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<b>let </b><font color="Maroon" title="c1">IIG</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> non <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">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> ;<br/><b>let </b><font color="Maroon" title="c2">A</font> be  <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">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/><b>let </b><font color="Maroon" title="c3">v</font>, <font color="Maroon" title="c4">v1</font> be   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">IIG</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:32"/><i><font color="Green" title="E16">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="E2:32"/><i><font color="Green" title="E17">A2</font></i>: 
<font color="Maroon" title="c4">v1</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>
 ;<br/>

<b>consider </b><font color="Maroon" title="c5">s</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="E4:32"/><i><font color="Green" title="E18">A3</font></i>: 
<font color="Maroon" title="c5">s</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="circuit1.html#K5" title="CIRCUIT1:func.5">depth</a> <font color="Olive" title="b1">t</font></span>)</span> where <font color="Olive" title="b1">t</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">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="E5:32"/><i><font color="Green" title="E19">A4</font></i>: 
 <a href="circuit1.html#K6" title="CIRCUIT1:func.6">depth</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="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c5">s</font>
 <b>by </b><i><a class="ref" href="circuit1.html#D6" target="_self" title="CIRCUIT1:def.6">Def6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">Y</font> = <font color="Maroon" title="c5">s</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#V2" title="MEMBERED:attr.2">real-membered</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<a NAME="E7:32"/>
 <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c6">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="circuit1.html#K5" title="CIRCUIT1:func.5">depth</a> <font color="Olive" title="b1">t</font></span>)</span> where <font color="Olive" title="b1">t</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">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>by </b><i><a class="txt" href="circuit1.html#E4:32"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="pre_circ.html#D1" title="PRE_CIRC:def.1">PRE_CIRC:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">t</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">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="E9:32"/><i><font color="Green" title="E20">A5</font></i>: 
 <a href="circuit1.html#K5" title="CIRCUIT1:func.5">depth</a> <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c6">Y</font>
 ;<br/>
<b>consider </b><font color="Maroon" title="c8">s1</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="E11:32"/><i><font color="Green" title="E21">A6</font></i>: 
<font color="Maroon" title="c8">s1</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="circuit1.html#K5" title="CIRCUIT1:func.5">depth</a> <font color="Olive" title="b1">t1</font></span>)</span> where <font color="Olive" title="b1">t1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">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">v1</font> : verum </span>}</span> 
 <b>and </b><br/><a NAME="E12:32"/><i><font color="Green" title="E22">A7</font></i>: 
 <a href="circuit1.html#K6" title="CIRCUIT1:func.6">depth</a> <font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c8">s1</font>
 <b>by </b><i><a class="ref" href="circuit1.html#D6" target="_self" title="CIRCUIT1:def.6">Def6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">Y1</font> = <font color="Maroon" title="c8">s1</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#V2" title="MEMBERED:attr.2">real-membered</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<a NAME="E14:32"/>
 <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c9">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="circuit1.html#K5" title="CIRCUIT1:func.5">depth</a> <font color="Olive" title="b1">t1</font></span>)</span> where <font color="Olive" title="b1">t1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">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">v1</font> : verum </span>}</span> 
 
<b>by </b><i><a class="txt" href="circuit1.html#E11:32"><i><font color="Green" title="E21">A6</font></i></a>, <a class="ref" href="pre_circ.html#D1" title="PRE_CIRC:def.1">PRE_CIRC:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">t1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">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">v1</font><b> such that </b><br/><a NAME="E16:32"/><i><font color="Green" title="E23">A8</font></i>: 
 <a href="circuit1.html#K5" title="CIRCUIT1:func.5">depth</a> <font color="Maroon" title="c10">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c9">Y1</font>
 ;<br/>
<b>consider </b><font color="Maroon" title="c11">e'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <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">v1</font><b> such that </b><br/><a NAME="E18:32"/><i><font color="Green" title="E24">A9</font></i>: 
<font color="Maroon" title="c10">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">e'</font>
 <b>and </b><br/><a NAME="E19:32"/><i><font color="Green" title="E25">A10</font></i>: 
 <a href="circuit1.html#K5" title="CIRCUIT1:func.5">depth</a> <font color="Maroon" title="c10">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msafree2.html#K9" title="MSAFREE2:func.9">depth</a> <font color="Maroon" title="c11">e'</font>
 <b>by </b><i><a class="ref" href="circuit1.html#D5" target="_self" title="CIRCUIT1:def.5">Def5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">dt1'</font> being  <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>, <font color="Maroon" title="c13">t1'</font> being  <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><b> such that </b><br/><a NAME="E21:32"/><i><font color="Green" title="E26">A11</font></i>: 
<font color="Maroon" title="c12">dt1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">e'</font>
 <b>and </b><br/><a NAME="E22:32"/><i><font color="Green" title="E27">A12</font></i>: 
<font color="Maroon" title="c13">t1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c12">dt1'</font>
 <b>and </b><br/><a NAME="E23:32"/><i><font color="Green" title="E28">A13</font></i>: 
 <a href="msafree2.html#K9" title="MSAFREE2:func.9">depth</a> <font color="Maroon" title="c11">e'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <font color="Maroon" title="c13">t1'</font>
 <b>by </b><i><a class="ref" href="msafree2.html#D14" title="MSAFREE2:def.14">MSAFREE2:def 14</a></i>;<br/>
<a NAME="E24:32"/><i><font color="Green" title="E29">A14</font></i>: 
 <a href="circuit1.html#K4" title="CIRCUIT1:func.4">size</a> <font color="Maroon" title="c4">v1</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="circuit1.html#E1:32"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="circuit1.html#E2:32"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="circuit1.html#T15" target="_self" title="CIRCUIT1:th.15">Th15</a></i>;<br/>
<a NAME="E25:32"/>
 <a href="circuit1.html#K4" title="CIRCUIT1:func.4">size</a> <font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c2">A</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="circuit1.html#T16" target="_self" title="CIRCUIT1:th.16">Th16</a></i>;<br/>
<a NAME="E26:32"/><b>then </b>
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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="c4">v1</font>,<font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E27:32"/><b>then </b>
not <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#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c1">IIG</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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="circuit1.html#E24:32"><i><font color="Green" title="E29">A14</font></i></a>, <a class="ref" href="circuit1.html#T11" target="_self" title="CIRCUIT1:th.11">Th11</a></i>;<br/>
<a NAME="E28:32"/><b>then </b><i><font color="Green" title="E30">A15</font></i>: 
( not <font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c1">IIG</font> &amp; not <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> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E29:32"/><b>then </b><i><font color="Green" title="E31">A16</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="circuit1.html#E1:32"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E30:32"/>
 <a href="msafree2.html#K6" title="MSAFREE2:func.6">FreeEnv</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#G3" title="MSUALG_1:aggr.3">MSAlgebra</a>(# <span class="p1">(<span class="default"><a href="msafree.html#K8" title="MSAFREE:func.8">FreeSort</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="msafree.html#K10" title="MSAFREE:func.10">FreeOper</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="msafree.html#D16" title="MSAFREE:def.16">MSAFREE:def 16</a></i>;<br/>
<a NAME="E31:32"/><b>then </b>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">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>  <a href="msafree.html#K7" title="MSAFREE:func.7">FreeSort</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">v</font>
 
<b>by </b><i><a class="ref" href="msafree.html#D13" title="MSAFREE:def.13">MSAFREE:def 13</a></i>;<br/>
<a NAME="E32:32"/><b>then </b>
<font color="Maroon" title="c7">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree.html#K7" title="MSAFREE:func.7">FreeSort</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">v</font>
 
;<br/>
<a NAME="E33:32"/><b>then </b>
<font color="Maroon" title="c7">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5" title="MSAFREE:func.5">DTConMSA</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font></span>)</span> : (  ex <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Maroon" title="c3">v</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) or  ex <font color="Olive" title="b2">o</font> being  <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">IIG</font> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">o</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Olive" title="b2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> ) ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="msafree.html#D12" title="MSAFREE:def.12">MSAFREE:def 12</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">a</font> being    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5" title="MSAFREE:func.5">DTConMSA</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font></span>)</span><b> such that </b><br/><a NAME="E35:32"/><i><font color="Green" title="E32">A17</font></i>: 
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font>
 <b>and </b><br/><a NAME="E36:32"/><i><font color="Green" title="E33">A18</font></i>: 
(  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font> &amp; <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">v</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) or  ex <font color="Olive" title="b1">o</font> being  <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">IIG</font> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">o</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Olive" title="b1">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> ) )
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c15">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E1:32_1"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font>
 <b>and </b><br/><a NAME="E2:32_1"/><i><font color="Green" title="E34">A19</font></i>: 
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c3">v</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ;<br/><b>consider </b><font color="Maroon" title="c16">e'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <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="E4:32_1"/><i><font color="Green" title="E35">A20</font></i>: 
<font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">e'</font>
 <b>and </b><br/><a NAME="E5:32_1"/><i><font color="Green" title="E36">A21</font></i>: 
 <a href="circuit1.html#K5" title="CIRCUIT1:func.5">depth</a> <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msafree2.html#K9" title="MSAFREE2:func.9">depth</a> <font color="Maroon" title="c16">e'</font>
 <b>by </b><i><a class="ref" href="circuit1.html#D5" target="_self" title="CIRCUIT1:def.5">Def5</a></i>;<br/><b>consider </b><font color="Maroon" title="c17">dta</font> being  <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>, <font color="Maroon" title="c18">ta</font> being  <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><b> such that </b><br/><a NAME="E7:32_1"/><i><font color="Green" title="E37">A22</font></i>: 
<font color="Maroon" title="c17">dta</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">e'</font>
 <b>and </b><br/><a NAME="E8:32_1"/><i><font color="Green" title="E38">A23</font></i>: 
<font color="Maroon" title="c18">ta</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c17">dta</font>
 <b>and </b><br/><a NAME="E9:32_1"/><i><font color="Green" title="E39">A24</font></i>: 
 <a href="msafree2.html#K9" title="MSAFREE2:func.9">depth</a> <font color="Maroon" title="c16">e'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <font color="Maroon" title="c18">ta</font>
 <b>by </b><i><a class="ref" href="msafree2.html#D14" title="MSAFREE2:def.14">MSAFREE2:def 14</a></i>;<br/><a NAME="E10:32_1"/>
 <a href="circuit1.html#K5" title="CIRCUIT1:func.5">depth</a> <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="circuit1.html#E35:32"><i><font color="Green" title="E32">A17</font></i></a>, <a class="txt" href="circuit1.html#E2:32_1"><i><font color="Green" title="E34">A19</font></i></a>, <a class="txt" href="circuit1.html#E4:32_1"><i><font color="Green" title="E35">A20</font></i></a>, <a class="txt" href="circuit1.html#E5:32_1"><i><font color="Green" title="E36">A21</font></i></a>, <a class="txt" href="circuit1.html#E7:32_1"><i><font color="Green" title="E37">A22</font></i></a>, <a class="txt" href="circuit1.html#E8:32_1"><i><font color="Green" title="E38">A23</font></i></a>, <a class="txt" href="circuit1.html#E9:32_1"><i><font color="Green" title="E39">A24</font></i></a>, <a class="ref" href="trees_1.html#T79" title="TREES_1:th.79">TREES_1:79</a>, <a class="ref" href="trees_4.html#T3" title="TREES_4:th.3">TREES_4:3</a></i>;<br/><b>hence </b><a NAME="E11:32_1"/>
contradiction
 <b>by </b><i><a class="txt" href="circuit1.html#E5:32"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="circuit1.html#E9:32"><i><font color="Green" title="E20">A5</font></i></a>, <a class="txt" href="circuit1.html#E28:32"><i><font color="Green" title="E30">A15</font></i></a>, <a class="ref" href="circuit1.html#T19" target="_self" title="CIRCUIT1:th.19">Th19</a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c15">o</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="E39:32"/><i><font color="Green" title="E34">A25</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">o</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>and </b><br/><a NAME="E40:32"/><i><font color="Green" title="E35">A26</font></i>: 
 <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c15">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font>
 <b>by </b><i><a class="txt" href="circuit1.html#E36:32"><i><font color="Green" title="E33">A18</font></i></a></i>;<br/>
<a NAME="E41:32"/><i><font color="Green" title="E36">A27</font></i>: 
 <a href="dtconstr.html#K7" title="DTCONSTR:func.7">NonTerminals</a> <span class="p1">(<span class="default"><a href="msafree.html#K5" title="MSAFREE:func.5">DTConMSA</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c1">IIG</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">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#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="msafree.html#T6" title="MSAFREE:th.6">MSAFREE:6</a></i>;<br/>
<a NAME="E42:32"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">IIG</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">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#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c16">o'</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">o</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> as   <a href="dtconstr.html#NM3" title="DTCONSTR:NM.3">NonTerminal</a> of <span class="p1">(<span class="default"><a href="msafree.html#K5" title="MSAFREE:func.5">DTConMSA</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font></span>)</span> <b>by </b><i><a class="txt" href="circuit1.html#E41:32"><i><font color="Green" title="E36">A27</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c17">q</font> being    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K5" title="MSAFREE:func.5">DTConMSA</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">A</font></span>)</span><b> such that </b><br/><a NAME="E45:32"/><i><font color="Green" title="E37">A28</font></i>: 
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">o'</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c17">q</font>
 <b>and </b><br/><a NAME="E46:32"/>
<font color="Maroon" title="c16">o'</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c17">q</font>
 <b>by </b><i><a class="txt" href="circuit1.html#E39:32"><i><font color="Green" title="E34">A25</font></i></a>, <a class="ref" href="dtconstr.html#T10" title="DTCONSTR:th.10">DTCONSTR:10</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">q'</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="E48:32"/><i><font color="Green" title="E38">A29</font></i>: 
<font color="Maroon" title="c17">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">q'</font>
 <b>and </b><br/><a NAME="E49:32"/><i><font color="Green" title="E39">A30</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c14">a</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="c18">q'</font></span>)</span>
 <b>by </b><i><a class="txt" href="circuit1.html#E45:32"><i><font color="Green" title="E37">A28</font></i></a>, <a class="ref" href="trees_4.html#D4" title="TREES_4:def.4">TREES_4:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">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="c20">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E52:32"/><i><font color="Green" title="E40">A31</font></i>: 
<font color="Maroon" title="c20">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="c19">av</font></span>)</span>
 <b>and </b><br/><a NAME="E53:32"/><i><font color="Green" title="E41">A32</font></i>: 
<font color="Maroon" title="c4">v1</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="c19">av</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font>
 <b>by </b><i><a class="txt" href="circuit1.html#E2:32"><i><font color="Green" title="E17">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="c21">k</font> = <font color="Maroon" title="c20">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="circuit1.html#E52:32"><i><font color="Green" title="E40">A31</font></i></a></i>;<br/>
<a NAME="E55:32"/><i><font color="Green" title="E42">A33</font></i>: 
<font color="Maroon" title="c15">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">av</font>
 
<b>by </b><i><a class="txt" href="circuit1.html#E1:32"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="circuit1.html#E40:32"><i><font color="Green" title="E35">A26</font></i></a>, <a class="ref" href="msafree2.html#D7" title="MSAFREE2:def.7">MSAFREE2:def 7</a></i>;<br/>
<a NAME="E56:32"/><b>then </b><i><font color="Green" title="E43">A34</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">q'</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="c19">av</font></span>)</span>
 
<b>by </b><i><a class="txt" href="circuit1.html#E35:32"><i><font color="Green" title="E32">A17</font></i></a>, <a class="txt" href="circuit1.html#E40:32"><i><font color="Green" title="E35">A26</font></i></a>, <a class="txt" href="circuit1.html#E45:32"><i><font color="Green" title="E37">A28</font></i></a>, <a class="txt" href="circuit1.html#E48:32"><i><font color="Green" title="E38">A29</font></i></a>, <a class="ref" href="msafree2.html#T13" title="MSAFREE2:th.13">MSAFREE2:13</a></i>;<br/>
<a NAME="E57:32"/><b>then </b><i><font color="Green" title="E44">A35</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c18">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c19">av</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E58:32"/><i><font color="Green" title="E45">A36</font></i>: 
<font color="Maroon" title="c21">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="c18">q'</font>
 
<b>by </b><i><a class="txt" href="circuit1.html#E52:32"><i><font color="Green" title="E40">A31</font></i></a>, <a class="txt" href="circuit1.html#E56:32"><i><font color="Green" title="E43">A34</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="c22">k1</font> = <font color="Maroon" title="c21">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</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="circuit1.html#E52:32"><i><font color="Green" title="E40">A31</font></i></a>, <a class="ref" href="finseq_3.html#T28" title="FINSEQ_3:th.28">FINSEQ_3:28</a></i>;<br/>
<a NAME="E60:32"/><i><font color="Green" title="E46">A37</font></i>: 
<font color="Maroon" title="c22">k1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">k</font>
 
;<br/>
<a NAME="E61:32"/><b>then </b><i><font color="Green" title="E47">A38</font></i>: 
<span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c22">k1</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c18">q'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="circuit1.html#E52:32"><i><font color="Green" title="E40">A31</font></i></a>, <a class="txt" href="circuit1.html#E57:32"><i><font color="Green" title="E44">A35</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="c23">f</font> = <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c22">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>consider </b><font color="Maroon" title="c24">qq</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="E64:32"/><i><font color="Green" title="E48">A39</font></i>: 
<font color="Maroon" title="c7">t</font> <a href="trees_2.html#K8" title="TREES_2:func.8">with-replacement</a> <font color="Maroon" title="c23">f</font>,<font color="Maroon" title="c10">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">o'</font> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c24">qq</font>
 <b>and </b><br/><a NAME="E65:32"/><i><font color="Green" title="E49">A40</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">qq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">q'</font>
 <b>and </b><br/><a NAME="E66:32"/>
<font color="Maroon" title="c24">qq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">k1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">t1</font>
 <b>and </b><br/><a NAME="E67:32"/>
for <font color="Olive" title="b1">i</font> being   <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>   st <font color="Olive" title="b1">i</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="c18">q'</font> &amp; <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c22">k1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 holds <br/><font color="Maroon" title="c24">qq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">q'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 <b>by </b><i><a class="txt" href="circuit1.html#E35:32"><i><font color="Green" title="E32">A17</font></i></a>, <a class="txt" href="circuit1.html#E45:32"><i><font color="Green" title="E37">A28</font></i></a>, <a class="txt" href="circuit1.html#E48:32"><i><font color="Green" title="E38">A29</font></i></a>, <a class="txt" href="circuit1.html#E49:32"><i><font color="Green" title="E39">A30</font></i></a>, <a class="txt" href="circuit1.html#E61:32"><i><font color="Green" title="E47">A38</font></i></a>, <a class="ref" href="pre_circ.html#T19" title="PRE_CIRC:th.19">PRE_CIRC:19</a></i>;<br/>
<a NAME="E68:32"/><i><font color="Green" title="E50">A41</font></i>: 
<font color="Maroon" title="c18">q'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">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">v1</font>
 
<b>by </b><i><a class="txt" href="circuit1.html#E35:32"><i><font color="Green" title="E32">A17</font></i></a>, <a class="txt" href="circuit1.html#E40:32"><i><font color="Green" title="E35">A26</font></i></a>, <a class="txt" href="circuit1.html#E45:32"><i><font color="Green" title="E37">A28</font></i></a>, <a class="txt" href="circuit1.html#E48:32"><i><font color="Green" title="E38">A29</font></i></a>, <a class="txt" href="circuit1.html#E52:32"><i><font color="Green" title="E40">A31</font></i></a>, <a class="txt" href="circuit1.html#E53:32"><i><font color="Green" title="E41">A32</font></i></a>, <a class="txt" href="circuit1.html#E55:32"><i><font color="Green" title="E42">A33</font></i></a>, <a class="ref" href="msafree2.html#T14" title="MSAFREE2:th.14">MSAFREE2:14</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c25">tft</font> = <font color="Maroon" title="c7">t</font> <a href="trees_2.html#K8" title="TREES_2:func.8">with-replacement</a> <font color="Maroon" title="c23">f</font>,<font color="Maroon" title="c10">t1</font> as   <a href="trees_2.html#NM4" title="TREES_2:NM.4">DecoratedTree</a> ;<br/>
<a NAME="E70:32"/>
<font color="Maroon" title="c7">t</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="c19">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="c18">q'</font>
 
<b>by </b><i><a class="txt" href="circuit1.html#E1:32"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="circuit1.html#E35:32"><i><font color="Green" title="E32">A17</font></i></a>, <a class="txt" href="circuit1.html#E40:32"><i><font color="Green" title="E35">A26</font></i></a>, <a class="txt" href="circuit1.html#E45:32"><i><font color="Green" title="E37">A28</font></i></a>, <a class="txt" href="circuit1.html#E48:32"><i><font color="Green" title="E38">A29</font></i></a>, <a class="ref" href="msafree2.html#D7" title="MSAFREE2:def.7">MSAFREE2:def 7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c26">tft</font> = <font color="Maroon" title="c25">tft</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">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="circuit1.html#E29:32"><i><font color="Green" title="E31">A16</font></i></a>, <a class="txt" href="circuit1.html#E58:32"><i><font color="Green" title="E45">A36</font></i></a>, <a class="txt" href="circuit1.html#E60:32"><i><font color="Green" title="E46">A37</font></i></a>, <a class="txt" href="circuit1.html#E68:32"><i><font color="Green" title="E50">A41</font></i></a>, <a class="ref" href="circuit1.html#T7" target="_self" title="CIRCUIT1:th.7">Th7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c27">dtft</font> =  <a href="circuit1.html#K5" title="CIRCUIT1:func.5">depth</a> <font color="Maroon" title="c26">tft</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<a NAME="E73:32"/>
<font color="Maroon" title="c27">dtft</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font>
 
<b>by </b><i><a class="txt" href="circuit1.html#E4:32"><i><font color="Green" title="E18">A3</font></i></a></i>;<br/>
<a NAME="E74:32"/><b>then </b><i><font color="Green" title="E51">A42</font></i>: 
<font color="Maroon" title="c27">dtft</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="circuit1.html#K5" title="CIRCUIT1:func.5">depth</a> <font color="Maroon" title="c7">t</font>
 
<b>by </b><i><a class="txt" href="circuit1.html#E9:32"><i><font color="Green" title="E20">A5</font></i></a>, <a class="ref" href="pre_circ.html#D1" title="PRE_CIRC:def.1">PRE_CIRC:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c28">e'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <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="E76:32"/><i><font color="Green" title="E52">A43</font></i>: 
<font color="Maroon" title="c26">tft</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">e'</font>
 <b>and </b><br/><a NAME="E77:32"/><i><font color="Green" title="E53">A44</font></i>: 
 <a href="circuit1.html#K5" title="CIRCUIT1:func.5">depth</a> <font color="Maroon" title="c26">tft</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msafree2.html#K9" title="MSAFREE2:func.9">depth</a> <font color="Maroon" title="c28">e'</font>
 <b>by </b><i><a class="ref" href="circuit1.html#D5" target="_self" title="CIRCUIT1:def.5">Def5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c29">dttft</font> being  <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>, <font color="Maroon" title="c30">ttft</font> being  <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><b> such that </b><br/><a NAME="E79:32"/><i><font color="Green" title="E54">A45</font></i>: 
<font color="Maroon" title="c29">dttft</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">e'</font>
 <b>and </b><br/><a NAME="E80:32"/><i><font color="Green" title="E55">A46</font></i>: 
<font color="Maroon" title="c30">ttft</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c29">dttft</font>
 <b>and </b><br/><a NAME="E81:32"/><i><font color="Green" title="E56">A47</font></i>: 
 <a href="msafree2.html#K9" title="MSAFREE2:func.9">depth</a> <font color="Maroon" title="c28">e'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <font color="Maroon" title="c30">ttft</font>
 <b>by </b><i><a class="ref" href="msafree2.html#D14" title="MSAFREE2:def.14">MSAFREE2:def 14</a></i>;<br/>
<a NAME="E82:32"/><i><font color="Green" title="E57">A48</font></i>: 
<font color="Maroon" title="c21">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="c24">qq</font>
 
<b>by </b><i><a class="txt" href="circuit1.html#E52:32"><i><font color="Green" title="E40">A31</font></i></a>, <a class="txt" href="circuit1.html#E56:32"><i><font color="Green" title="E43">A34</font></i></a>, <a class="txt" href="circuit1.html#E65:32"><i><font color="Green" title="E49">A40</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c31">qq'</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="E84:32"/><i><font color="Green" title="E58">A49</font></i>: 
<font color="Maroon" title="c24">qq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">qq'</font>
 <b>and </b><br/><a NAME="E85:32"/><i><font color="Green" title="E59">A50</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c26">tft</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="c31">qq'</font></span>)</span>
 <b>by </b><i><a class="txt" href="circuit1.html#E64:32"><i><font color="Green" title="E48">A39</font></i></a>, <a class="ref" href="trees_4.html#D4" title="TREES_4:def.4">TREES_4:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c32">f'</font> = <font color="Maroon" title="c23">f</font> as    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c30">ttft</font> <b>by </b><i><a class="txt" href="circuit1.html#E60:32"><i><font color="Green" title="E46">A37</font></i></a>, <a class="txt" href="circuit1.html#E76:32"><i><font color="Green" title="E52">A43</font></i></a>, <a class="txt" href="circuit1.html#E79:32"><i><font color="Green" title="E54">A45</font></i></a>, <a class="txt" href="circuit1.html#E80:32"><i><font color="Green" title="E55">A46</font></i></a>, <a class="txt" href="circuit1.html#E82:32"><i><font color="Green" title="E57">A48</font></i></a>, <a class="txt" href="circuit1.html#E84:32"><i><font color="Green" title="E58">A49</font></i></a>, <a class="txt" href="circuit1.html#E85:32"><i><font color="Green" title="E59">A50</font></i></a>, <a class="ref" href="pre_circ.html#T17" title="PRE_CIRC:th.17">PRE_CIRC:17</a></i>;<br/>
<a NAME="E87:32"/><i><font color="Green" title="E60">A51</font></i>: 
<font color="Maroon" title="c32">f'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<a NAME="E88:32"/><i><font color="Green" title="E61">A52</font></i>: 
<font color="Maroon" title="c32">f'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">t</font>
 
<b>by </b><i><a class="txt" href="circuit1.html#E35:32"><i><font color="Green" title="E32">A17</font></i></a>, <a class="txt" href="circuit1.html#E49:32"><i><font color="Green" title="E39">A30</font></i></a>, <a class="txt" href="circuit1.html#E58:32"><i><font color="Green" title="E45">A36</font></i></a>, <a class="txt" href="circuit1.html#E60:32"><i><font color="Green" title="E46">A37</font></i></a>, <a class="ref" href="pre_circ.html#T17" title="PRE_CIRC:th.17">PRE_CIRC:17</a></i>;<br/>
<a NAME="E89:32"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c26">tft</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">t</font></span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> <font color="Maroon" title="c23">f</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">t1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="circuit1.html#E35:32"><i><font color="Green" title="E32">A17</font></i></a>, <a class="txt" href="circuit1.html#E49:32"><i><font color="Green" title="E39">A30</font></i></a>, <a class="txt" href="circuit1.html#E61:32"><i><font color="Green" title="E47">A38</font></i></a>, <a class="ref" href="trees_2.html#D12" title="TREES_2:def.12">TREES_2:def 12</a></i>;<br/>
<a NAME="E90:32"/><b>then </b>
<font color="Maroon" title="c30">ttft</font> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c23">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">t1'</font>
 
<b>by </b><i><a class="txt" href="circuit1.html#E18:32"><i><font color="Green" title="E24">A9</font></i></a>, <a class="txt" href="circuit1.html#E21:32"><i><font color="Green" title="E26">A11</font></i></a>, <a class="txt" href="circuit1.html#E22:32"><i><font color="Green" title="E27">A12</font></i></a>, <a class="txt" href="circuit1.html#E76:32"><i><font color="Green" title="E52">A43</font></i></a>, <a class="txt" href="circuit1.html#E79:32"><i><font color="Green" title="E54">A45</font></i></a>, <a class="txt" href="circuit1.html#E80:32"><i><font color="Green" title="E55">A46</font></i></a>, <a class="txt" href="circuit1.html#E88:32"><i><font color="Green" title="E61">A52</font></i></a>, <a class="ref" href="trees_1.html#T66" title="TREES_1:th.66">TREES_1:66</a></i>;<br/>
<a NAME="E91:32"/><b>then </b>
 <a href="circuit1.html#K5" title="CIRCUIT1:func.5">depth</a> <font color="Maroon" title="c10">t1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="circuit1.html#K5" title="CIRCUIT1:func.5">depth</a> <font color="Maroon" title="c26">tft</font>
 
<b>by </b><i><a class="txt" href="circuit1.html#E19:32"><i><font color="Green" title="E25">A10</font></i></a>, <a class="txt" href="circuit1.html#E23:32"><i><font color="Green" title="E28">A13</font></i></a>, <a class="txt" href="circuit1.html#E77:32"><i><font color="Green" title="E53">A44</font></i></a>, <a class="txt" href="circuit1.html#E81:32"><i><font color="Green" title="E56">A47</font></i></a>, <a class="txt" href="circuit1.html#E87:32"><i><font color="Green" title="E60">A51</font></i></a>, <a class="ref" href="trees_1.html#T85" title="TREES_1:th.85">TREES_1:85</a></i>;<br/>
<b>hence </b><a NAME="E92:32"/>
 <a href="circuit1.html#K6" title="CIRCUIT1:func.6">depth</a> <font color="Maroon" title="c4">v1</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#K6" title="CIRCUIT1:func.6">depth</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="circuit1.html#E5:32"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="circuit1.html#E9:32"><i><font color="Green" title="E20">A5</font></i></a>, <a class="txt" href="circuit1.html#E12:32"><i><font color="Green" title="E22">A7</font></i></a>, <a class="txt" href="circuit1.html#E16:32"><i><font color="Green" title="E23">A8</font></i></a>, <a class="txt" href="circuit1.html#E74:32"><i><font color="Green" title="E51">A42</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


</div>
