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<b>let </b><font color="Maroon" title="c1">f</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="relat_1.html#V2" title="RELAT_1:attr.2">with_zero</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>let </b><font color="Maroon" title="c2">D</font> be  <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>set </b><font color="Maroon" title="c3">fa</font> =  <a href="freealg.html#K15" title="FREEALG:func.15">FreeUnivAlgZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font>;<br/>
<b>set </b><font color="Maroon" title="c4">A</font> =  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span>;<br/>
<a NAME="E1:40"/><i><font color="Green" title="E23">A1</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <span class="p2">(<span class="default"><a href="freealg.html#K15" title="FREEALG:func.15">FreeUnivAlgZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span></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> 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#K15" title="FREEALG:func.15">FreeUnivAlgZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span>
 
<b>by </b><i><a class="ref" href="unialg_1.html#D11" title="UNIALG_1:def.11">UNIALG_1:def 11</a></i>;<br/>
<a NAME="E2:40"/><i><font color="Green" title="E24">A2</font></i>: 
 <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#K15" title="FREEALG:func.15">FreeUnivAlgZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</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="c1">f</font>
 
<b>by </b><i><a class="ref" href="freealg.html#D18" target="_self" title="FREEALG:def.18">Def18</a></i>;<br/>
<a NAME="E3:40"/><b>then </b><i><font color="Green" title="E25">X</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <span class="p2">(<span class="default"><a href="freealg.html#K15" title="FREEALG:func.15">FreeUnivAlgZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="freealg.html#E1:40"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:40_1"/>
<font color="Maroon" title="c5">n</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> <span class="p1">(<span class="default"><a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <span class="p2">(<span class="default"><a href="freealg.html#K15" title="FREEALG:func.15">FreeUnivAlgZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:40_1"/><b>then </b><i><font color="Green" title="E26">A3</font></i>: 
( <font color="Maroon" title="c5">n</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="c1">f</font> &amp; <font color="Maroon" title="c5">n</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> <span class="p1">(<span class="default"><a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <span class="p2">(<span class="default"><a href="freealg.html#K15" title="FREEALG:func.15">FreeUnivAlgZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c5">n</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> <span class="p1">(<span class="default"><a href="freealg.html#K14" title="FREEALG:func.14">FreeOpSeqZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> )
 <b>by </b><i><a class="txt" href="freealg.html#E2:40"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="txt" href="freealg.html#E3:40"><i><font color="Green" title="E25">X</font></i></a></i>;<br/><a NAME="E3:40_1"/><b>then </b><i><font color="Green" title="E27">A4</font></i>: 
( 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#K15" title="FREEALG:func.15">FreeUnivAlgZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="freealg.html#K13" title="FREEALG:func.13">FreeOpZAO</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font> &amp;  <a href="freealg.html#K13" title="FREEALG:func.13">FreeOpZAO</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a> <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a> <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <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="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> , <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> )
 <b>by </b><i><a class="ref" href="freealg.html#D18" target="_self" title="FREEALG:def.18">Def18</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">h</font> =  <a href="freealg.html#K13" title="FREEALG:func.13">FreeOpZAO</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a> <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a> <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="freealg.html#K15" title="FREEALG:func.15">FreeUnivAlgZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="freealg.html#K15" title="FREEALG:func.15">FreeUnivAlgZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> ;<br/><a NAME="E5:40_1"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">n</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="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c6">h</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="freealg.html#K15" title="FREEALG:func.15">FreeUnivAlgZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> )
 <b>by </b><i><a class="txt" href="freealg.html#E2:40_1"><i><font color="Green" title="E26">A3</font></i></a>, <a class="ref" href="freealg.html#D17" target="_self" title="FREEALG:def.17">Def17</a>, <a class="ref" href="unialg_2.html#T2" title="UNIALG_2:th.2">UNIALG_2:2</a></i>;<br/><a NAME="E6:40_1"/><b>then </b><i><font color="Green" title="E28">A5</font></i>: 
 <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c6">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="ref" href="pralg_1.html#T1" title="PRALG_1:th.1">PRALG_1:1</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <span class="p2">(<span class="default"><a href="freealg.html#K15" title="FREEALG:func.15">FreeUnivAlgZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> = 
 <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c6">h</font>
<b>by </b><i><a class="txt" href="freealg.html#E2:40_1"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="freealg.html#E3:40_1"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="unialg_1.html#D11" title="UNIALG_1:def.11">UNIALG_1:def 11</a></i>
<br/>.= 
<font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font>
<b>by </b><i><a class="txt" href="freealg.html#E2:40_1"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="freealg.html#E6:40_1"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:40"/>
 <a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <span class="p1">(<span class="default"><a href="freealg.html#K15" title="FREEALG:func.15">FreeUnivAlgZAO</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font>
 <b>by </b><i><a class="txt" href="freealg.html#E1:40"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="freealg.html#E2:40"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></i>;<br/>


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