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<b>let </b><font color="Maroon">M1</font> be    <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">A</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">A</font></span>)</span>, <font color="Maroon">M2</font> be    <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">A</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">A</font></span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:3_1_2"/><i><font color="Green">E110</font></i>: 
for <font color="Olive">v</font> being  <a href="msafree2.html#NM1">Vertex</a> of <font color="Maroon">IIG</font> holds <br/> ( ( <font color="Olive">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">IIG</font> implies <font color="Maroon">M1</font> <a href="msualg_3.html#K1">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K12">FreeGen</a> <font color="Olive">v</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</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">iv</font> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font></span>)</span>,<font color="Olive">v</font></span><a href="tarski.html#K4">]</a></span></span>)</span> ) &amp; ( <font color="Olive">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">IIG</font> implies <font color="Maroon">M1</font> <a href="msualg_3.html#K1">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K12">FreeGen</a> <font color="Olive">v</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</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">v</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">IIG</font></span><a href="tarski.html#K4">]</a></span></span>)</span> ) &amp; ( <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">IIG</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">IIG</font></span>)</span> implies <font color="Maroon">M1</font> <a href="msualg_3.html#K1">.</a> <font color="Olive">v</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">v</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font></span>)</span> ) )
 <b>and </b><br/><a NAME="E2:3_1_2"/><i><font color="Green">E111</font></i>: 
for <font color="Olive">v</font> being  <a href="msafree2.html#NM1">Vertex</a> of <font color="Maroon">IIG</font> holds <br/> ( ( <font color="Olive">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">IIG</font> implies <font color="Maroon">M2</font> <a href="msualg_3.html#K1">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K12">FreeGen</a> <font color="Olive">v</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</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">iv</font> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font></span>)</span>,<font color="Olive">v</font></span><a href="tarski.html#K4">]</a></span></span>)</span> ) &amp; ( <font color="Olive">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">IIG</font> implies <font color="Maroon">M2</font> <a href="msualg_3.html#K1">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K12">FreeGen</a> <font color="Olive">v</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</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">v</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">IIG</font></span><a href="tarski.html#K4">]</a></span></span>)</span> ) &amp; ( <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">IIG</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">IIG</font></span>)</span> implies <font color="Maroon">M2</font> <a href="msualg_3.html#K1">.</a> <font color="Olive">v</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">v</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font></span>)</span> ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:3_1_2_1"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">IIG</font>
 ;<br/><b>then reconsider </b><font color="Maroon">v</font> = <font color="Maroon">i</font> as   <a href="msafree2.html#NM1">Vertex</a> of <font color="Maroon">IIG</font> ;<br/><a NAME="E3:3_1_2_1"/>
 <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">IIG</font> <a href="tarski.html#R1">c=</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">IIG</font>
 <b>by </b><i><a class="ref" href="msafree2.html#T5">MSAFREE2:5</a></i>;<br/><a NAME="E4:3_1_2_1"/><b>then </b><i><font color="Green">E113</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">IIG</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">IIG</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">IIG</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">IIG</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>;<br/><a NAME="E5:3_1_2_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">IIG</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">IIG</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>;<br/><a NAME="E6:3_1_2_1"/><b>then </b><i><font color="Green">E114</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">IIG</font> or <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">IIG</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_2_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_2_1_1"/>
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">IIG</font> or <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">IIG</font> or <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">IIG</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">IIG</font></span>)</span> )
 </span><b>by </b><i>, , <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_2_1_1_1"><b>suppose </b></a><a NAME="E1:3_1_2_1_1_1"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">IIG</font>
 ;<br/><div class="add"><a NAME="E2:3_1_2_1_1_1"/><b>then </b>
<font color="Maroon">M1</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K12">FreeGen</a> <font color="Maroon">v</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</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">iv</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font></span>)</span>,<font color="Maroon">v</font></span><a href="tarski.html#K4">]</a></span></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:3_1_2_1_1_1"/>
<font color="Maroon">M1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:3_1_2_1_1_2"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">IIG</font>
 ;<br/><div class="add"><a NAME="E2:3_1_2_1_1_2"/><b>then </b>
<font color="Maroon">M1</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K12">FreeGen</a> <font color="Maroon">v</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</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">v</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">IIG</font></span><a href="tarski.html#K4">]</a></span></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:3_1_2_1_1_2"/>
<font color="Maroon">M1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:3_1_2_1_1_3"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">IIG</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">IIG</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:3_1_2_1_1_3"/><b>then </b>
<font color="Maroon">M1</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">v</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">v</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:3_1_2_1_1_3"/>
<font color="Maroon">M1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:3_1_2"/>
<font color="Maroon">M1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">M2</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


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