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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">ds</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">k</font></span>)</span> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> );<br/>
<a NAME="E1:2_1_1"/><span class="lab"><font color="Green" title="E1">X</font></span>: 
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] holds <br/><font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></span>;<br/>
<a NAME="E2:2_1_1"/><span class="lab"><font color="Green" title="E2">Y</font></span>: 
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y1</font>, <font color="Olive" title="b3">y2</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>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">y2</font>] holds <br/><font color="Olive" title="b2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">y2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y1</font>, <font color="Maroon" title="c5">y2</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="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">y2</font>] implies <font color="Maroon" title="c4">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>
 <span class="kw">and </span><br/><a NAME="E2:2_1_1_1"/><span class="lab"><font color="Green" title="E3">W2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y1</font>]
 <span class="kw">and </span><br/><a NAME="E3:2_1_1_1"/><span class="lab"><font color="Green" title="E4">W3</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">y2</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="c4">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y2</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">k1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E5:2_1_1_1"/><span class="lab"><font color="Green" title="E5">W4</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">ds</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">k1</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:2_1_1_1"/><span class="lab"><font color="Green" title="E6">W5</font></span>: 
<font color="Maroon" title="c4">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k1</font> <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="#E2:2_1_1_1"><span class="lab"><font color="Green" title="E3">W2</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">k2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E8:2_1_1_1"/><span class="lab"><font color="Green" title="E7">W6</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">ds</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">k2</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E9:2_1_1_1"/><span class="lab"><font color="Green" title="E8">W7</font></span>: 
<font color="Maroon" title="c5">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k2</font> <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="#E3:2_1_1_1"><span class="lab"><font color="Green" title="E4">W3</font></span></a></span>;<br/>
<a NAME="E10:2_1_1_1"/>
<font color="Maroon" title="c2">ds</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">ds</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">k2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_1"><span class="lab"><font color="Green" title="E5">W4</font></span></a>, <a class="txt" href="#E8:2_1_1_1"><span class="lab"><font color="Green" title="E7">W6</font></span></a>, <a class="ref" href="ami_3.html#T10" title="AMI_3:th.10">AMI_3:10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:2_1_1_1"/>
<font color="Maroon" title="c4">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_1_1"><span class="lab"><font color="Green" title="E8">W7</font></span></a>, <a class="txt" href="#E6:2_1_1_1"><span class="lab"><font color="Green" title="E6">W5</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">consider </span><font color="Maroon" title="c3">p</font> being   <a href="struct_0.html#NM10" title="STRUCT_0:NM.10">PartFunc</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a><span class="kw"> such that </span><br/><a NAME="E4:2_1_1"/><span class="lab"><font color="Green" title="E3">W0</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">p</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> &amp;  ex <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] ) )
 <span class="kw">and </span><br/><a NAME="E5:2_1_1"/><span class="lab"><font color="Green" title="E4">W1</font></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="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">p</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="partfun1.html#S2" title="PARTFUN1:sch.2">PARTFUN1:sch 2</a>(<a class="txt" href="#E1:2_1_1"><span class="lab"><font color="Green" title="E1">X</font></span></a>, <a class="txt" href="#E2:2_1_1"><span class="lab"><font color="Green" title="E2">Y</font></span></a>);<br/></span>
<a NAME="E6:2_1_1"/><span class="lab"><font color="Green" title="E5">C</font></span>: 
<font color="Maroon" title="c3">p</font> is  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">e</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="ref" href="relat_1.html#D18" title="RELAT_1:def.18">RELAT_1:def 18</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="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c3">p</font> or <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> )</span><br/>

<span class="kw">thus </span><a NAME="E1:2_1_1_2"/>
(  not <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c3">p</font> or <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1"><span class="lab"><font color="Green" title="E3">W0</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="E7:2_1_1"/>
<font color="Maroon" title="c3">p</font> is  the <a href="memstr_0.html#U1" title="MEMSTR_0:sel.1">Object-Kind</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> <a href="funct_1.html#V5" title="FUNCT_1:attr.5">-compatible</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">e</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="funct_1.html#D14" title="FUNCT_1:def.14">FUNCT_1:def 14</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="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c3">p</font> or <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="memstr_0.html#U1" title="MEMSTR_0:sel.1">Object-Kind</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">e</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:2_1_1_3"/><span class="lab"><font color="Green" title="E6">Z</font></span>: 
<font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">p</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="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="memstr_0.html#U1" title="MEMSTR_0:sel.1">Object-Kind</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">e</font></span><br/>

<a NAME="E2:2_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">AA</font></span>: 
 ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">e</font>,<font color="Olive" title="b1">y</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1"><span class="lab"><font color="Green" title="E3">W0</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">o</font> = <font color="Maroon" title="c4">e</font> as   <a href="memstr_0.html#NM3" title="MEMSTR_0:NM.3">Object</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_3"><span class="lab"><font color="Green" title="E6">Z</font></span></a>, <a class="txt" href="#E4:2_1_1"><span class="lab"><font color="Green" title="E3">W0</font></span></a></span>;<br/>
<a NAME="E4:2_1_1_3"/><span class="lab"><font color="Green" title="E8">B</font></span>: 
 <a href="memstr_0.html#K2" title="MEMSTR_0:func.2">ObjectKind</a> <font color="Maroon" title="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_3"><span class="lab"><font color="Green" title="E7">AA</font></span></a>, <a class="ref" href="ami_3.html#T11" title="AMI_3:th.11">AMI_3:11</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E6:2_1_1_3"/><span class="lab"><font color="Green" title="E9">W3</font></span>: 
( <font color="Maroon" title="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">ds</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">k</font></span>)</span> &amp; <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <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="#E5:2_1_1"><span class="lab"><font color="Green" title="E4">W1</font></span></a>, <a class="txt" href="#E1:2_1_1_3"><span class="lab"><font color="Green" title="E6">Z</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E7:2_1_1_3"/>
<font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="memstr_0.html#U1" title="MEMSTR_0:sel.1">Object-Kind</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_3"><span class="lab"><font color="Green" title="E8">B</font></span></a>, <a class="txt" href="#E6:2_1_1_3"><span class="lab"><font color="Green" title="E9">W3</font></span></a>, <a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 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>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">p</font> = <font color="Maroon" title="c3">p</font> as   <a href="memstr_0.html#NM1" title="MEMSTR_0:NM.1">PartState</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_1"><span class="lab"><font color="Green" title="E5">C</font></span></a></span>;<br/>
<span class="kw">take </span><font color="Maroon" title="c5">s</font> =  the   <a href="memstr_0.html#NM2" title="MEMSTR_0:NM.2">State</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c4">p</font>; <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">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">D</font> holds <br/><font color="Maroon" title="c5">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">ds</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">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" title="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">D</font> implies <font color="Maroon" title="c5">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">ds</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E9:2_1_1"/>
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">D</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="c5">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">ds</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/>

<a NAME="E10:2_1_1"/>
 ex <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>(  <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">ds</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">ds</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">i</font></span>)</span> &amp; <font color="Maroon" title="c1">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</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="c1">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 
;<br/>
<a NAME="E11:2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E6">X</font></span>: 
 <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">ds</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1"><span class="lab"><font color="Green" title="E3">W0</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E13:2_1_1"/><span class="lab"><font color="Green" title="E7">W2</font></span>: 
(  <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">ds</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">ds</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">i</font></span>)</span> &amp; <font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">ds</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <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="#E5:2_1_1"><span class="lab"><font color="Green" title="E4">W1</font></span></a></span>;<br/>
<a NAME="E14:2_1_1"/><span class="lab"><font color="Green" title="E8">A</font></span>: 
<font color="Maroon" title="c2">ds</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">ds</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:2_1_1"><span class="lab"><font color="Green" title="E7">W2</font></span></a>, <a class="ref" href="ami_3.html#T10" title="AMI_3:th.10">AMI_3:10</a></span>;<br/>
<a NAME="E15:2_1_1"/>
<font color="Maroon" title="c4">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T25" title="FUNCT_4:th.25">FUNCT_4:25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:2_1_1"/>
<font color="Maroon" title="c5">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">ds</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <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="#E14:2_1_1"><span class="lab"><font color="Green" title="E8">A</font></span></a>, <a class="txt" href="#E13:2_1_1"><span class="lab"><font color="Green" title="E7">W2</font></span></a>, <a class="txt" href="#E11:2_1_1"><span class="lab"><font color="Green" title="E6">X</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</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>
