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

<a NAME="E1:51_1_1"/>
 <a href="compos_1.html#NK5" title="COMPOS_1:NK.5">JumpPart</a> <font color="Maroon" title="c2">I</font> is  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">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="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a>,<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="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">proj2</a> <span class="p1">(<span class="default"><a href="compos_1.html#NK5" title="COMPOS_1:NK.5">JumpPart</a> <font color="Maroon" title="c2">I</font></span>)</span> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:51_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="compos_1.html#NK5" title="COMPOS_1:NK.5">JumpPart</a> <font color="Maroon" title="c2">I</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="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">k</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:51_1_1_1"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="compos_1.html#NK5" title="COMPOS_1:NK.5">JumpPart</a> <font color="Maroon" title="c2">I</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:51_1_1_1"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
<span class="p1">(<span class="default"><a href="compos_1.html#NK5" title="COMPOS_1:NK.5">JumpPart</a> <font color="Maroon" title="c2">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">X</font> =  <a href="compos_1.html#K7" title="COMPOS_1:func.7">JumpParts</a> <span class="p1">(<span class="default"><a href="compos_1.html#K5" title="COMPOS_1:func.5">InsCode</a> <font color="Maroon" title="c2">I</font></span>)</span>;<br/>
<a NAME="E5:51_1_1_1"/><span class="lab"><font color="Green" title="E21">A3</font></span>: 
 <a href="compos_1.html#NK5" title="COMPOS_1:NK.5">JumpPart</a> <font color="Maroon" title="c2">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="compos_1.html#K7" title="COMPOS_1:func.7">JumpParts</a> <span class="p1">(<span class="default"><a href="compos_1.html#K5" title="COMPOS_1:func.5">InsCode</a> <font color="Maroon" title="c2">I</font></span>)</span>
 
;<br/>
<a NAME="E6:51_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="card_3.html#K10" title="CARD_3:func.10">product"</a> <span class="p2">(<span class="default"><a href="compos_1.html#K7" title="COMPOS_1:func.7">JumpParts</a> <span class="p3">(<span class="default"><a href="compos_1.html#K5" title="COMPOS_1:func.5">InsCode</a> <font color="Maroon" title="c2">I</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1_1_1"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="card_3.html#T100" title="CARD_3:th.100">CARD_3:100</a></span>;<br/>
<a NAME="E7:51_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="compos_1.html#NK5" title="COMPOS_1:NK.5">JumpPart</a> <font color="Maroon" title="c2">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="card_3.html#K10" title="CARD_3:func.10">product"</a> <span class="p2">(<span class="default"><a href="compos_1.html#K7" title="COMPOS_1:func.7">JumpParts</a> <span class="p3">(<span class="default"><a href="compos_1.html#K5" title="COMPOS_1:func.5">InsCode</a> <font color="Maroon" title="c2">I</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:51_1_1_1"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="card_3.html#T101" title="CARD_3:th.101">CARD_3:101</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:51_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1_1_1"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E4:51_1_1_1"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="compos_1.html#D14" target="_self" title="COMPOS_1:def.14">Def35</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">hence </span><a NAME="E2:51_1_1"/>
 for <font color="Olive">b<sub>1</sub></font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="compos_1.html#NK5" title="COMPOS_1:NK.5">JumpPart</a> <font color="Maroon" title="c2">I</font> holds <br/><font color="Olive">b<sub>1</sub></font> is  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</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>
