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<b>let </b><font color="Maroon" title="c1">IIG</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a> <a href="msafree2.html#V2" title="MSAFREE2:attr.2">Circuit-like</a> <a href="msafree2.html#V5" title="MSAFREE2:attr.5">monotonic</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font> being <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a> <a href="msafree2.html#V4" title="MSAFREE2:attr.4">locally-finite</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">IIG</font><br/> for <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">w</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">IIG</font>  st <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon" title="c1">IIG</font> &amp; <font color="Olive" title="b3">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <span class="p2">(<span class="default"><a href="msafree2.html#K5" title="MSAFREE2:func.5">action_at</a> <font color="Olive" title="b2">v</font></span>)</span></span>)</span> holds <br/> <a href="circuit1.html#K4" title="CIRCUIT1:func.4">size</a> <font color="Olive" title="b3">w</font>,<font color="Olive" title="b1">A</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="circuit1.html#K4" title="CIRCUIT1:func.4">size</a> <font color="Olive" title="b2">v</font>,<font color="Olive" title="b1">A</font></span><br/><b>let </b><font color="Maroon" title="c2">A</font> be  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a> <a href="msafree2.html#V4" title="MSAFREE2:attr.4">locally-finite</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">IIG</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">IIG</font>  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon" title="c1">IIG</font> &amp; <font color="Olive" title="b2">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <span class="p2">(<span class="default"><a href="msafree2.html#K5" title="MSAFREE2:func.5">action_at</a> <font color="Olive" title="b1">v</font></span>)</span></span>)</span> holds <br/> <a href="circuit1.html#K4" title="CIRCUIT1:func.4">size</a> <font color="Olive" title="b2">w</font>,<font color="Maroon" title="c2">A</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="circuit1.html#K4" title="CIRCUIT1:func.4">size</a> <font color="Olive" title="b1">v</font>,<font color="Maroon" title="c2">A</font></span><br/><b>let </b><font color="Maroon" title="c3">v</font>, <font color="Maroon" title="c4">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">IIG</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon" title="c1">IIG</font> &amp; <font color="Maroon" title="c4">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <span class="p2">(<span class="default"><a href="msafree2.html#K5" title="MSAFREE2:func.5">action_at</a> <font color="Maroon" title="c3">v</font></span>)</span></span>)</span> implies  <a href="circuit1.html#K4" title="CIRCUIT1:func.4">size</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c2">A</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="circuit1.html#K4" title="CIRCUIT1:func.4">size</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c2">A</font> )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:25"/><i><font color="Green" title="E12">A1</font></i>: 
<font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon" title="c1">IIG</font>
 <b>and </b><br/><a NAME="E2:25"/><i><font color="Green" title="E13">A2</font></i>: 
<font color="Maroon" title="c4">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <span class="p2">(<span class="default"><a href="msafree2.html#K5" title="MSAFREE2:func.5">action_at</a> <font color="Maroon" title="c3">v</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="circuit1.html#K4" title="CIRCUIT1:func.4">size</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c2">A</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="circuit1.html#K4" title="CIRCUIT1:func.4">size</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c2">A</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c5">av</font> =  <a href="msafree2.html#K5" title="MSAFREE2:func.5">action_at</a> <font color="Maroon" title="c3">v</font> as   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">IIG</font> ;<br/>
<b>consider </b><font color="Maroon" title="c6">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:25"/><i><font color="Green" title="E14">A3</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c5">av</font></span>)</span>
 <b>and </b><br/><a NAME="E6:25"/><i><font color="Green" title="E15">A4</font></i>: 
<font color="Maroon" title="c4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c5">av</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="#E2:25"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">k</font> = <font color="Maroon" title="c6">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E5:25"><i><font color="Green" title="E14">A3</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">sv</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E9:25"/><i><font color="Green" title="E16">A5</font></i>: 
<font color="Maroon" title="c8">sv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">tv</font></span>)</span> where <font color="Olive" title="b1">tv</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6" title="MSAFREE2:func.6">FreeEnv</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> : verum </span>}</span> 
 <b>and </b><br/><a NAME="E10:25"/><i><font color="Green" title="E17">A6</font></i>: 
 <a href="circuit1.html#K4" title="CIRCUIT1:func.4">size</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c8">sv</font>
 <b>by </b><i><a class="ref" href="circuit1.html#D4" target="_self" title="CIRCUIT1:def.4">Def4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">Yv</font> = <font color="Maroon" title="c8">sv</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="membered.html#V3" title="MEMBERED:attr.3">real-membered</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<a NAME="E12:25"/>
 <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c9">Yv</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Yv</font>
 
<b>by </b><i><a class="ref" href="xxreal_2.html#D8" title="XXREAL_2:def.8">XXREAL_2:def 8</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">tv</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6" title="MSAFREE2:func.6">FreeEnv</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font><b> such that </b><br/><a NAME="E14:25"/><i><font color="Green" title="E18">A7</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">tv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c9">Yv</font>
 <b>by </b><i><a class="txt" href="#E9:25"><i><font color="Green" title="E16">A5</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">e1</font> = <font color="Maroon" title="c10">tv</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="trees_2.html#NM4" title="TREES_2:NM.4">DecoratedTree</a> ;<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" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K1" title="MSAFREE2:func.1">SortsWithConstants</a> <font color="Maroon" title="c1">IIG</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:25_1"/><b>then </b>
<font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">v'</font> where <font color="Olive" title="b1">v'</font> is   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">IIG</font> : <font color="Olive" title="b1">v'</font> is <a href="msualg_2.html#V1" title="MSUALG_2:attr.1">with_const_op</a> </span>}</span> 
 <b>by </b><i><a class="ref" href="msafree2.html#D1" title="MSAFREE2:def.1">MSAFREE2:def 1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c12">v'</font> being   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">IIG</font><b> such that </b><br/><a NAME="E4:25_1"/><i><font color="Green" title="E19">A8</font></i>: 
<font color="Maroon" title="c12">v'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font>
 <b>and </b><br/><a NAME="E5:25_1"/><i><font color="Green" title="E20">A9</font></i>: 
<font color="Maroon" title="c12">v'</font> is <a href="msualg_2.html#V1" title="MSUALG_2:attr.1">with_const_op</a>
 ;<br/><b>consider </b><font color="Maroon" title="c13">oo</font> being   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">IIG</font><b> such that </b><br/><a NAME="E7:25_1"/><i><font color="Green" title="E21">A10</font></i>: 
the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c1">IIG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">oo</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>and </b><br/><a NAME="E8:25_1"/><i><font color="Green" title="E22">A11</font></i>: 
the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c1">IIG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">oo</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">v'</font>
 <b>by </b><i><a class="txt" href="#E5:25_1"><i><font color="Green" title="E20">A9</font></i></a>, <a class="ref" href="msualg_2.html#D2" title="MSUALG_2:def.2">MSUALG_2:def 2</a></i>;<br/> <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c13">oo</font> = 
<font color="Maroon" title="c3">v</font>
<b>by </b><i><a class="txt" href="#E4:25_1"><i><font color="Green" title="E19">A8</font></i></a>, <a class="txt" href="#E8:25_1"><i><font color="Green" title="E22">A11</font></i></a>, <a class="ref" href="msualg_1.html#D7" title="MSUALG_1:def.7">MSUALG_1:def 7</a></i>
<br/>.= 
 <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c5">av</font>
<b>by </b><i><a class="txt" href="#E1:25"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="msafree2.html#D7" title="MSAFREE2:def.7">MSAFREE2:def 7</a></i>
;<br/><a NAME="E10:25_1"/><b>then </b><i><font color="Green" title="E23">A12</font></i>: 
<font color="Maroon" title="c5">av</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">oo</font>
 <b>by </b><i><a class="ref" href="msafree2.html#D6" title="MSAFREE2:def.6">MSAFREE2:def 6</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">aoo</font> = the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c1">IIG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">oo</font> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <b>by </b><i><a class="txt" href="#E7:25_1"><i><font color="Green" title="E21">A10</font></i></a></i>;<br/><a NAME="E12:25_1"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">aoo</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="#E7:25_1"><i><font color="Green" title="E21">A10</font></i></a></i>;<br/><b>hence </b><a NAME="E13:25_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E5:25"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="#E10:25_1"><i><font color="Green" title="E23">A12</font></i></a>, <a class="ref" href="msualg_1.html#D6" title="MSUALG_1:def.6">MSUALG_1:def 6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E17:25"/><b>then </b><i><font color="Green" title="E19">A13</font></i>: 
<font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon" title="c1">IIG</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="msafree2.html#K1" title="MSAFREE2:func.1">SortsWithConstants</a> <font color="Maroon" title="c1">IIG</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:25"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">p</font> being  <a href="trees_3.html#V6" title="TREES_3:attr.6">DTree-yielding</a> <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E19:25"/><i><font color="Green" title="E20">A14</font></i>: 
<font color="Maroon" title="c10">tv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">av</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">IIG</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c12">p</font>
 <b>by </b><i><a class="txt" href="#E10:25"><i><font color="Green" title="E17">A6</font></i></a>, <a class="txt" href="#E14:25"><i><font color="Green" title="E18">A7</font></i></a>, <a class="ref" href="circuit1.html#T13" target="_self" title="CIRCUIT1:th.13">Th13</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">sw</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E21:25"/><i><font color="Green" title="E21">A15</font></i>: 
<font color="Maroon" title="c13">sw</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">tw</font></span>)</span> where <font color="Olive" title="b1">tw</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6" title="MSAFREE2:func.6">FreeEnv</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">w</font> : verum </span>}</span> 
 <b>and </b><br/><a NAME="E22:25"/><i><font color="Green" title="E22">A16</font></i>: 
 <a href="circuit1.html#K4" title="CIRCUIT1:func.4">size</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c13">sw</font>
 <b>by </b><i><a class="ref" href="circuit1.html#D4" target="_self" title="CIRCUIT1:def.4">Def4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">Yw</font> = <font color="Maroon" title="c13">sw</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="membered.html#V3" title="MEMBERED:attr.3">real-membered</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<a NAME="E24:25"/>
 <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c14">Yw</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Yw</font>
 
<b>by </b><i><a class="ref" href="xxreal_2.html#D8" title="XXREAL_2:def.8">XXREAL_2:def 8</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">tw</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6" title="MSAFREE2:func.6">FreeEnv</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">w</font><b> such that </b><br/><a NAME="E26:25"/><i><font color="Green" title="E23">A17</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">tw</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c14">Yw</font>
 <b>by </b><i><a class="txt" href="#E21:25"><i><font color="Green" title="E21">A15</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">e2</font> = <font color="Maroon" title="c15">tw</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="trees_2.html#NM4" title="TREES_2:NM.4">DecoratedTree</a> ;<br/>
<b>consider </b><font color="Maroon" title="c17">p'</font> being  <a href="trees_3.html#V6" title="TREES_3:attr.6">DTree-yielding</a> <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E29:25"/><i><font color="Green" title="E24">A18</font></i>: 
<font color="Maroon" title="c17">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">p</font>
 <b>and </b><br/><a NAME="E30:25"/><i><font color="Green" title="E25">A19</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_3.html#K13" title="TREES_3:func.13">tree</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c17">p'</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E19:25"><i><font color="Green" title="E20">A14</font></i></a>, <a class="ref" href="trees_4.html#D4" title="TREES_4:def.4">TREES_4:def 4</a></i>;<br/>
<a NAME="E31:25"/><i><font color="Green" title="E26">A20</font></i>: 
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6" title="MSAFREE2:func.6">FreeEnv</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6" title="MSAFREE2:func.6">FreeEnv</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c5">av</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:25"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="msafree2.html#D7" title="MSAFREE2:def.7">MSAFREE2:def 7</a></i>;<br/>
<a NAME="E32:25"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c5">av</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E19:25"><i><font color="Green" title="E20">A14</font></i></a>, <a class="ref" href="msafree2.html#T13" title="MSAFREE2:th.13">MSAFREE2:13</a></i>;<br/>
<a NAME="E33:25"/><b>then </b><i><font color="Green" title="E27">A21</font></i>: 
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">p</font>
 
<b>by </b><i><a class="txt" href="#E5:25"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">k1</font> = <font color="Maroon" title="c7">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E5:25"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="finseq_3.html#T28" title="FINSEQ_3:th.28">FINSEQ_3:28</a></i>;<br/>
<a NAME="E35:25"/><i><font color="Green" title="E28">A22</font></i>: 
<font color="Maroon" title="c18">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font>
 
;<br/>
<b>reconsider </b><font color="Maroon" title="c19">o</font> = <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c18">k1</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> as    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c20">o</font> = <font color="Maroon" title="c19">o</font> as    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">e1</font> <b>by </b><i><a class="txt" href="#E29:25"><i><font color="Green" title="E24">A18</font></i></a>, <a class="txt" href="#E30:25"><i><font color="Green" title="E25">A19</font></i></a>, <a class="txt" href="#E33:25"><i><font color="Green" title="E27">A21</font></i></a>, <a class="txt" href="#E35:25"><i><font color="Green" title="E28">A22</font></i></a>, <a class="ref" href="pre_circ.html#T17" title="PRE_CIRC:th.17">PRE_CIRC:17</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c21">eoe</font> = <font color="Maroon" title="c11">e1</font> <a href="trees_2.html#K8" title="TREES_2:func.8">with-replacement</a> <font color="Maroon" title="c20">o</font>,<font color="Maroon" title="c16">e2</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c22">de1</font> =  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">e1</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c23">de2</font> =  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c16">e2</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c24">o</font> = <font color="Maroon" title="c20">o</font> as    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c22">de1</font> ;<br/>
<a NAME="E42:25"/><i><font color="Green" title="E29">A23</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c21">eoe</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">de1</font> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <font color="Maroon" title="c24">o</font>,<font color="Maroon" title="c23">de2</font>
 
<b>by </b><i><a class="ref" href="trees_2.html#D12" title="TREES_2:def.12">TREES_2:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c25">deoe</font> =  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c21">eoe</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> ;<br/>
<a NAME="E44:25"/>
<font color="Maroon" title="c12">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6" title="MSAFREE2:func.6">FreeEnv</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c5">av</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E5:25"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="#E19:25"><i><font color="Green" title="E20">A14</font></i></a>, <a class="txt" href="#E31:25"><i><font color="Green" title="E26">A20</font></i></a>, <a class="ref" href="msafree2.html#T14" title="MSAFREE2:th.14">MSAFREE2:14</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c26">eoe</font> = <font color="Maroon" title="c21">eoe</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6" title="MSAFREE2:func.6">FreeEnv</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> <b>by </b><i><a class="txt" href="#E6:25"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="#E17:25"><i><font color="Green" title="E19">A13</font></i></a>, <a class="txt" href="#E19:25"><i><font color="Green" title="E20">A14</font></i></a>, <a class="txt" href="#E33:25"><i><font color="Green" title="E27">A21</font></i></a>, <a class="txt" href="#E35:25"><i><font color="Green" title="E28">A22</font></i></a>, <a class="ref" href="circuit1.html#T7" target="_self" title="CIRCUIT1:th.7">Th7</a></i>;<br/>
<a NAME="E46:25"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c26">eoe</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Yv</font>
 
<b>by </b><i><a class="txt" href="#E9:25"><i><font color="Green" title="E16">A5</font></i></a></i>;<br/>
<a NAME="E47:25"/><b>then </b><i><font color="Green" title="E30">A24</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c26">eoe</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="circuit1.html#K4" title="CIRCUIT1:func.4">size</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="#E10:25"><i><font color="Green" title="E17">A6</font></i></a>, <a class="ref" href="xxreal_2.html#D8" title="XXREAL_2:def.8">XXREAL_2:def 8</a></i>;<br/>
<a NAME="E48:25"/><i><font color="Green" title="E31">A25</font></i>: 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c25">deoe</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">de1</font> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c24">o</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c22">de1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c23">de2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E42:25"><i><font color="Green" title="E29">A23</font></i></a>, <a class="ref" href="pre_circ.html#T22" title="PRE_CIRC:th.22">PRE_CIRC:22</a></i>;<br/>
<a NAME="E49:25"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">de1</font> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c24">o</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c22">de1</font>
 
<b>by </b><i><a class="ref" href="pre_circ.html#T20" title="PRE_CIRC:th.20">PRE_CIRC:20</a></i>;<br/>
<a NAME="E50:25"/><b>then </b>
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">de1</font> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c24">o</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c23">de2</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c25">deoe</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">de1</font> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c24">o</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E48:25"><i><font color="Green" title="E31">A25</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E51:25"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c23">de2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c25">deoe</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E52:25"/><b>then </b><i><font color="Green" title="E32">A26</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c16">e2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c25">deoe</font>
 
<b>by </b><i><a class="ref" href="pre_circ.html#T21" title="PRE_CIRC:th.21">PRE_CIRC:21</a></i>;<br/>
<a NAME="E53:25"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c25">deoe</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="circuit1.html#K4" title="CIRCUIT1:func.4">size</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="#E47:25"><i><font color="Green" title="E30">A24</font></i></a>, <a class="ref" href="pre_circ.html#T21" title="PRE_CIRC:th.21">PRE_CIRC:21</a></i>;<br/>
<b>hence </b><a NAME="E54:25"/>
 <a href="circuit1.html#K4" title="CIRCUIT1:func.4">size</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c2">A</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="circuit1.html#K4" title="CIRCUIT1:func.4">size</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="#E22:25"><i><font color="Green" title="E22">A16</font></i></a>, <a class="txt" href="#E26:25"><i><font color="Green" title="E23">A17</font></i></a>, <a class="txt" href="#E52:25"><i><font color="Green" title="E32">A26</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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