<?xml version="1.0"?>
<div class="add">

<span class="lab"><font color="Green" title="E37">A1</font></span>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="card_3.html#K12" title="CARD_3:func.12">proj</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span>,<font color="Maroon" title="c2">i</font></span>)</span> = 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D17" title="CARD_3:def.17">CARD_3:def 17</a></span>
<br/>.= 
<font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_6.html#T9" title="FUNCT_6:th.9">FUNCT_6:9</a></span>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c3">P</font> =  <a href="card_3.html#K12" title="CARD_3:func.12">proj</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span>,<font color="Maroon" title="c2">i</font>;<br/><a NAME="E1:71_1_1_2"/><span class="lab"><font color="Green" title="E38">A2</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="card_3.html#K12" title="CARD_3:func.12">proj</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="card_3.html#K12" title="CARD_3:func.12">proj</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span>,<font color="Maroon" title="c2">i</font></span>)</span> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:71_1_1_2_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="card_3.html#K12" title="CARD_3:func.12">proj</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span>,<font color="Maroon" title="c2">i</font></span>)</span>
 ; <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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">d</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:71_1_1_2_1"/><span class="lab"><font color="Green" title="E39">A3</font></span>: 
<font color="Maroon" title="c5">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="card_3.html#K12" title="CARD_3:func.12">proj</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span>,<font color="Maroon" title="c2">i</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:71_1_1_2_1"/><span class="lab"><font color="Green" title="E40">A4</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_3.html#K12" title="CARD_3:func.12">proj</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">d</font> = <font color="Maroon" title="c5">d</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71_1_1"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="txt" href="#E3:71_1_1_2_1"><span class="lab"><font color="Green" title="E39">A3</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">d</font> = <font color="Maroon" title="c6">d</font> as    <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/>
<a NAME="E7:71_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="card_3.html#K12" title="CARD_3:func.12">proj</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:71_1_1_2_1"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="ref" href="card_3.html#D17" title="CARD_3:def.17">CARD_3:def 17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:71_1_1_2_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:71_1_1_2_1"><span class="lab"><font color="Green" title="E40">A4</font></span></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/>


</div><span class="kw">end;</span></div><a NAME="E2:71_1_1_2"/><span class="lab"><font color="Green" title="E39">A5</font></span>: 
 <a href="card_3.html#K12" title="CARD_3:func.12">proj</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span>,<font color="Maroon" title="c2">i</font> is  <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:71_1_1_2_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="card_3.html#K12" title="CARD_3:func.12">proj</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span>,<font color="Maroon" title="c2">i</font></span>)</span> is  <a href="card_3.html#V2" title="CARD_3:attr.2">with_common_domain</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71_1_1"><span class="lab"><font color="Green" title="E37">A1</font></span></a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="unialg_1.html#D1" title="UNIALG_1:def.1">UNIALG_1:def 1</a> <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">reconsider </span><font color="Maroon" title="c4">P</font> =  <a href="card_3.html#K12" title="CARD_3:func.12">proj</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span>,<font color="Maroon" title="c2">i</font> as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71_1_1"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="txt" href="#E1:71_1_1_2"><span class="lab"><font color="Green" title="E38">A2</font></span></a>, <a class="ref" href="finseq_2.html#T162" title="FINSEQ_2:th.162">FINSEQ_2:162</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></span>;<br/><a NAME="E4:71_1_1_2"/>
<font color="Maroon" title="c4">P</font> is    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="rfunct_3.html#K4" title="RFUNCT_3:func.4">PFuncs</a> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T119" title="PARTFUN1:th.119">PARTFUN1:119</a></span>;<br/><a NAME="E5:71_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="comput_1.html#K2" title="COMPUT_1:func.2">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71_1_1_2"><span class="lab"><font color="Green" title="E39">A5</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:71_1_1_2"/>
 <a href="card_3.html#K12" title="CARD_3:func.12">proj</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span>,<font color="Maroon" title="c2">i</font> is    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="comput_1.html#K2" title="COMPUT_1:func.2">HFuncs</a> <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"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:71_1_1"/>
 <a href="card_3.html#K12" title="CARD_3:func.12">proj</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span>,<font color="Maroon" title="c2">i</font> is  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>   <a href="comput_1.html#NV2" title="COMPUT_1:NV.2">to-naturals</a>   <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>
 ; <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>
