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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="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> ; <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">SCS</font> being <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/> for <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">w</font> being  <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c1">IIG</font><br/> for <font color="Olive" title="b4">e1</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="Olive" title="b1">SCS</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">v</font><br/> for <font color="Olive" title="b5">q1</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>  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="b4">e1</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"><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>,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="Olive" title="b5">q1</font> holds <br/>for <font color="Olive" title="b6">k</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="b6">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="Olive" title="b5">q1</font> &amp; <font color="Olive" title="b5">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">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="Olive" title="b1">SCS</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">w</font> holds <br/><font color="Olive" title="b3">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> <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> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b6">k</font></span><br/><b>let </b><font color="Maroon" title="c2">SCS</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>; <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="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c1">IIG</font><br/> for <font color="Olive" title="b3">e1</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">SCS</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v</font><br/> for <font color="Olive" title="b4">q1</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>  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="b3">e1</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"><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>,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="Olive" title="b4">q1</font> holds <br/>for <font color="Olive" title="b5">k</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="b5">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="Olive" title="b4">q1</font> &amp; <font color="Olive" title="b4">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">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">SCS</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">w</font> holds <br/><font color="Olive" title="b2">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> <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> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b5">k</font></span><br/><b>let </b><font color="Maroon" title="c3">v</font>, <font color="Maroon" title="c4">w</font> be   <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</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">e1</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">SCS</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font><br/> for <font color="Olive" title="b2">q1</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>  st <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="Olive" title="b1">e1</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"><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>,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="Olive" title="b2">q1</font> holds <br/>for <font color="Olive" title="b3">k</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="b3">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="Olive" title="b2">q1</font> &amp; <font color="Olive" title="b2">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">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">SCS</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">w</font> holds <br/><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> <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 href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">k</font></span><br/><b>let </b><font color="Maroon" title="c5">e1</font> be    <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">SCS</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</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">q1</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>  st <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="c5">e1</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"><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>,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="Olive" title="b1">q1</font> holds <br/>for <font color="Olive" title="b2">k</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="b2">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="Olive" title="b1">q1</font> &amp; <font color="Olive" title="b1">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">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">SCS</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">w</font> holds <br/><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> <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 href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">k</font></span><br/><b>let </b><font color="Maroon" title="c6">q1</font> be  <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>; <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="c5">e1</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"><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>,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="c6">q1</font> implies for <font color="Olive" title="b1">k</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">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="c6">q1</font> &amp; <font color="Maroon" title="c6">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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">SCS</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">w</font> holds <br/><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> <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 href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font> )</span><br/>









<b>assume </b><b>that </b><br/><a NAME="E1:11"/><i><font color="Green" title="E4">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="E2:11"/><i><font color="Green" title="E5">A2</font></i>: 
<font color="Maroon" title="c5">e1</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"><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>,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="c6">q1</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">k</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">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="c6">q1</font> &amp; <font color="Maroon" title="c6">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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">SCS</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">w</font> holds <br/><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> <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 href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font></span><br/>

<b>thus </b><a NAME="E3:11"/>
for <font color="Olive" title="b1">k</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">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="c6">q1</font> &amp; <font color="Maroon" title="c6">q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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">SCS</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">w</font> holds <br/><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> <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 href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">k</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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="c6">q1</font> &amp; <font color="Maroon" title="c6">q1</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#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">SCS</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">w</font> implies <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> <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 href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">k</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:11_1"/><i><font color="Green" title="E6">A3</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="c6">q1</font>
 <b>and </b><br/><a NAME="E2:11_1"/><i><font color="Green" title="E7">A4</font></i>: 
<font color="Maroon" title="c6">q1</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#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">SCS</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">w</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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> <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 href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">k</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c8">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/>
<a NAME="E4:11_1"/>
<font color="Maroon" title="c5">e1</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">SCS</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">v</font>
 
;<br/>
<a NAME="E5:11_1"/><b>then </b><i><font color="Green" title="E8">A5</font></i>: 
<font color="Maroon" title="c5">e1</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">SCS</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="c8">av</font></span>)</span>
 
<b>by </b><i><a class="txt" href="circuit1.html#E1:11"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="msafree2.html#D7" title="MSAFREE2:def.7">MSAFREE2:def 7</a></i>;<br/>
<a NAME="E6:11_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">q1</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="c8">av</font></span>)</span>
 
<b>by </b><i><a class="txt" href="circuit1.html#E2:11"><i><font color="Green" title="E5">A2</font></i></a>, <a class="ref" href="msafree2.html#T13" title="MSAFREE2:th.13">MSAFREE2:13</a></i>;<br/>
<a NAME="E7:11_1"/><b>then </b><i><font color="Green" title="E9">A6</font></i>: 
<font color="Maroon" title="c7">k</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="c8">av</font></span>)</span>
 
<b>by </b><i><a class="txt" href="circuit1.html#E1:11_1"><i><font color="Green" title="E6">A3</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E8:11_1"/><b>then </b><i><font color="Green" title="E10">A7</font></i>: 
<font color="Maroon" title="c6">q1</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#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">SCS</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="c8">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="circuit1.html#E2:11"><i><font color="Green" title="E5">A2</font></i></a>, <a class="txt" href="circuit1.html#E5:11_1"><i><font color="Green" title="E8">A5</font></i></a>, <a class="ref" href="msafree2.html#T14" title="MSAFREE2:th.14">MSAFREE2:14</a></i>;<br/>
<a NAME="E9:11_1"/><i><font color="Green" title="E11">A8</font></i>: 
 <a href="msafree2.html#K6" title="MSAFREE2:func.6">FreeEnv</a> <font color="Maroon" title="c2">SCS</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">SCS</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">SCS</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="E10:11_1"/><b>then </b><i><font color="Green" title="E12">A9</font></i>: 
<font color="Maroon" title="c6">q1</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> <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">SCS</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="c8">av</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="circuit1.html#E7:11_1"><i><font color="Green" title="E9">A6</font></i></a>, <a class="txt" href="circuit1.html#E8:11_1"><i><font color="Green" title="E10">A7</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_1_1"/>
<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="c8">av</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">w</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="E2:11_1_1"/><b>then </b>
<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">SCS</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="c8">av</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</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">SCS</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">w</font>
 <b>by </b><i><a class="ref" href="msafree.html#T13" title="MSAFREE:th.13">MSAFREE:13</a></i>;<br/><a NAME="E3:11_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<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">SCS</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c8">av</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">k</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<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">SCS</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">w</font></span>)</span> <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="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E4:11_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="circuit1.html#E2:11_1"><i><font color="Green" title="E7">A4</font></i></a>, <a class="txt" href="circuit1.html#E9:11_1"><i><font color="Green" title="E11">A8</font></i></a>, <a class="txt" href="circuit1.html#E10:11_1"><i><font color="Green" title="E12">A9</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</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>
<b>hence </b><a NAME="E12:11_1"/>
<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> <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 href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">k</font>
 ; <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>


</div>
