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<span class="kw">let </span><font color="Maroon" title="c1">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">e1</font>, <font color="Olive" title="b2">e2</font> being  <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="margrel1.html#V2" title="MARGREL1:attr.2">homogeneous</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><br/>  for <font color="Olive" title="b3">p</font>, <font color="Olive" title="b4">q</font> being    <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>  holds   <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Olive" title="b1">e1</font>,<font color="Olive" title="b2">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b3">p</font>) <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a>  <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Olive" title="b1">e1</font>,<font color="Olive" title="b2">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b4">q</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">e1</font>, <font color="Maroon" title="c3">e2</font> be  <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="margrel1.html#V2" title="MARGREL1:attr.2">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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being    <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>  holds   <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b1">p</font>) <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a>  <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b2">q</font>)</span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">f1</font> = <font color="Maroon" title="c2">e1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">f2</font> = <font color="Maroon" title="c3">e2</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">p1</font>, <font color="Maroon" title="c7">p2</font> be    <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="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c6">p1</font>) <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a>  <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c7">p2</font>)</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:91_1"/>
(  not <font color="Maroon" title="c1">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="c6">p1</font> or  not <font color="Maroon" title="c1">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="c7">p2</font> or ( <font color="Maroon" title="c1">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="c6">p1</font> &amp; <font color="Maroon" title="c1">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="c7">p2</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1"><span class="kw">suppose </span></a><a NAME="E1:91_1_1"/>
(  not <font color="Maroon" title="c1">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="c6">p1</font> or  not <font color="Maroon" title="c1">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="c7">p2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c6">p1</font>) <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a>  <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c7">p2</font>)</span><br/><div class="add"><a NAME="E2:91_1_1"/><span class="kw">then </span>
(  <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c6">p1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c7">p2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="comput_1.html#T50" target="_self" title="COMPUT_1:th.50">Th54</a></span>;<br/><span class="kw">hence </span><a NAME="E3:91_1_1"/>
 <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c6">p1</font>) <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a>  <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c7">p2</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T59" title="PARTFUN1:th.59">PARTFUN1:59</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_2"><span class="kw">suppose </span></a><a NAME="E1:91_1_2"/><span class="lab"><font color="Green" title="E54">A1</font></span>: 
( <font color="Maroon" title="c1">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="c6">p1</font> &amp; <font color="Maroon" title="c1">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="c7">p2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c6">p1</font>) <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a>  <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c7">p2</font>)</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c8">m</font> = <font color="Maroon" title="c6">p1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c9">n</font> = <font color="Maroon" title="c7">p2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><span class="kw">consider </span><font color="Maroon" title="c10">F2</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<span class="p1">(<span class="default"><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>)</span><span class="kw"> such that </span><br/><a NAME="E4:91_1_2"/><span class="lab"><font color="Green" title="E55">A2</font></span>: 
 <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c7">p2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">i</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:91_1_2"/><span class="lab"><font color="Green" title="E56">A3</font></span>: 
( <font color="Maroon" title="c1">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="c7">p2</font> &amp;  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">i</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">e1</font> implies <font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">e1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">i</font>)</span>)</span></span>)</span> )
 <span class="kw">and </span><br/><a NAME="E6:91_1_2"/><span class="lab"><font color="Green" title="E57">A4</font></span>: 
( (  not <font color="Maroon" title="c1">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="c7">p2</font> or  not  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">i</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">e1</font> ) implies <font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <span class="kw">and </span><br/><a NAME="E7:91_1_2"/><span class="lab"><font color="Green" title="E58">A5</font></span>: 
 for <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>,<font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">m</font>,<font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c3">e2</font>]
 <span class="kw">by</span> <span class="lab"><a class="ref" href="comput_1.html#D10" target="_self" title="COMPUT_1:def.10">Def13</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c11">F1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<span class="p1">(<span class="default"><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>)</span><span class="kw"> such that </span><br/><a NAME="E9:91_1_2"/><span class="lab"><font color="Green" title="E59">A6</font></span>: 
 <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c6">p1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">i</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E10:91_1_2"/><span class="lab"><font color="Green" title="E60">A7</font></span>: 
( <font color="Maroon" title="c1">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="c6">p1</font> &amp;  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c1">i</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">e1</font> implies <font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">e1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c1">i</font>)</span>)</span></span>)</span> )
 <span class="kw">and </span><br/><a NAME="E11:91_1_2"/><span class="lab"><font color="Green" title="E61">A8</font></span>: 
( (  not <font color="Maroon" title="c1">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="c6">p1</font> or  not  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c1">i</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">e1</font> ) implies <font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <span class="kw">and </span><br/><a NAME="E12:91_1_2"/><span class="lab"><font color="Green" title="E62">A9</font></span>: 
 for <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>,<font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">m</font>,<font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c3">e2</font>]
 <span class="kw">by</span> <span class="lab"><a class="ref" href="comput_1.html#D10" target="_self" title="COMPUT_1:def.10">Def13</a></span>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span> for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1</span>)</span> holds <br/> ex <font color="Olive" title="b2">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b2">n</font>) &amp; <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 );<br/><div><span class="lab"><font color="Green" title="E63">A10</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">m</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:91_1_2_1"/><span class="lab"><font color="Green" title="E64">A11</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">m</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><span class="kw">thus </span><a NAME="E2:91_1_2_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c13">p1c</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c12">m</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c12">m</font>)</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> implies  ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b1">n</font>) &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:91_1_2_1_1"/><span class="lab"><font color="Green" title="E65">A12</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b1">n</font>) &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )</span><br/>

<a NAME="E2:91_1_2_1_1"/><span class="lab"><font color="Green" title="E66">A13</font></span>: 
<font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_2_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:91_1_2_1_1_1"/>
( ( <font color="Maroon" title="c1">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="c6">p1</font> &amp; <font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c12">m</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">m</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c12">m</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c12">m</font>)</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">e2</font> ) or  not <font color="Maroon" title="c1">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="c6">p1</font> or  not <font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c12">m</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">m</font></span>)</span> or  not <span class="p1">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c12">m</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c12">m</font>)</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">e2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_2_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:91_1_2_1_1_1_1"/>
( <font color="Maroon" title="c1">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="c6">p1</font> &amp; <font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c12">m</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">m</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c12">m</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c12">m</font>)</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">e2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b1">n</font>) &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )</span><br/><div class="add"><a NAME="E2:91_1_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">e2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c12">m</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c12">m</font>)</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:91_1_2"><span class="lab"><font color="Green" title="E62">A9</font></span></a></span>;<br/><span class="kw">then </span> <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">m</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">e2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c12">m</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p5"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c12">m</font>)</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">m</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
;<br/><a NAME="E4:91_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A14</font></span>: 
( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">m</font></span>)</span> or <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91_1_2_1_1"><span class="lab"><font color="Green" title="E65">A12</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><span class="kw">thus </span><a NAME="E5:91_1_2_1_1_1_1"/>
 ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b1">n</font>) &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_2_1_1_1_1_2"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_2_1_1_1_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:91_1_2_1_1_1_1_2_1"/>
( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">m</font></span>)</span> or <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:91_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E67">A14</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_2_1_1_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:91_1_2_1_1_1_1_2_1_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">m</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b1">n</font>) &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )</span><br/><div class="add"><a NAME="E2:91_1_2_1_1_1_1_2_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b1">n</font>) &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91_1_2_1"><span class="lab"><font color="Green" title="E64">A11</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:91_1_2_1_1_1_1_2_1_1"/>
 ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b1">n</font>) &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:91_1_2_1_1"><span class="lab"><font color="Green" title="E66">A13</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_2_1_1_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:91_1_2_1_1_1_1_2_1_2"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b1">n</font>) &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:91_1_2_1_1_1_1_2_1_2"/>
 ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b1">n</font>) &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_2_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:91_1_2_1_1_1_2"/>
(  not <font color="Maroon" title="c1">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="c6">p1</font> or  not <font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c12">m</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">m</font></span>)</span> or  not <span class="p1">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c12">m</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c12">m</font>)</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">e2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b1">n</font>) &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )</span><br/><div class="add"><a NAME="E2:91_1_2_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:91_1_2"><span class="lab"><font color="Green" title="E62">A9</font></span></a></span>;<br/><a NAME="E3:91_1_2_1_1_1_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b1">n</font>) &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91_1_2_1"><span class="lab"><font color="Green" title="E64">A11</font></span></a>, <a class="txt" href="#E1:91_1_2_1_1"><span class="lab"><font color="Green" title="E65">A12</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:91_1_2_1_1_1_2"/>
 ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b1">n</font>) &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:91_1_2_1_1"><span class="lab"><font color="Green" title="E66">A13</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E64">A15</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1;<br/><span class="kw">assume </span><a NAME="E1:91_1_2_2"/><span class="lab"><font color="Green" title="E65">A16</font></span>: 
<font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">m</font>]</span><br/><a NAME="E2:91_1_2_2"/><span class="lab"><font color="Green" title="E66">A17</font></span>: 
<font color="Maroon">S<sub>3</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_2_2_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_2_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:91_1_2_2_1_1"/>
( ( <font color="Maroon" title="c1">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="c6">p1</font> &amp;  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c1">i</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">e1</font> ) or  not <font color="Maroon" title="c1">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="c6">p1</font> or  not  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c1">i</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">e1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_2_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:91_1_2_2_1_1_1"/>
( <font color="Maroon" title="c1">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="c6">p1</font> &amp;  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c1">i</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">e1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:91_1_2_2_1_1_1"/>
<font color="Maroon">S<sub>3</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91_1_2"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="txt" href="#E10:91_1_2"><span class="lab"><font color="Green" title="E60">A7</font></span></a>, <a class="txt" href="#E5:91_1_2"><span class="lab"><font color="Green" title="E56">A3</font></span></a>, <a class="txt" href="#E1:91_1_2_2"><span class="lab"><font color="Green" title="E65">A16</font></span></a>, <a class="ref" href="comput_1.html#T4" target="_self" title="COMPUT_1:th.4">Th7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_2_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:91_1_2_2_1_1_2"/>
(  not <font color="Maroon" title="c1">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="c6">p1</font> or  not  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> (<font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c1">i</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">e1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:91_1_2_2_1_1_2"/>
<font color="Maroon">S<sub>3</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91_1_2"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="txt" href="#E11:91_1_2"><span class="lab"><font color="Green" title="E61">A8</font></span></a>, <a class="txt" href="#E6:91_1_2"><span class="lab"><font color="Green" title="E57">A4</font></span></a>, <a class="txt" href="#E1:91_1_2_2"><span class="lab"><font color="Green" title="E65">A16</font></span></a>, <a class="ref" href="comput_1.html#T4" target="_self" title="COMPUT_1:th.4">Th7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E67">A18</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_2_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c12">m</font>] implies <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:91_1_2_2_2"/><span class="lab"><font color="Green" title="E68">A19</font></span>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c12">m</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><a NAME="E2:91_1_2_2_2"/><span class="lab"><font color="Green" title="E69">A20</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">m</font>,<font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">m</font>,<font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c3">e2</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:91_1_2"><span class="lab"><font color="Green" title="E62">A9</font></span></a></span>;<br/><a NAME="E3:91_1_2_2_2"/><span class="lab"><font color="Green" title="E70">A21</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">m</font>,<font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">m</font>,<font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c3">e2</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:91_1_2"><span class="lab"><font color="Green" title="E58">A5</font></span></a></span>;<br/><a NAME="E4:91_1_2_2_2"/>
<font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c12">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c12">m</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91_1_2_2"><span class="lab"><font color="Green" title="E65">A16</font></span></a>, <a class="ref" href="comput_1.html#T2" target="_self" title="COMPUT_1:th.2">Th5</a></span>;<br/><span class="kw">hence </span><a NAME="E5:91_1_2_2_2"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91_1_2"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="txt" href="#E1:91_1_2_2_2"><span class="lab"><font color="Green" title="E68">A19</font></span></a>, <a class="txt" href="#E2:91_1_2_2_2"><span class="lab"><font color="Green" title="E69">A20</font></span></a>, <a class="txt" href="#E3:91_1_2_2_2"><span class="lab"><font color="Green" title="E70">A21</font></span></a>, <a class="ref" href="comput_1.html#T2" target="_self" title="COMPUT_1:th.2">Th5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E4:91_1_2_2"/>
 for <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">m</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E2:91_1_2_2"><span class="lab"><font color="Green" title="E66">A17</font></span></a>, <a class="txt" href="#E3:91_1_2_2"><span class="lab"><font color="Green" title="E67">A18</font></span></a>);</span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E15:91_1_2"/><span class="lab"><font color="Green" title="E65">A22</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> implies  ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b1">n</font>) &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:91_1_2_3"/><span class="lab"><font color="Green" title="E66">A23</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b1">n</font>) &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<a NAME="E2:91_1_2_3"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:91_1_2"><span class="lab"><font color="Green" title="E60">A7</font></span></a>, <a class="txt" href="#E11:91_1_2"><span class="lab"><font color="Green" title="E61">A8</font></span></a>, <a class="txt" href="#E1:91_1_2_3"><span class="lab"><font color="Green" title="E66">A23</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/>
<a NAME="E3:91_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91_1_2_3"><span class="lab"><font color="Green" title="E66">A23</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:91_1_2_3"/>
 ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b1">n</font>) &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E16:91_1_2"/><span class="lab"><font color="Green" title="E66">A24</font></span>: 
 for <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">m</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E15:91_1_2"><span class="lab"><font color="Green" title="E65">A22</font></span></a>, <a class="txt" href="#E13:91_1_2"><span class="lab"><font color="Green" title="E63">A10</font></span></a>);<br/></span><a NAME="E17:91_1_2"/><span class="lab"><font color="Green" title="E67">A25</font></span>: 
 for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">n</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_2_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">n</font></span>)</span></span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span>;<br/>
<div><span class="lab"><font color="Green" title="E68">A26</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_2_4_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c13">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c13">n</font>] implies <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:91_1_2_4_1"/><span class="lab"><font color="Green" title="E69">A27</font></span>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c13">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><span class="kw">set </span><font color="Maroon" title="c14">k</font> = <font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font>;<br/><a NAME="E2:91_1_2_4_1"/>
<font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_2_4_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c15">p1c</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span>)</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c16">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: 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="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span> or <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:91_1_2_4_1_1"/><span class="lab"><font color="Green" title="E70">A28</font></span>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_2_4_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:91_1_2_4_1_1_1"/>
( ( <font color="Maroon" title="c1">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="c6">p1</font> &amp; <font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span>)</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">e2</font> ) or  not <font color="Maroon" title="c1">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="c6">p1</font> or  not <font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span> or  not <span class="p1">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span>)</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">e2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_2_4_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:91_1_2_4_1_1_1_1"/><span class="lab"><font color="Green" title="E71">A29</font></span>: 
( <font color="Maroon" title="c1">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="c6">p1</font> &amp; <font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span>)</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">e2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/><div class="add"><a NAME="E2:91_1_2_4_1_1_1_1"/><span class="lab"><font color="Green" title="E72">A30</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">e2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span>)</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_2_4_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:91_1_2_4_1_1_1_1_1"/>
 not  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">e2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span>)</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:91_1_2_4_1_1_1_1_1"/><span class="lab"><font color="Green" title="E73">A31</font></span>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">e2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p5"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span>)</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T4" title="XBOOLE_0:th.4">XBOOLE_0:4</a></span>;<br/>
<a NAME="E4:91_1_2_4_1_1_1_1_1"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:91_1_2_4_1_1_1_1_1"><span class="lab"><font color="Green" title="E73">A31</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">n1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E6:91_1_2_4_1_1_1_1_1"/><span class="lab"><font color="Green" title="E74">A32</font></span>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c18">n1</font>)
 <span class="kw">and </span><br/><a NAME="E7:91_1_2_4_1_1_1_1_1"/><span class="lab"><font color="Green" title="E75">A33</font></span>: 
<font color="Maroon" title="c18">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:91_1_2"><span class="lab"><font color="Green" title="E66">A24</font></span></a></span>;<br/>
<a NAME="E8:91_1_2_4_1_1_1_1_1"/><span class="lab"><font color="Green" title="E76">A34</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91_1_2"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="ref" href="funct_7.html#T31" title="FUNCT_7:th.31">FUNCT_7:31</a></span>;<br/>
<a NAME="E9:91_1_2_4_1_1_1_1_1"/><span class="lab"><font color="Green" title="E77">A35</font></span>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:91_1_2_4_1_1_1_1_1"><span class="lab"><font color="Green" title="E73">A31</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E10:91_1_2_4_1_1_1_1_1"/>
<font color="Maroon" title="c18">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c18">n1</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91_1_2"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="ref" href="funct_7.html#T31" title="FUNCT_7:th.31">FUNCT_7:31</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:91_1_2_4_1_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:91_1_2_4_1_1_1_1_1"><span class="lab"><font color="Green" title="E77">A35</font></span></a>, <a class="txt" href="#E6:91_1_2_4_1_1_1_1_1"><span class="lab"><font color="Green" title="E74">A32</font></span></a>, <a class="txt" href="#E7:91_1_2_4_1_1_1_1_1"><span class="lab"><font color="Green" title="E75">A33</font></span></a>, <a class="txt" href="#E8:91_1_2_4_1_1_1_1_1"><span class="lab"><font color="Green" title="E76">A34</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:91_1_2_4_1_1_1_1"/>
<font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">e2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span>)</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:91_1_2"><span class="lab"><font color="Green" title="E62">A9</font></span></a>, <a class="txt" href="#E1:91_1_2_4_1_1_1_1"><span class="lab"><font color="Green" title="E71">A29</font></span></a></span>;<br/><a NAME="E4:91_1_2_4_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:91_1_2_4_1_1_1_1"><span class="lab"><font color="Green" title="E72">A30</font></span></a>, <a class="ref" href="funct_4.html#T32" title="FUNCT_4:th.32">FUNCT_4:32</a></span>;<br/><span class="kw">hence </span><a NAME="E5:91_1_2_4_1_1_1_1"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91_1_2_4_1_1"><span class="lab"><font color="Green" title="E70">A28</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_2_4_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:91_1_2_4_1_1_1_2"/>
(  not <font color="Maroon" title="c1">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="c6">p1</font> or  not <font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span> or  not <span class="p1">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span>)</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">e2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:91_1_2_4_1_1_1_2"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:91_1_2"><span class="lab"><font color="Green" title="E62">A9</font></span></a>, <a class="txt" href="#E1:91_1_2_4_1_1"><span class="lab"><font color="Green" title="E70">A28</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:91_1_2_4_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91_1_2_4_1"><span class="lab"><font color="Green" title="E69">A27</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E2:91_1_2_4"/><span class="lab"><font color="Green" title="E69">A36</font></span>: 
<font color="Maroon">S<sub>3</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<span class="kw">thus </span><a NAME="E3:91_1_2_4"/>
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E2:91_1_2_4"><span class="lab"><font color="Green" title="E69">A36</font></span></a>, <a class="txt" href="#E1:91_1_2_4"><span class="lab"><font color="Green" title="E68">A26</font></span></a>);</span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span> for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1</span>)</span> holds <br/> ex <font color="Olive" title="b2">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b2">n</font>) &amp; <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 );<br/><div><span class="lab"><font color="Green" title="E68">A37</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_2_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c12">m</font>] implies <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:91_1_2_5"/><span class="lab"><font color="Green" title="E69">A38</font></span>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c12">m</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><span class="kw">thus </span><a NAME="E2:91_1_2_5"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_2_5_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c13">p2c</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c12">m</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c12">m</font>)</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> implies  ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b1">n</font>) &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:91_1_2_5_1"/><span class="lab"><font color="Green" title="E70">A39</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b1">n</font>) &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )</span><br/>

<a NAME="E2:91_1_2_5_1"/><span class="lab"><font color="Green" title="E71">A40</font></span>: 
<font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_2_5_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:91_1_2_5_1_1"/>
( ( <font color="Maroon" title="c1">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="c7">p2</font> &amp; <font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c12">m</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">m</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c12">m</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c12">m</font>)</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">e2</font> ) or  not <font color="Maroon" title="c1">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="c7">p2</font> or  not <font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c12">m</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">m</font></span>)</span> or  not <span class="p1">(<span class="default"><font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c12">m</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c12">m</font>)</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">e2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_2_5_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:91_1_2_5_1_1_1"/>
( <font color="Maroon" title="c1">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="c7">p2</font> &amp; <font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c12">m</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">m</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c12">m</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c12">m</font>)</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">e2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b1">n</font>) &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )</span><br/><div class="add"><a NAME="E2:91_1_2_5_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">e2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c12">m</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c12">m</font>)</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:91_1_2"><span class="lab"><font color="Green" title="E58">A5</font></span></a></span>;<br/><span class="kw">then </span> <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">m</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">e2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c12">m</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p5"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c12">m</font>)</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">m</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
;<br/><a NAME="E4:91_1_2_5_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A41</font></span>: 
( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">m</font></span>)</span> or <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91_1_2_5_1"><span class="lab"><font color="Green" title="E70">A39</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><span class="kw">thus </span><a NAME="E5:91_1_2_5_1_1_1"/>
 ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b1">n</font>) &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_2_5_1_1_1_2"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_2_5_1_1_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:91_1_2_5_1_1_1_2_1"/>
( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">m</font></span>)</span> or <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:91_1_2_5_1_1_1"><span class="lab"><font color="Green" title="E72">A41</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_2_5_1_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:91_1_2_5_1_1_1_2_1_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">m</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b1">n</font>) &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )</span><br/><div class="add"><a NAME="E2:91_1_2_5_1_1_1_2_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b1">n</font>) &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91_1_2_5"><span class="lab"><font color="Green" title="E69">A38</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:91_1_2_5_1_1_1_2_1_1"/>
 ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b1">n</font>) &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:91_1_2_5_1"><span class="lab"><font color="Green" title="E71">A40</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_2_5_1_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:91_1_2_5_1_1_1_2_1_2"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b1">n</font>) &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:91_1_2_5_1_1_1_2_1_2"/>
 ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b1">n</font>) &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_2_5_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:91_1_2_5_1_1_2"/>
(  not <font color="Maroon" title="c1">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="c7">p2</font> or  not <font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c12">m</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">m</font></span>)</span> or  not <span class="p1">(<span class="default"><font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c12">m</font>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c12">m</font>)</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">e2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b1">n</font>) &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )</span><br/><div class="add"><a NAME="E2:91_1_2_5_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:91_1_2"><span class="lab"><font color="Green" title="E58">A5</font></span></a></span>;<br/><a NAME="E3:91_1_2_5_1_1_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b1">n</font>) &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91_1_2_5"><span class="lab"><font color="Green" title="E69">A38</font></span></a>, <a class="txt" href="#E1:91_1_2_5_1"><span class="lab"><font color="Green" title="E70">A39</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:91_1_2_5_1_1_2"/>
 ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b1">n</font>) &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:91_1_2_5_1"><span class="lab"><font color="Green" title="E71">A40</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E19:91_1_2"/><span class="lab"><font color="Green" title="E69">A42</font></span>: 
<font color="Maroon">S<sub>3</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_2_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> implies  ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b1">n</font>) &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:91_1_2_6"/><span class="lab"><font color="Green" title="E70">A43</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b1">n</font>) &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<a NAME="E2:91_1_2_6"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:91_1_2"><span class="lab"><font color="Green" title="E56">A3</font></span></a>, <a class="txt" href="#E6:91_1_2"><span class="lab"><font color="Green" title="E57">A4</font></span></a>, <a class="txt" href="#E1:91_1_2_6"><span class="lab"><font color="Green" title="E70">A43</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/>
<a NAME="E3:91_1_2_6"/><span class="kw">then </span>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91_1_2_6"><span class="lab"><font color="Green" title="E70">A43</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:91_1_2_6"/>
 ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b1">n</font>) &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E20:91_1_2"/><span class="lab"><font color="Green" title="E70">A44</font></span>: 
 for <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">m</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E19:91_1_2"><span class="lab"><font color="Green" title="E69">A42</font></span></a>, <a class="txt" href="#E18:91_1_2"><span class="lab"><font color="Green" title="E68">A37</font></span></a>);<br/></span><a NAME="E21:91_1_2"/><span class="lab"><font color="Green" title="E71">A45</font></span>: 
 for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">n</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_2_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">n</font></span>)</span></span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span>;<br/>
<div><span class="lab"><font color="Green" title="E72">A46</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_2_7_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c13">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c13">n</font>] implies <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:91_1_2_7_1"/><span class="lab"><font color="Green" title="E73">A47</font></span>: 
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c13">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><span class="kw">set </span><font color="Maroon" title="c14">k</font> = <font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font>;<br/><a NAME="E2:91_1_2_7_1"/>
<font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_2_7_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c15">p2c</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span>)</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c16">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: 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="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span> or <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:91_1_2_7_1_1"/><span class="lab"><font color="Green" title="E74">A48</font></span>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_2_7_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:91_1_2_7_1_1_1"/>
( ( <font color="Maroon" title="c1">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="c7">p2</font> &amp; <font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span>)</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">e2</font> ) or  not <font color="Maroon" title="c1">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="c7">p2</font> or  not <font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span> or  not <span class="p1">(<span class="default"><font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span>)</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">e2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_2_7_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:91_1_2_7_1_1_1_1"/><span class="lab"><font color="Green" title="E75">A49</font></span>: 
( <font color="Maroon" title="c1">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="c7">p2</font> &amp; <font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span>)</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">e2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/><div class="add"><a NAME="E2:91_1_2_7_1_1_1_1"/><span class="lab"><font color="Green" title="E76">A50</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">e2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span>)</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_2_7_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:91_1_2_7_1_1_1_1_1"/>
 not  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">e2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span>)</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:91_1_2_7_1_1_1_1_1"/><span class="lab"><font color="Green" title="E77">A51</font></span>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">e2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p5"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span>)</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T4" title="XBOOLE_0:th.4">XBOOLE_0:4</a></span>;<br/>
<a NAME="E4:91_1_2_7_1_1_1_1_1"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:91_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E77">A51</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">n2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E6:91_1_2_7_1_1_1_1_1"/><span class="lab"><font color="Green" title="E78">A52</font></span>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c18">n2</font>)
 <span class="kw">and </span><br/><a NAME="E7:91_1_2_7_1_1_1_1_1"/><span class="lab"><font color="Green" title="E79">A53</font></span>: 
<font color="Maroon" title="c18">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:91_1_2"><span class="lab"><font color="Green" title="E70">A44</font></span></a></span>;<br/>
<a NAME="E8:91_1_2_7_1_1_1_1_1"/><span class="lab"><font color="Green" title="E80">A54</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91_1_2"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="ref" href="funct_7.html#T31" title="FUNCT_7:th.31">FUNCT_7:31</a></span>;<br/>
<a NAME="E9:91_1_2_7_1_1_1_1_1"/><span class="lab"><font color="Green" title="E81">A55</font></span>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:91_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E77">A51</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E10:91_1_2_7_1_1_1_1_1"/>
<font color="Maroon" title="c18">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c18">n2</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91_1_2"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="ref" href="funct_7.html#T31" title="FUNCT_7:th.31">FUNCT_7:31</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:91_1_2_7_1_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:91_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E81">A55</font></span></a>, <a class="txt" href="#E6:91_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E78">A52</font></span></a>, <a class="txt" href="#E7:91_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E79">A53</font></span></a>, <a class="txt" href="#E8:91_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E80">A54</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:91_1_2_7_1_1_1_1"/>
<font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">e2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span>)</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:91_1_2"><span class="lab"><font color="Green" title="E58">A5</font></span></a>, <a class="txt" href="#E1:91_1_2_7_1_1_1_1"><span class="lab"><font color="Green" title="E75">A49</font></span></a></span>;<br/><a NAME="E4:91_1_2_7_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:91_1_2_7_1_1_1_1"><span class="lab"><font color="Green" title="E76">A50</font></span></a>, <a class="ref" href="funct_4.html#T32" title="FUNCT_4:th.32">FUNCT_4:32</a></span>;<br/><span class="kw">hence </span><a NAME="E5:91_1_2_7_1_1_1_1"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91_1_2_7_1_1"><span class="lab"><font color="Green" title="E74">A48</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_2_7_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:91_1_2_7_1_1_1_2"/>
(  not <font color="Maroon" title="c1">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="c7">p2</font> or  not <font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span> or  not <span class="p1">(<span class="default"><font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span>)</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span>)</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">e2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:91_1_2_7_1_1_1_2"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:91_1_2"><span class="lab"><font color="Green" title="E58">A5</font></span></a>, <a class="txt" href="#E1:91_1_2_7_1_1"><span class="lab"><font color="Green" title="E74">A48</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:91_1_2_7_1"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91_1_2_7_1"><span class="lab"><font color="Green" title="E73">A47</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E2:91_1_2_7"/><span class="lab"><font color="Green" title="E73">A56</font></span>: 
<font color="Maroon">S<sub>4</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<span class="kw">thus </span><a NAME="E3:91_1_2_7"/>
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">n</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E2:91_1_2_7"><span class="lab"><font color="Green" title="E73">A56</font></span></a>, <a class="txt" href="#E1:91_1_2_7"><span class="lab"><font color="Green" title="E72">A46</font></span></a>);</span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="c12">F1m</font> = <font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">m</font>, <font color="Maroon" title="c13">F1n</font> = <font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">n</font>, <font color="Maroon" title="c14">F2m</font> = <font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">m</font>, <font color="Maroon" title="c15">F2n</font> = <font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">n</font> as    <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> ;<br/><div><span class="lab"><font color="Green" title="E72">A57</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_2_8"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:91_1_2_8"/><span class="lab"><font color="Green" title="E73">A58</font></span>: 
<font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds   not <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c16">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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="firebrick">::  thesis: </font></span></a><span class="hide">  not <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">m</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">m</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><span class="kw">assume </span><a NAME="E2:91_1_2_8"/>
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">m</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">m</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:91_1_2_8"/><span class="lab"><font color="Green" title="E74">A59</font></span>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">m</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">m</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/><a NAME="E5:91_1_2_8"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:91_1_2_8"><span class="lab"><font color="Green" title="E74">A59</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E6:91_1_2_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A60</font></span>: 
 ex <font color="Olive" title="b1">n2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b1">n2</font>) &amp; <font color="Olive" title="b1">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">m</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:91_1_2"><span class="lab"><font color="Green" title="E70">A44</font></span></a></span>;<br/><a NAME="E7:91_1_2_8"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c16">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:91_1_2_8"><span class="lab"><font color="Green" title="E74">A59</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E8:91_1_2_8"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">n1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p1</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> (<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b1">n1</font>) &amp; <font color="Olive" title="b1">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">m</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:91_1_2"><span class="lab"><font color="Green" title="E66">A24</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E9:91_1_2_8"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91_1_2_8"><span class="lab"><font color="Green" title="E73">A58</font></span></a>, <a class="txt" href="#E6:91_1_2_8"><span class="lab"><font color="Green" title="E75">A60</font></span></a>, <a class="ref" href="comput_1.html#T2" target="_self" title="COMPUT_1:th.2">Th5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E24:91_1_2"/>
 <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c6">p1</font>) <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a>  <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c7">p2</font>)
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_2_9"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_2_9_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:91_1_2_9_1"/>
( <font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> or <font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_2_9_1_1"><span class="kw">suppose </span></a><a NAME="E1:91_1_2_9_1_1"/>
<font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c6">p1</font>) <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a>  <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c7">p2</font>)</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:91_1_2_9_1_1"/><span class="lab"><font color="Green" title="E73">A61</font></span>: 
<font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c17">k</font> = <font color="Maroon" title="c16">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><a NAME="E5:91_1_2_9_1_1"/><span class="lab"><font color="Green" title="E74">A62</font></span>: 
<font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:91_1_2_9_1_1"><span class="lab"><font color="Green" title="E73">A61</font></span></a></span>;<br/><span class="kw">thus </span><a NAME="E6:91_1_2_9_1_1"/>
 <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c6">p1</font>) <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a>  <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c7">p2</font>)
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_2_9_1_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_2_9_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:91_1_2_9_1_1_1_1"/>
( <font color="Maroon" title="c13">F1n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">F2n</font> or <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">F1n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c15">F2n</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>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:91_1_2"><span class="lab"><font color="Green" title="E64">A15</font></span></a>, <a class="txt" href="#E23:91_1_2"><span class="lab"><font color="Green" title="E72">A57</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_2_9_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:91_1_2_9_1_1_1_1_1"/>
<font color="Maroon" title="c13">F1n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">F2n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c6">p1</font>) <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a>  <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c7">p2</font>)</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:91_1_2_9_1_1_1_1_1"/>
 <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c6">p1</font>) <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a>  <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c7">p2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:91_1_2"><span class="lab"><font color="Green" title="E59">A6</font></span></a>, <a class="txt" href="#E4:91_1_2"><span class="lab"><font color="Green" title="E55">A2</font></span></a>, <a class="txt" href="#E17:91_1_2"><span class="lab"><font color="Green" title="E67">A25</font></span></a>, <a class="txt" href="#E5:91_1_2_9_1_1"><span class="lab"><font color="Green" title="E74">A62</font></span></a>, <a class="ref" href="partfun1.html#T58" title="PARTFUN1:th.58">PARTFUN1:58</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_2_9_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:91_1_2_9_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">F1n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c15">F2n</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c6">p1</font>) <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a>  <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c7">p2</font>)</span><br/><div class="add"><a NAME="E2:91_1_2_9_1_1_1_1_2"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">F1n</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c15">F2n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><a NAME="E3:91_1_2_9_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">F1n</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> <font color="Maroon" title="c15">F2n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T56" title="PARTFUN1:th.56">PARTFUN1:56</a></span>;<br/><span class="kw">hence </span><a NAME="E4:91_1_2_9_1_1_1_1_2"/>
 <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c6">p1</font>) <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a>  <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c7">p2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:91_1_2"><span class="lab"><font color="Green" title="E59">A6</font></span></a>, <a class="txt" href="#E4:91_1_2"><span class="lab"><font color="Green" title="E55">A2</font></span></a>, <a class="txt" href="#E17:91_1_2"><span class="lab"><font color="Green" title="E67">A25</font></span></a>, <a class="txt" href="#E5:91_1_2_9_1_1"><span class="lab"><font color="Green" title="E74">A62</font></span></a>, <a class="ref" href="partfun1.html#T58" title="PARTFUN1:th.58">PARTFUN1:58</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_2_9_1_2"><span class="kw">suppose </span></a><a NAME="E1:91_1_2_9_1_2"/>
<font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c6">p1</font>) <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a>  <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c7">p2</font>)</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:91_1_2_9_1_2"/><span class="lab"><font color="Green" title="E73">A63</font></span>: 
<font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c17">k</font> = <font color="Maroon" title="c16">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><a NAME="E5:91_1_2_9_1_2"/><span class="lab"><font color="Green" title="E74">A64</font></span>: 
<font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:91_1_2_9_1_2"><span class="lab"><font color="Green" title="E73">A63</font></span></a></span>;<br/><span class="kw">thus </span><a NAME="E6:91_1_2_9_1_2"/>
 <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c6">p1</font>) <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a>  <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c7">p2</font>)
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_2_9_1_2_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_2_9_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:91_1_2_9_1_2_1_1"/>
( <font color="Maroon" title="c12">F1m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F2m</font> or <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">F1m</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c14">F2m</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>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:91_1_2"><span class="lab"><font color="Green" title="E64">A15</font></span></a>, <a class="txt" href="#E23:91_1_2"><span class="lab"><font color="Green" title="E72">A57</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_2_9_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:91_1_2_9_1_2_1_1_1"/>
<font color="Maroon" title="c12">F1m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F2m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c6">p1</font>) <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a>  <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c7">p2</font>)</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:91_1_2_9_1_2_1_1_1"/>
 <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c6">p1</font>) <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a>  <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c7">p2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:91_1_2"><span class="lab"><font color="Green" title="E59">A6</font></span></a>, <a class="txt" href="#E4:91_1_2"><span class="lab"><font color="Green" title="E55">A2</font></span></a>, <a class="txt" href="#E21:91_1_2"><span class="lab"><font color="Green" title="E71">A45</font></span></a>, <a class="txt" href="#E5:91_1_2_9_1_2"><span class="lab"><font color="Green" title="E74">A64</font></span></a>, <a class="ref" href="partfun1.html#T58" title="PARTFUN1:th.58">PARTFUN1:58</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_2_9_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:91_1_2_9_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">F1m</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c14">F2m</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c6">p1</font>) <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a>  <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c7">p2</font>)</span><br/><div class="add"><a NAME="E2:91_1_2_9_1_2_1_1_2"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">F1m</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c14">F2m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><a NAME="E3:91_1_2_9_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">F1m</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> <font color="Maroon" title="c14">F2m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T56" title="PARTFUN1:th.56">PARTFUN1:56</a></span>;<br/><span class="kw">hence </span><a NAME="E4:91_1_2_9_1_2_1_1_2"/>
 <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c6">p1</font>) <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a>  <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> (<font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c7">p2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:91_1_2"><span class="lab"><font color="Green" title="E59">A6</font></span></a>, <a class="txt" href="#E4:91_1_2"><span class="lab"><font color="Green" title="E55">A2</font></span></a>, <a class="txt" href="#E21:91_1_2"><span class="lab"><font color="Green" title="E71">A45</font></span></a>, <a class="txt" href="#E5:91_1_2_9_1_2"><span class="lab"><font color="Green" title="E74">A64</font></span></a>, <a class="ref" href="partfun1.html#T58" title="PARTFUN1:th.58">PARTFUN1:58</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
