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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="freealg.html#V1" title="FREEALG:attr.1">disjoint_with_NAT</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">S</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>reconsider </b><font color="Maroon" title="c3">S'</font> = <font color="Maroon" title="c2">S</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  ;<br/>
<b>set </b><font color="Maroon" title="c4">G</font> =  <a href="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">X</font>;<br/>
<b>set </b><font color="Maroon" title="c5">A</font> =  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">X</font>;<br/>
<b>let </b><font color="Maroon" title="c6">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E2:36"/><i><font color="Green" title="E20">A1</font></i>: 
<font color="Maroon" title="c6">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="c2">S</font>
 ;<br/>

<a NAME="E3:36"/><b>then </b><i><font color="Green" title="E21">A2</font></i>: 
<font color="Maroon" title="c3">S'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 
<b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<b>let </b><font color="Maroon" title="c7">p</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">X</font></span>)</span>;<br/>

<b>assume </b><a NAME="E4:36"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 ;<br/>

<a NAME="E5:36"/><b>then </b>
<font color="Maroon" title="c7">p</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">S'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p2">(<span class="default"><a href="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">X</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="aofa_000.html#E3:36"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></i>;<br/>
<a NAME="E6:36"/><b>then </b>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">S'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p2">(<span class="default"><a href="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">X</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E7:36"/><b>then </b><i><font color="Green" title="E22">A3</font></i>: 
<font color="Maroon" title="c7">p</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="freealg.html#K5" title="FREEALG:func.5">FreeOpNSG</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="aofa_000.html#E2:36"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="freealg.html#D11" title="FREEALG:def.11">FREEALG:def 11</a></i>;<br/>
<a NAME="E8:36"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p1">(<span class="default"><a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">X</font></span>)</span> <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="c2">S</font>
 
<b>by </b><i><a class="ref" href="freealg.html#D12" title="FREEALG:def.12">FREEALG:def 12</a></i>;<br/>
<a NAME="E9:36"/><b>then </b>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p1">(<span class="default"><a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">S</font>
 
<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="E10:36"/><b>then </b>
 <a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <font color="Maroon" title="c6">i</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p2">(<span class="default"><a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i</font>
 
<b>by </b><i><a class="txt" href="aofa_000.html#E2:36"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <span class="p2">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <font color="Maroon" title="c6">i</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p4">(<span class="default"><a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">X</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">p</font> = 
<span class="p1">(<span class="default"><a href="freealg.html#K5" title="FREEALG:func.5">FreeOpNSG</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">p</font>
<b>by </b><i><a class="txt" href="aofa_000.html#E2:36"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="freealg.html#D12" title="FREEALG:def.12">FREEALG:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="freealg.html#K4" title="FREEALG:func.4">Sym</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c7">p</font>
<b>by </b><i><a class="txt" href="aofa_000.html#E2:36"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="aofa_000.html#E7:36"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="freealg.html#D11" title="FREEALG:def.11">FREEALG:def 11</a></i>
<br/>.= 
<font color="Maroon" title="c6">i</font> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c7">p</font>
<b>by </b><i><a class="txt" href="aofa_000.html#E2:36"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="freealg.html#D10" title="FREEALG:def.10">FREEALG:def 10</a></i>
;<br/><br/>


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