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K13">FreeGen</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="msafree.html#D18">MSAFREE:def 18</a></i>;<br/><i><font color="Green">E10</font></i>:  <a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">c<sub>2</sub></font> = 
 <a href="msafree.html#K11">FreeMSA</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font>

<br/>.= 
 <a href="msualg_1.html#G3">MSAlgebra</a>(# <span class="p1">(<span class="default"><a href="msafree.html#K8">FreeSort</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msafree.html#K10">FreeOper</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> #)
<b>by </b><i><a class="ref" href="msafree.html#D16">MSAFREE:def 16</a></i>
;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_2_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:3_1_1_2_2"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 </span><b>by </b><i><a class="txt" href="circuit2.html#E4:3_1_1_2"><i><font color="Green">E7</font></i></a>, <a class="txt" href="circuit2.html#E6:3_1_1_2"><i><font color="Green">E8</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_2_2_1"><b>suppose </b></a><a NAME="E1:3_1_1_2_2_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:3_1_1_2_2_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> 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>6</sub></font>
 <b>by </b><i><a class="ref" href="msafree2.html#D5">MSAFREE2:def 5</a></i>;<br/><a NAME="E3:3_1_1_2_2_1"/><b>then </b><i><font color="Green">E12</font></i>: 
 <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="msafree.html#K12">FreeGen</a> <font color="Maroon">c<sub>6</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="msafree.html#D17">MSAFREE:def 17</a></i>;<br/><a NAME="E4:3_1_1_2_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>]
 <b>by </b><i><a class="txt" href="circuit2.html#E3:3_1_1"><i><font color="Green">E4</font></i></a></i>;<br/><b>hence </b><a NAME="E5:3_1_1_2_2_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><a href="msafree.html#K13">FreeGen</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="circuit2.html#E7:3_1_1_2"><i><font color="Green">E9</font></i></a>, <a class="txt" href="circuit2.html#E8:3_1_1_2"><i><font color="Green">E10</font></i></a>, <a class="txt" href="circuit2.html#E1:3_1_1_2_2_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="circuit2.html#E3:3_1_1_2_2_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="funcop_1.html#T57">FUNCOP_1:57</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_2_2_2"><b>suppose </b></a><a NAME="E1:3_1_1_2_2_2"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:3_1_1_2_2_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="msualg_2.html#V1">with_const_op</a> </span>}</span> 
 <b>by </b><i><a class="ref" href="msafree2.html#D1">MSAFREE2:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:3_1_1_2_2_2"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E5:3_1_1_2_2_2"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is <a href="msualg_2.html#V1">with_const_op</a>
 ;<br/><b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:3_1_1_2_2_2"/><i><font color="Green">E14</font></i>: 
the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E8:3_1_1_2_2_2"/><i><font color="Green">E15</font></i>: 
the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="circuit2.html#E4:3_1_1_2_2_2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="circuit2.html#E5:3_1_1_2_2_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="msualg_2.html#D2">MSUALG_2:def 2</a></i>;<br/><b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,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/><a NAME="E9:3_1_1_2_2_2"/><i><font color="Green">E16</font></i>: 
 <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="msafree2.html#T5">MSAFREE2:5</a></i>;<br/><a NAME="E10:3_1_1_2_2_2"/>
 <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="circuit2.html#E8:3_1_1_2_2_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="msualg_1.html#D7">MSUALG_1:def 7</a></i>;<br/><a NAME="E11:3_1_1_2_2_2"/><b>then </b><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="circuit2.html#E1:3_1_1_2_2_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="circuit2.html#E9:3_1_1_2_2_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="msafree2.html#D7">MSAFREE2:def 7</a></i>;<br/><a NAME="E12:3_1_1_2_2_2"/><i><font color="Green">E18</font></i>: 
 <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,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="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,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> <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="trees_4.html#T20">TREES_4:20</a></i>;<br/><a NAME="E13:3_1_1_2_2_2"/><i><font color="Green">E19</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="lang1.html#K2">&lt;*&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="msaterm.html#K1">-Terms</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> =  <a href="xboole_0.html#K1">{}</a>  as    <a href="trees_4.html#M1">FinSequence</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="ref" href="finseq_1.html#T29">FINSEQ_1:29</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> =  <a href="msaterm.html#K2">Sym</a> <span class="p1">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> as   <a href="dtconstr.html#NM3">NonTerminal</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/><a NAME="E16:3_1_1_2_2_2"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="circuit2.html#E7:3_1_1_2_2_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="msualg_1.html#D6">MSUALG_1:def 6</a></i>;<br/><a NAME="E17:3_1_1_2_2_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> 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>10</sub></font> holds <br/><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree.html#K7">FreeSort</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T26">FINSEQ_1:26</a></i>;<br/><a NAME="E18:3_1_1_2_2_2"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msafree.html#K8">FreeSort</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="circuit2.html#E16:3_1_1_2_2_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="msafree.html#T9">MSAFREE:9</a></i>;<br/><a NAME="E19:3_1_1_2_2_2"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K1">roots</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="circuit2.html#E11:3_1_1_2_2_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="msafree.html#T10">MSAFREE:10</a></i>;<br/><a NAME="E20:3_1_1_2_2_2"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> is    <a href="dtconstr.html#M2">SubtreeSeq</a> of  <a href="msaterm.html#K2">Sym</a> <span class="p1">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="dtconstr.html#D9">DTCONSTR:def 9</a></i>;<br/><a NAME="E21:3_1_1_2_2_2"/><b>then </b>
 <a href="xboole_0.html#K1">{}</a>  is    <a href="msaterm.html#M1">ArgumentSeq</a> of <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="circuit2.html#E13:3_1_1_2_2_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="msaterm.html#D2">MSATERM:def 2</a></i>;<br/><a NAME="E22:3_1_1_2_2_2"/><b>then </b>
 <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,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="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="msaterm.html#K1">-Terms</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="circuit2.html#E12:3_1_1_2_2_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="msaterm.html#T1">MSATERM:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> =  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,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> as    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="ref" href="msaterm.html#D1">MSATERM:def 1</a></i>;<br/><a NAME="E24:3_1_1_2_2_2"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <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"><span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,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#T3">TREES_4:3</a></i>;<br/><a NAME="E25:3_1_1_2_2_2"/>
 <a href="msualg_1.html#K2">the_result_sort_of</a> <span class="p1">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="circuit2.html#E1:3_1_1_2_2_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="circuit2.html#E9:3_1_1_2_2_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="msafree2.html#D7">MSAFREE2:def 7</a></i>;<br/><a NAME="E26:3_1_1_2_2_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5">DTConMSA</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> : ( ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> 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>6</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> ) or ex <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</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="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="circuit2.html#E24:3_1_1_2_2_2"><i><font color="Green">E21</font></i></a></i>;<br/><a NAME="E27:3_1_1_2_2_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree.html#K7">FreeSort</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="msafree.html#D12">MSAFREE:def 12</a></i>;<br/><a NAME="E28:3_1_1_2_2_2"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="circuit2.html#E8:3_1_1_2"><i><font color="Green">E10</font></i></a>, <a class="ref" href="msafree.html#D13">MSAFREE:def 13</a></i>;<br/><a NAME="E29:3_1_1_2_2_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>]
 <b>by </b><i><a class="txt" href="circuit2.html#E3:3_1_1"><i><font color="Green">E4</font></i></a></i>;<br/><b>hence </b><a NAME="E30:3_1_1_2_2_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><a href="msafree.html#K13">FreeGen</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="circuit2.html#E7:3_1_1_2"><i><font color="Green">E9</font></i></a>, <a class="txt" href="circuit2.html#E1:3_1_1_2_2_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="circuit2.html#E28:3_1_1_2_2_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="funcop_1.html#T57">FUNCOP_1:57</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_2_2_3"><b>suppose </b></a><a NAME="E1:3_1_1_2_2_3"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:3_1_1_2_2_3"/><i><font color="Green">E12</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <span class="p2">(<span class="default"><a href="msafree.html#K12">FreeGen</a> <font color="Maroon">c<sub>6</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msafree.html#K12">FreeGen</a> <font color="Maroon">c<sub>6</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <span class="p2">(<span class="default"><a href="msafree.html#K12">FreeGen</a> <font color="Maroon">c<sub>6</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msafree.html#K12">FreeGen</a> <font color="Maroon">c<sub>6</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T71">RELAT_1:71</a></i>;<br/><a NAME="E3:3_1_1_2_2_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>]
 <b>by </b><i><a class="txt" href="circuit2.html#E3:3_1_1"><i><font color="Green">E4</font></i></a></i>;<br/><b>hence </b><a NAME="E4:3_1_1_2_2_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><a href="msafree.html#K13">FreeGen</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="circuit2.html#E7:3_1_1_2"><i><font color="Green">E9</font></i></a>, <a class="txt" href="circuit2.html#E8:3_1_1_2"><i><font color="Green">E10</font></i></a>, <a class="txt" href="circuit2.html#E1:3_1_1_2_2_3"><i><font color="Green">E11</font></i></a>, <a class="txt" href="circuit2.html#E2:3_1_1_2_2_3"><i><font color="Green">E12</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:3_1_1_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:3_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</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="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><b>hence </b><a NAME="E3:3_1_1_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> is   <a href="funct_1.html#NM1">Function</a>
 <b>by </b><i><a class="txt" href="circuit2.html#E4:3_1_1"><i><font color="Green">E5</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="pboole.html#NM2">ManySortedFunction</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="funcop_1.html#D6">FUNCOP_1:def 6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="pboole.html#M3">ManySortedFunction</a> of  <a href="msafree.html#K13">FreeGen</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="txt" href="circuit2.html#E4:3_1_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="msafree2.html#NM1">Vertex</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:3_1_1_4"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:3_1_1_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>]
 <b>by </b><i><a class="txt" href="circuit2.html#E3:3_1_1"><i><font color="Green">E4</font></i></a></i>;<br/><b>hence </b><a NAME="E3:3_1_1_4"/>
<font color="Maroon">c<sub>6</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K12">FreeGen</a> <font color="Maroon">c<sub>7</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p1">(<span class="default"><a href="trees_4.html#K1">root-tree</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="circuit2.html#E1:3_1_1_4"><i><font color="Green">E6</font></i></a></i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:3_1_1_5"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:3_1_1_5"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>]
 <b>by </b><i><a class="txt" href="circuit2.html#E3:3_1_1"><i><font color="Green">E4</font></i></a></i>;<br/><b>hence </b><a NAME="E3:3_1_1_5"/>
<font color="Maroon">c<sub>6</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K12">FreeGen</a> <font color="Maroon">c<sub>7</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p1">(<span class="default"><a href="trees_4.html#K1">root-tree</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,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></span>)</span>
 <b>by </b><i><a class="txt" href="circuit2.html#E1:3_1_1_5"><i><font color="Green">E6</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E10:3_1_1"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>

<a NAME="E11:3_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>]
 
<b>by </b><i><a class="txt" href="circuit2.html#E3:3_1_1"><i><font color="Green">E4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E12:3_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K6">id</a> <span class="p1">(<span class="default"><a href="msafree.html#K12">FreeGen</a> <font color="Maroon">c<sub>7</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="circuit2.html#E10:3_1_1"><i><font color="Green">E6</font></i></a></i>;<br/>


</div>
