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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="msafree2.html#V2">Circuit-like</a> <a href="msafree2.html#V5">monotonic</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="msafree2.html#V4">locally-finite</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>, <font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:25"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>3</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>and </b><br/><a NAME="E2:25"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>3</sub></font> as   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:25"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E6:25"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>5</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="circuit1.html#E2:25"><i><font color="Green">E13</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="circuit1.html#E5:25"><i><font color="Green">E14</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E9:25"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="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>3</sub></font> : verum </span>}</span> 
 <b>and </b><br/><a NAME="E10:25"/><i><font color="Green">E17</font></i>: 
 <a href="circuit1.html#K4" target="_self">size</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="circuit1.html#D4" target="_self">Def4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="membered.html#V2">real-membered</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E12:25"/>
 <a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="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>3</sub></font><b> such that </b><br/><a NAME="E14:25"/><i><font color="Green">E18</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="circuit1.html#E9:25"><i><font color="Green">E16</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as  <a href="finset_1.html#V1">finite</a> <a href="trees_2.html#NM4">DecoratedTree</a> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_1"/>
<font color="Maroon">c<sub>3</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="E2:25_1"/><b>then </b>
<font color="Maroon">c<sub>3</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>12</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="E4:25_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E5:25_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>12</sub></font> is <a href="msualg_2.html#V1">with_const_op</a>
 ;<br/><b>consider </b><font color="Maroon">c<sub>13</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="E7:25_1"/><i><font color="Green">E21</font></i>: 
the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E8:25_1"/><i><font color="Green">E22</font></i>: 
the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="circuit1.html#E5:25_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="msualg_2.html#D2">MSUALG_2:def 2</a></i>;<br/> <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>13</sub></font> = 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="circuit1.html#E4:25_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="circuit1.html#E8:25_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="msualg_1.html#D7">MSUALG_1:def 7</a></i>
<br/>.= 
 <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="circuit1.html#E1:25"><i><font color="Green">E12</font></i></a>, <a class="ref" href="msafree2.html#D7">MSAFREE2:def 7</a></i>
;<br/><a NAME="E10:25_1"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="msafree2.html#D6">MSAFREE2:def 6</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> as   <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i><a class="txt" href="circuit1.html#E7:25_1"><i><font color="Green">E21</font></i></a></i>;<br/><a NAME="E12:25_1"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="circuit1.html#E7:25_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="finseq_1.html#T26">FINSEQ_1:26</a></i>;<br/><b>hence </b><a NAME="E13:25_1"/>
contradiction
 <b>by </b><i><a class="txt" href="circuit1.html#E5:25"><i><font color="Green">E14</font></i></a>, <a class="txt" href="circuit1.html#E10:25_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="msualg_1.html#D6">MSUALG_1:def 6</a></i>;<br/></div><b>end;</b></div>
<a NAME="E17:25"/><b>then </b><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>3</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>
 
<b>by </b><i><a class="txt" href="circuit1.html#E1:25"><i><font color="Green">E12</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being  <a href="trees_3.html#V6">DTree-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E19:25"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>10</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>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>12</sub></font>
 <b>by </b><i><a class="txt" href="circuit1.html#E10:25"><i><font color="Green">E17</font></i></a>, <a class="txt" href="circuit1.html#E14:25"><i><font color="Green">E18</font></i></a>, <a class="ref" href="circuit1.html#T13" target="_self">Th13</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E21:25"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="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>4</sub></font> : verum </span>}</span> 
 <b>and </b><br/><a NAME="E22:25"/><i><font color="Green">E22</font></i>: 
 <a href="circuit1.html#K4" target="_self">size</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="circuit1.html#D4" target="_self">Def4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="membered.html#V2">real-membered</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E24:25"/>
 <a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>15</sub></font> being    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="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>4</sub></font><b> such that </b><br/><a NAME="E26:25"/><i><font color="Green">E23</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="circuit1.html#E21:25"><i><font color="Green">E21</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> as  <a href="finset_1.html#V1">finite</a> <a href="trees_2.html#NM4">DecoratedTree</a> ;<br/>
<b>consider </b><font color="Maroon">c<sub>17</sub></font> being  <a href="trees_3.html#V6">DTree-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E29:25"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>and </b><br/><a NAME="E30:25"/><i><font color="Green">E25</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="trees_3.html#K13">tree</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="circuit1.html#E19:25"><i><font color="Green">E20</font></i></a>, <a class="ref" href="trees_4.html#D4">TREES_4:def 4</a></i>;<br/>
<a NAME="E31:25"/><i><font color="Green">E26</font></i>: 
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>3</sub></font> <a href="hidden.html#R1">=</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> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="circuit1.html#E1:25"><i><font color="Green">E12</font></i></a>, <a class="ref" href="msafree2.html#D7">MSAFREE2:def 7</a></i>;<br/>
<a NAME="E32:25"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="circuit1.html#E19:25"><i><font color="Green">E20</font></i></a>, <a class="ref" href="msafree2.html#T13">MSAFREE2:13</a></i>;<br/>
<a NAME="E33:25"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="circuit1.html#E5:25"><i><font color="Green">E14</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K5">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="circuit1.html#E5:25"><i><font color="Green">E14</font></i></a>, <a class="ref" href="finseq_3.html#T28">FINSEQ_3:28</a></i>;<br/>
<a NAME="E35:25"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 
;<br/>
<b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>18</sub></font></span><a href="lang1.html#K3">*&gt;</a></span> as    <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>19</sub></font> as    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font> <b>by </b><i><a class="txt" href="circuit1.html#E29:25"><i><font color="Green">E24</font></i></a>, <a class="txt" href="circuit1.html#E30:25"><i><font color="Green">E25</font></i></a>, <a class="txt" href="circuit1.html#E33:25"><i><font color="Green">E27</font></i></a>, <a class="txt" href="circuit1.html#E35:25"><i><font color="Green">E28</font></i></a>, <a class="ref" href="pre_circ.html#T17">PRE_CIRC:17</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="trees_2.html#K8">with-replacement</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>16</sub></font> as  <a href="finset_1.html#V1">finite</a> <a href="funct_1.html#NM1">Function</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>22</sub></font> =  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font> as  <a href="finset_1.html#V1">finite</a> <a href="trees_1.html#NM1">Tree</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>23</sub></font> =  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>16</sub></font> as  <a href="finset_1.html#V1">finite</a> <a href="trees_1.html#NM1">Tree</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>20</sub></font> as    <a href="trees_1.html#M1">Element</a> of <font color="Maroon">c<sub>22</sub></font> ;<br/>
<a NAME="E42:25"/><i><font color="Green">E29</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="trees_1.html#K5">with-replacement</a> <font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>23</sub></font>
 
<b>by </b><i><a class="ref" href="trees_2.html#D12">TREES_2:def 12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>25</sub></font> =  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>21</sub></font> as  <a href="finset_1.html#V1">finite</a> <a href="trees_1.html#NM1">Tree</a> ;<br/>
<a NAME="E44:25"/>
<font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</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> <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>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="circuit1.html#E5:25"><i><font color="Green">E14</font></i></a>, <a class="txt" href="circuit1.html#E19:25"><i><font color="Green">E20</font></i></a>, <a class="txt" href="circuit1.html#E31:25"><i><font color="Green">E26</font></i></a>, <a class="ref" href="msafree2.html#T14">MSAFREE2:14</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>26</sub></font> = <font color="Maroon">c<sub>21</sub></font> as    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="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>3</sub></font> <b>by </b><i><a class="txt" href="circuit1.html#E6:25"><i><font color="Green">E15</font></i></a>, <a class="txt" href="circuit1.html#E17:25"><i><font color="Green">E19</font></i></a>, <a class="txt" href="circuit1.html#E19:25"><i><font color="Green">E20</font></i></a>, <a class="txt" href="circuit1.html#E33:25"><i><font color="Green">E27</font></i></a>, <a class="txt" href="circuit1.html#E35:25"><i><font color="Green">E28</font></i></a>, <a class="ref" href="circuit1.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E46:25"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="circuit1.html#E9:25"><i><font color="Green">E16</font></i></a></i>;<br/>
<a NAME="E47:25"/><b>then </b><i><font color="Green">E30</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>26</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="circuit1.html#K4" target="_self">size</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="circuit1.html#E10:25"><i><font color="Green">E17</font></i></a>, <a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/>
<a NAME="E48:25"/><i><font color="Green">E31</font></i>: 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="trees_1.html#K4">|</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="circuit1.html#E42:25"><i><font color="Green">E29</font></i></a>, <a class="ref" href="pre_circ.html#T22">PRE_CIRC:22</a></i>;<br/>
<a NAME="E49:25"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="trees_1.html#K4">|</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>22</sub></font>
 
<b>by </b><i><a class="ref" href="pre_circ.html#T20">PRE_CIRC:20</a></i>;<br/>
<a NAME="E50:25"/><b>then </b>
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="trees_1.html#K4">|</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="trees_1.html#K4">|</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="circuit1.html#E48:25"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E51:25"/><b>then </b>
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>23</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>25</sub></font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E52:25"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>25</sub></font>
 
<b>by </b><i><a class="ref" href="pre_circ.html#T21">PRE_CIRC:21</a></i>;<br/>
<a NAME="E53:25"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>25</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="circuit1.html#K4" target="_self">size</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="circuit1.html#E47:25"><i><font color="Green">E30</font></i></a>, <a class="ref" href="pre_circ.html#T21">PRE_CIRC:21</a></i>;<br/>
<b>hence </b><a NAME="E54:25"/>
 <a href="circuit1.html#K4" target="_self">size</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="circuit1.html#K4" target="_self">size</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="circuit1.html#E22:25"><i><font color="Green">E22</font></i></a>, <a class="txt" href="circuit1.html#E26:25"><i><font color="Green">E23</font></i></a>, <a class="txt" href="circuit1.html#E52:25"><i><font color="Green">E32</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div>
