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<span class="kw">let </span><font color="Maroon" title="c1">f</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">D</font> being 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>  holds   <a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <span class="p1">(<span class="default"><a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Olive" title="b1">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">D</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <span class="p1">(<span class="default"><a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</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></span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">fa</font> =  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font>;<br/>
<span class="kw">set </span><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="E1:28"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
 <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="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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="freealg.html#D12" target="_self" title="FREEALG:def.12">Def12</a></span>;<br/>
<a NAME="E2:28"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
 <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#K7" title="FREEALG:func.7">FreeUnivAlgNSG</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#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_1.html#D11" title="UNIALG_1:def.11">UNIALG_1:def 11</a></span>;<br/>
<a NAME="E3:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A3</font></span>: 
 <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#K7" title="FREEALG:func.7">FreeUnivAlgNSG</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> implies <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#K7" title="FREEALG:func.7">FreeUnivAlgNSG</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c6">h</font> =  <a href="freealg.html#K5" title="FREEALG:func.5">FreeOpNSG</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 <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</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> </span>)</span>,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#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> ;<br/><a NAME="E2:28_1"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#T2" title="UNIALG_2:th.2">UNIALG_2:2</a></span>;<br/><span class="kw">assume </span><a NAME="E3:28_1"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
<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#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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#K7" title="FREEALG:func.7">FreeUnivAlgNSG</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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></span><br/><a NAME="E4:28_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A6</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E5:28_1"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="freealg.html#D11" target="_self" title="FREEALG:def.11">Def11</a></span>;<br/><a NAME="E6:28_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A7</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="finseq_2.html#T130" title="FINSEQ_2:th.130">FINSEQ_2:130</a></span>;<br/><a NAME="E7:28_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="freealg.html#K6" title="FREEALG:func.6">FreeOpSeqNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E3:28_1"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E8:28_1"/><span class="kw">then </span>
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="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#K5" title="FREEALG:func.5">FreeOpNSG</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="freealg.html#D12" target="_self" title="FREEALG:def.12">Def12</a></span>;<br/><span class="kw">hence </span><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#K7" title="FREEALG:func.7">FreeUnivAlgNSG</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28_1"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="ref" href="unialg_1.html#D11" title="UNIALG_1:def.11">UNIALG_1:def 11</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28_1"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="txt" href="#E6:28_1"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:28"/>
 <a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <span class="p1">(<span class="default"><a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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