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

<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="algstr_0.html#V7" title="ALGSTR_0:attr.7">add-cancelable</a>   <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>   <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>   <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>   <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a>   <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a>   <a href="algstr_1.html#V1" title="ALGSTR_1:attr.1">left_zeroed</a>   <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</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">F</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font><br/>  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> <a href="ideal_1.html#K8" title="IDEAL_1:func.8">-LeftIdeal</a>  iff  ex <font color="Olive" title="b3">f</font> being    <a href="ideal_1.html#M2" title="IDEAL_1:mode.2">LeftLinearCombination</a> of <font color="Olive" title="b1">F</font> st <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b3">f</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">F</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</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">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> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K8" title="IDEAL_1:func.8">-LeftIdeal</a>  iff  ex <font color="Olive" title="b2">f</font> being    <a href="ideal_1.html#M2" title="IDEAL_1:mode.2">LeftLinearCombination</a> of <font color="Maroon" title="c2">F</font> st <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b2">f</font> )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">I</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b2">lc</font> being    <a href="ideal_1.html#M2" title="IDEAL_1:mode.2">LeftLinearCombination</a> of <font color="Maroon" title="c2">F</font> st  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b2">lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> </span> } </span> ;<br/>
<a NAME="E1:88"/><span class="lab"><font color="Green" title="E43">A1</font></span>: 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b2">lc</font> being    <a href="ideal_1.html#M2" title="IDEAL_1:mode.2">LeftLinearCombination</a> of <font color="Maroon" title="c2">F</font> st  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b2">lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b2">lc</font> being    <a href="ideal_1.html#M2" title="IDEAL_1:mode.2">LeftLinearCombination</a> of <font color="Maroon" title="c2">F</font> st  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b2">lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> </span> } </span>  or <font color="Maroon" title="c4">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 <font color="Maroon" title="c1">L</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:88_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b2">lc</font> being    <a href="ideal_1.html#M2" title="IDEAL_1:mode.2">LeftLinearCombination</a> of <font color="Maroon" title="c2">F</font> st  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b2">lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</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="c4">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 <font color="Maroon" title="c1">L</font></span><br/>

<a NAME="E2:88_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> &amp;  ex <font color="Olive" title="b2">lc</font> being    <a href="ideal_1.html#M2" title="IDEAL_1:mode.2">LeftLinearCombination</a> of <font color="Maroon" title="c2">F</font> st  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b2">lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x9</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:88_1"/>
<font color="Maroon" title="c4">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 <font color="Maroon" title="c1">L</font>
 ; <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">let </span><font color="Maroon" title="c4">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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K8" title="IDEAL_1:func.8">-LeftIdeal</a>  iff  ex <font color="Olive" title="b1">f</font> being    <a href="ideal_1.html#M2" title="IDEAL_1:mode.2">LeftLinearCombination</a> of <font color="Maroon" title="c2">F</font> st <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b1">f</font> )</span><br/>

<a NAME="E2:88"/><span class="lab"><font color="Green" title="E44">A2</font></span>: 
<font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b2">lc</font> being    <a href="ideal_1.html#M2" title="IDEAL_1:mode.2">LeftLinearCombination</a> of <font color="Maroon" title="c2">F</font> st  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b2">lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> </span> } </span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">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="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b2">lc</font> being    <a href="ideal_1.html#M2" title="IDEAL_1:mode.2">LeftLinearCombination</a> of <font color="Maroon" title="c2">F</font> st  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b2">lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> </span> } </span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:88_2"/><span class="lab"><font color="Green" title="E45">A3</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b2">lc</font> being    <a href="ideal_1.html#M2" title="IDEAL_1:mode.2">LeftLinearCombination</a> of <font color="Maroon" title="c2">F</font> st  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b2">lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> </span> } </span> </span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c5">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">lc</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">i</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="c8">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> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> implies  ex <font color="Olive" title="b1">u</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ex <font color="Olive" title="b2">a</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">F</font> st <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">u</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">a</font> )</span><br/><span class="kw">assume </span><a NAME="E1:88_2_1"/><span class="lab"><font color="Green" title="E46">A4</font></span>: 
<font color="Maroon" title="c8">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> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></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">u</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ex <font color="Olive" title="b2">a</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">F</font> st <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">u</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">a</font></span><br/><a NAME="E2:88_2_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T2" title="FINSEQ_1:th.2">FINSEQ_1:2</a>, <a class="ref" href="finseq_1.html#T38" title="FINSEQ_1:th.38">FINSEQ_1:38</a></span>;<br/><a NAME="E3:88_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:88_2_1"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">then </span><span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> = 
<font color="Maroon" title="c6">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D8" title="VECTSP_1:def.8">VECTSP_1:def 8</a></span>
;<br/><span class="kw">hence </span><a NAME="E5:88_2_1"/>
 ex <font color="Olive" title="b1">u</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ex <font color="Olive" title="b2">a</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">F</font> st <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">u</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:88_2"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="txt" href="#E1:88_2_1"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</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="c8">lc</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> as    <a href="ideal_1.html#M2" title="IDEAL_1:mode.2">LeftLinearCombination</a> of <font color="Maroon" title="c2">F</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="ideal_1.html#D9" target="_self" title="IDEAL_1:def.9">Def10</a></span>;<br/>
<a NAME="E5:88_2"/>
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c8">lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="binom.html#T3" title="BINOM:th.3">BINOM:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:88_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b2">lc</font> being    <a href="ideal_1.html#M2" title="IDEAL_1:mode.2">LeftLinearCombination</a> of <font color="Maroon" title="c2">F</font> st  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b2">lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> </span> } </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="E3:88"/><span class="lab"><font color="Green" title="E45">A5</font></span>: 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b2">lc</font> being    <a href="ideal_1.html#M2" title="IDEAL_1:mode.2">LeftLinearCombination</a> of <font color="Maroon" title="c2">F</font> st  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b2">lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K8" title="IDEAL_1:func.8">-LeftIdeal</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span> for <font color="Olive" title="b1">lc</font> being    <a href="ideal_1.html#M2" title="IDEAL_1:mode.2">LeftLinearCombination</a> of <font color="Maroon" title="c2">F</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">lc</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 holds <br/> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b1">lc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K8" title="IDEAL_1:func.8">-LeftIdeal</a> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">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="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b2">lc</font> being    <a href="ideal_1.html#M2" title="IDEAL_1:mode.2">LeftLinearCombination</a> of <font color="Maroon" title="c2">F</font> st  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b2">lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> </span> } </span>  or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K8" title="IDEAL_1:func.8">-LeftIdeal</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:88_3"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b2">lc</font> being    <a href="ideal_1.html#M2" title="IDEAL_1:mode.2">LeftLinearCombination</a> of <font color="Maroon" title="c2">F</font> st  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b2">lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</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="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K8" title="IDEAL_1:func.8">-LeftIdeal</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">x9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E3:88_3"/><span class="lab"><font color="Green" title="E46">A6</font></span>: 
<font color="Maroon" title="c6">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font>
 <span class="kw">and </span><br/><a NAME="E4:88_3"/><span class="lab"><font color="Green" title="E47">A7</font></span>: 
 ex <font color="Olive" title="b1">lc</font> being    <a href="ideal_1.html#M2" title="IDEAL_1:mode.2">LeftLinearCombination</a> of <font color="Maroon" title="c2">F</font> st  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b1">lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x9</font>
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">lc</font> being    <a href="ideal_1.html#M2" title="IDEAL_1:mode.2">LeftLinearCombination</a> of <font color="Maroon" title="c2">F</font><span class="kw"> such that </span><br/><a NAME="E6:88_3"/><span class="lab"><font color="Green" title="E48">A8</font></span>: 
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c7">lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:88_3"><span class="lab"><font color="Green" title="E47">A7</font></span></a></span>;<br/>
<a NAME="E7:88_3"/><span class="lab"><font color="Green" title="E49">A9</font></span>: 
 for <font color="Olive" title="b1">k</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 <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">k</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>1</sub></font>[<font color="Maroon" title="c8">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:88_3_1"/><span class="lab"><font color="Green" title="E50">A10</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</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>1</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<span class="kw">thus </span><a NAME="E2:88_3_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</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="88_3_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">lc</font> be    <a href="ideal_1.html#M2" title="IDEAL_1:mode.2">LeftLinearCombination</a> of <font color="Maroon" title="c2">F</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">lc</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c9">lc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K8" title="IDEAL_1:func.8">-LeftIdeal</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:88_3_1_1"/><span class="lab"><font color="Green" title="E51">A11</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">lc</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</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">  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c9">lc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K8" title="IDEAL_1:func.8">-LeftIdeal</a> </span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="88_3_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="E1:88_3_1_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">lc</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:88_3_1_1"><span class="lab"><font color="Green" title="E51">A11</font></span></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="88_3_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:88_3_1_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">lc</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c9">lc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K8" title="IDEAL_1:func.8">-LeftIdeal</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:88_3_1_1_1_1"/>
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c9">lc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K8" title="IDEAL_1:func.8">-LeftIdeal</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:88_3_1"><span class="lab"><font color="Green" title="E50">A10</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="88_3_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:88_3_1_1_1_2"/><span class="lab"><font color="Green" title="E52">A12</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</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">  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c9">lc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K8" title="IDEAL_1:func.8">-LeftIdeal</a> </span><br/><div class="add"><a NAME="E2:88_3_1_1_1_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c9">lc</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">q</font> being    <a href="ideal_1.html#M2" title="IDEAL_1:mode.2">LeftLinearCombination</a> of <font color="Maroon" title="c2">F</font>, <font color="Maroon" title="c11">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E4:88_3_1_1_1_2"/><span class="lab"><font color="Green" title="E53">A13</font></span>: 
<font color="Maroon" title="c9">lc</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c10">q</font> <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"><font color="Maroon" title="c11">r</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E5:88_3_1_1_1_2"/><span class="lab"><font color="Green" title="E54">A14</font></span>: 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c11">r</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> is    <a href="ideal_1.html#M2" title="IDEAL_1:mode.2">LeftLinearCombination</a> of <font color="Maroon" title="c2">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ideal_1.html#T33" target="_self" title="IDEAL_1:th.33">Th33</a></span>;<br/><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c11">r</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:88_3_1_1_1_2"><span class="lab"><font color="Green" title="E52">A12</font></span></a>, <a class="txt" href="#E4:88_3_1_1_1_2"><span class="lab"><font color="Green" title="E53">A13</font></span></a>, <a class="ref" href="finseq_1.html#T22" title="FINSEQ_1:th.22">FINSEQ_1:22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">q</font></span>)</span> <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="finseq_1.html#T39" title="FINSEQ_1:th.39">FINSEQ_1:39</a></span>
;<br/><a NAME="E7:88_3_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A15</font></span>: 
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K8" title="IDEAL_1:func.8">-LeftIdeal</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:88_3_1"><span class="lab"><font color="Green" title="E50">A10</font></span></a></span>;<br/><a NAME="E8:88_3_1_1_1_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c11">r</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T2" title="FINSEQ_1:th.2">FINSEQ_1:2</a>, <a class="ref" href="finseq_1.html#T38" title="FINSEQ_1:th.38">FINSEQ_1:38</a></span>;<br/><a NAME="E9:88_3_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A16</font></span>: 
1 <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"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c11">r</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">u</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>, <font color="Maroon" title="c13">a</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">F</font><span class="kw"> such that </span><br/><a NAME="E11:88_3_1_1_1_2"/><span class="lab"><font color="Green" title="E57">A17</font></span>: 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c11">r</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">u</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:88_3_1_1_1_2"><span class="lab"><font color="Green" title="E54">A14</font></span></a>, <a class="ref" href="ideal_1.html#D9" target="_self" title="IDEAL_1:def.9">Def10</a></span>;<br/><a NAME="E12:88_3_1_1_1_2"/>
<font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K8" title="IDEAL_1:func.8">-LeftIdeal</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ideal_1.html#D15" target="_self" title="IDEAL_1:def.15">Def16</a></span>;<br/><a NAME="E13:88_3_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A18</font></span>: 
<font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K8" title="IDEAL_1:func.8">-LeftIdeal</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><a NAME="E14:88_3_1_1_1_2"/><span class="lab"><font color="Green" title="E59">A19</font></span>: 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c11">r</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c11">r</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:88_3_1_1_1_2"><span class="lab"><font color="Green" title="E56">A16</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c11">r</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> = 
<font color="Maroon" title="c11">r</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binom.html#T3" title="BINOM:th.3">BINOM:3</a></span>
<br/>.= 
<font color="Maroon" title="c12">u</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:88_3_1_1_1_2"><span class="lab"><font color="Green" title="E57">A17</font></span></a>, <a class="txt" href="#E14:88_3_1_1_1_2"><span class="lab"><font color="Green" title="E59">A19</font></span></a>, <a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>
;<br/><a NAME="E16:88_3_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A20</font></span>: 
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c11">r</font></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> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K8" title="IDEAL_1:func.8">-LeftIdeal</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:88_3_1_1_1_2"><span class="lab"><font color="Green" title="E58">A18</font></span></a>, <a class="ref" href="ideal_1.html#D2" target="_self" title="IDEAL_1:def.2">Def2</a></span>;<br/><a NAME="E17:88_3_1_1_1_2"/>
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c9">lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c10">q</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c11">r</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:88_3_1_1_1_2"><span class="lab"><font color="Green" title="E53">A13</font></span></a>, <a class="ref" href="rlvect_1.html#T41" title="RLVECT_1:th.41">RLVECT_1:41</a></span>;<br/><span class="kw">hence </span><a NAME="E18:88_3_1_1_1_2"/>
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c9">lc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K8" title="IDEAL_1:func.8">-LeftIdeal</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:88_3_1_1_1_2"><span class="lab"><font color="Green" title="E55">A15</font></span></a>, <a class="txt" href="#E16:88_3_1_1_1_2"><span class="lab"><font color="Green" title="E60">A20</font></span></a>, <a class="ref" href="ideal_1.html#D1" target="_self" title="IDEAL_1:def.1">Def1</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="E8:88_3"/><span class="lab"><font color="Green" title="E50">A21</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_3_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c8">y</font> =  the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">F</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c9">lc</font> be    <a href="ideal_1.html#M2" title="IDEAL_1:mode.2">LeftLinearCombination</a> of <font color="Maroon" title="c2">F</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">lc</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>  implies  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c9">lc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K8" title="IDEAL_1:func.8">-LeftIdeal</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:88_3_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">lc</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">  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c9">lc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K8" title="IDEAL_1:func.8">-LeftIdeal</a> </span><br/>

<a NAME="E2:88_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">lc</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 
;<br/>
<a NAME="E3:88_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A22</font></span>: 
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c9">lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T43" title="RLVECT_1:th.43">RLVECT_1:43</a></span>;<br/>
<a NAME="E4:88_3_2"/>
<font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K8" title="IDEAL_1:func.8">-LeftIdeal</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ideal_1.html#D15" target="_self" title="IDEAL_1:def.15">Def16</a></span>;<br/>
<a NAME="E5:88_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A23</font></span>: 
 the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K8" title="IDEAL_1:func.8">-LeftIdeal</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E6:88_3_2"/>
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a>  the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="binom.html#T1" title="BINOM:th.1">BINOM:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:88_3_2"/>
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c9">lc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K8" title="IDEAL_1:func.8">-LeftIdeal</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:88_3_2"><span class="lab"><font color="Green" title="E51">A22</font></span></a>, <a class="txt" href="#E5:88_3_2"><span class="lab"><font color="Green" title="E52">A23</font></span></a>, <a class="ref" href="ideal_1.html#D2" target="_self" title="IDEAL_1:def.2">Def2</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="E9:88_3"/>
 for <font color="Olive" title="b1">k</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">k</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="#E8:88_3"><span class="lab"><font color="Green" title="E50">A21</font></span></a>, <a class="txt" href="#E7:88_3"><span class="lab"><font color="Green" title="E49">A9</font></span></a>);<br/></span>
<a NAME="E10:88_3"/><span class="kw">then </span>
<font color="Maroon">S<sub>1</sub></font>[ <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">lc</font>]
 
;<br/>
<span class="kw">hence </span><a NAME="E11:88_3"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K8" title="IDEAL_1:func.8">-LeftIdeal</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:88_3"><span class="lab"><font color="Green" title="E46">A6</font></span></a>, <a class="txt" href="#E6:88_3"><span class="lab"><font color="Green" title="E48">A8</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="c5">I</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b2">lc</font> being    <a href="ideal_1.html#M2" title="IDEAL_1:mode.2">LeftLinearCombination</a> of <font color="Maroon" title="c2">F</font> st  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b2">lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> </span> } </span>  as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:88"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="txt" href="#E1:88"><span class="lab"><font color="Green" title="E43">A1</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">I9</font> = <font color="Maroon" title="c5">I</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E6:88"/><span class="lab"><font color="Green" title="E46">A24</font></span>: 
<font color="Maroon" title="c6">I9</font> is  <a href="ideal_1.html#V1" title="IDEAL_1:attr.1">add-closed</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="ideal_1.html#D1" target="_self" title="IDEAL_1:def.1">IDEAL_1:def 1</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="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">I9</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">I9</font> implies <font color="Maroon" title="c7">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">I9</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:88_4"/><span class="lab"><font color="Green" title="E47">A25</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">I9</font>
 <span class="kw">and </span><br/><a NAME="E2:88_4"/><span class="lab"><font color="Green" title="E48">A26</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">I9</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="c7">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">I9</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c9">x9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E4:88_4"/><span class="lab"><font color="Green" title="E49">A27</font></span>: 
<font color="Maroon" title="c9">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">and </span><br/><a NAME="E5:88_4"/><span class="lab"><font color="Green" title="E50">A28</font></span>: 
 ex <font color="Olive" title="b1">lc</font> being    <a href="ideal_1.html#M2" title="IDEAL_1:mode.2">LeftLinearCombination</a> of <font color="Maroon" title="c2">F</font> st  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b1">lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:88_4"><span class="lab"><font color="Green" title="E47">A25</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">lcx</font> being    <a href="ideal_1.html#M2" title="IDEAL_1:mode.2">LeftLinearCombination</a> of <font color="Maroon" title="c2">F</font><span class="kw"> such that </span><br/><a NAME="E7:88_4"/><span class="lab"><font color="Green" title="E51">A29</font></span>: 
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c10">lcx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:88_4"><span class="lab"><font color="Green" title="E50">A28</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">y9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E9:88_4"/><span class="lab"><font color="Green" title="E52">A30</font></span>: 
<font color="Maroon" title="c11">y9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font>
 <span class="kw">and </span><br/><a NAME="E10:88_4"/><span class="lab"><font color="Green" title="E53">A31</font></span>: 
 ex <font color="Olive" title="b1">lc</font> being    <a href="ideal_1.html#M2" title="IDEAL_1:mode.2">LeftLinearCombination</a> of <font color="Maroon" title="c2">F</font> st  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b1">lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:88_4"><span class="lab"><font color="Green" title="E48">A26</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">lcy</font> being    <a href="ideal_1.html#M2" title="IDEAL_1:mode.2">LeftLinearCombination</a> of <font color="Maroon" title="c2">F</font><span class="kw"> such that </span><br/><a NAME="E12:88_4"/><span class="lab"><font color="Green" title="E54">A32</font></span>: 
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c12">lcy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:88_4"><span class="lab"><font color="Green" title="E53">A31</font></span></a></span>;<br/>
<a NAME="E13:88_4"/>
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">lcx</font> <a href="ideal_1.html#K5" title="IDEAL_1:func.5">^</a> <font color="Maroon" title="c12">lcy</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x9</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c11">y9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:88_4"><span class="lab"><font color="Green" title="E51">A29</font></span></a>, <a class="txt" href="#E12:88_4"><span class="lab"><font color="Green" title="E54">A32</font></span></a>, <a class="ref" href="rlvect_1.html#T41" title="RLVECT_1:th.41">RLVECT_1:41</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:88_4"/>
<font color="Maroon" title="c7">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">I9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:88_4"><span class="lab"><font color="Green" title="E49">A27</font></span></a>, <a class="txt" href="#E9:88_4"><span class="lab"><font color="Green" title="E52">A30</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="E7:88"/>
<font color="Maroon" title="c6">I9</font> is  <a href="ideal_1.html#V2" title="IDEAL_1:attr.2">left-ideal</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">p</font>, <font color="Maroon" title="c8">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="ideal_1.html#D2" target="_self" title="IDEAL_1:def.2">IDEAL_1:def 2</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="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">I9</font> implies <font color="Maroon" title="c7">p</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">I9</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:88_5"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">I9</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="c7">p</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">I9</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">x9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E3:88_5"/><span class="lab"><font color="Green" title="E47">A33</font></span>: 
<font color="Maroon" title="c9">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font>
 <span class="kw">and </span><br/><a NAME="E4:88_5"/><span class="lab"><font color="Green" title="E48">A34</font></span>: 
 ex <font color="Olive" title="b1">lc</font> being    <a href="ideal_1.html#M2" title="IDEAL_1:mode.2">LeftLinearCombination</a> of <font color="Maroon" title="c2">F</font> st  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b1">lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x9</font>
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">lcx</font> being    <a href="ideal_1.html#M2" title="IDEAL_1:mode.2">LeftLinearCombination</a> of <font color="Maroon" title="c2">F</font><span class="kw"> such that </span><br/><a NAME="E6:88_5"/><span class="lab"><font color="Green" title="E49">A35</font></span>: 
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c10">lcx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:88_5"><span class="lab"><font color="Green" title="E48">A34</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">plcx</font> = <font color="Maroon" title="c7">p</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c10">lcx</font> as    <a href="ideal_1.html#M2" title="IDEAL_1:mode.2">LeftLinearCombination</a> of <font color="Maroon" title="c2">F</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="ideal_1.html#T26" target="_self" title="IDEAL_1:th.26">Th26</a></span>;<br/>
<a NAME="E8:88_5"/>
<font color="Maroon" title="c7">p</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c11">plcx</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:88_5"><span class="lab"><font color="Green" title="E47">A33</font></span></a>, <a class="txt" href="#E6:88_5"><span class="lab"><font color="Green" title="E49">A35</font></span></a>, <a class="ref" href="binom.html#T4" title="BINOM:th.4">BINOM:4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:88_5"/>
<font color="Maroon" title="c7">p</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">I9</font>
 ; <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="E8:88"/><span class="kw">then </span>
<font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K8" title="IDEAL_1:func.8">-LeftIdeal</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:88"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="txt" href="#E6:88"><span class="lab"><font color="Green" title="E46">A24</font></span></a>, <a class="ref" href="ideal_1.html#D15" target="_self" title="IDEAL_1:def.15">Def16</a></span>;<br/>
<a NAME="E9:88"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A36</font></span>: 
<font color="Maroon" title="c5">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K8" title="IDEAL_1:func.8">-LeftIdeal</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:88"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<div><span class="kw">hereby </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">f</font> being    <a href="ideal_1.html#M2" title="IDEAL_1:mode.2">LeftLinearCombination</a> of <font color="Maroon" title="c2">F</font> st <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b1">f</font> implies <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K8" title="IDEAL_1:func.8">-LeftIdeal</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:88_6"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K8" title="IDEAL_1:func.8">-LeftIdeal</a> 
 ; <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">f</font> being    <a href="ideal_1.html#M2" title="IDEAL_1:mode.2">LeftLinearCombination</a> of <font color="Maroon" title="c2">F</font> st <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b1">f</font></span><br/><a NAME="E2:88_6"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> &amp;  ex <font color="Olive" title="b2">lc</font> being    <a href="ideal_1.html#M2" title="IDEAL_1:mode.2">LeftLinearCombination</a> of <font color="Maroon" title="c2">F</font> st  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b2">lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:88"><span class="lab"><font color="Green" title="E47">A36</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:88_6"/>
 ex <font color="Olive" title="b1">f</font> being    <a href="ideal_1.html#M2" title="IDEAL_1:mode.2">LeftLinearCombination</a> of <font color="Maroon" title="c2">F</font> st <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b1">f</font>
 ; <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">assume </span><a NAME="E11:88"/>
 ex <font color="Olive" title="b1">f</font> being    <a href="ideal_1.html#M2" title="IDEAL_1:mode.2">LeftLinearCombination</a> of <font color="Maroon" title="c2">F</font> st <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b1">f</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K8" title="IDEAL_1:func.8">-LeftIdeal</a> </span><br/>

<span class="kw">hence </span><a NAME="E12:88"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K8" title="IDEAL_1:func.8">-LeftIdeal</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:88"><span class="lab"><font color="Green" title="E47">A36</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>
