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<div class="add">

<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">L</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</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="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_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> <br/> for <font color="Olive" title="b2">b1</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>  st  ex <font color="Olive" title="b3">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Olive" title="b2">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b3">u</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/>for <font color="Olive" title="b3">b2</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font><br/> for <font color="Olive" title="b4">x</font> being  <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">L</font> holds   <a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Olive" title="b3">b2</font></span>)</span>,<font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Olive" title="b2">b1</font>,<font color="Olive" title="b4">x</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Olive" title="b3">b2</font>,<font color="Olive" title="b4">x</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</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="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_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">b1</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>  st  ex <font color="Olive" title="b2">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Olive" title="b1">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">u</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/>for <font color="Olive" title="b2">b2</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font><br/> for <font color="Olive" title="b3">x</font> being  <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font> holds   <a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Olive" title="b2">b2</font></span>)</span>,<font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Olive" title="b1">b1</font>,<font color="Olive" title="b3">x</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Olive" title="b2">b2</font>,<font color="Olive" title="b3">x</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">b1</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c3">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">u</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies for <font color="Olive" title="b1">b2</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font><br/> for <font color="Olive" title="b2">x</font> being  <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font> holds   <a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Olive" title="b1">b2</font></span>)</span>,<font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c3">b1</font>,<font color="Olive" title="b2">x</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Olive" title="b1">b2</font>,<font color="Olive" title="b2">x</font></span>)</span> )</span><br/>





<b>assume </b><a NAME="E1:35"/><i><font color="Green" title="E26">A1</font></i>: 
 ex <font color="Olive" title="b1">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c3">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">u</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">b2</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font><br/> for <font color="Olive" title="b2">x</font> being  <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font> holds   <a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Olive" title="b1">b2</font></span>)</span>,<font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c3">b1</font>,<font color="Olive" title="b2">x</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Olive" title="b1">b2</font>,<font color="Olive" title="b2">x</font></span>)</span></span><br/>

<b>let </b><font color="Maroon" title="c4">b2</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font> holds   <a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span>,<font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c3">b1</font>,<font color="Olive" title="b1">x</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c4">b2</font>,<font color="Olive" title="b1">x</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c5">x</font> be   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span>,<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c4">b2</font>,<font color="Maroon" title="c5">x</font></span>)</span></span><br/>



<a NAME="E2:35"/><i><font color="Green" title="E27">A2</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">x</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:35"/><i><font color="Green" title="E28">A3</font></i>: 
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c3">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">u</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="#E1:35"><i><font color="Green" title="E26">A1</font></i></a></i>;<br/>
<a NAME="E5:35"/><i><font color="Green" title="E29">A4</font></i>: 
for <font color="Olive" title="b1">u'</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">u'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">u</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">u'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">u'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">u'</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">u'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">u</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">u'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">u'</font> )</span><br/>

<b>assume </b><a NAME="E1:35_1"/>
<font color="Maroon" title="c7">u'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">u</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">u'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">u'</font></span><br/>

<a NAME="E2:35_1"/><b>then </b><i><font color="Green" title="E30">A5</font></i>: 
not <font color="Maroon" title="c7">u'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c3">b1</font>
 
<b>by </b><i><a class="txt" href="#E4:35"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">u'</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">u'</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">u'</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D5" title="POLYNOM1:def.5">POLYNOM1:def 5</a></i>
<br/>.= 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">u'</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:35_1"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="polynom1.html#D7" title="POLYNOM1:def.7">POLYNOM1:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c4">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">u'</font>

; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c7">sb2</font> =  <a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c8">sb1b2</font> =  <a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c9">yb1b2</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font><b> such that </b><br/><a NAME="E7:35"/><i><font color="Green" title="E30">A6</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">yb1b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span>,<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c9">yb1b2</font> &amp; ( for <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 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">yb1b2</font> holds <br/><font color="Maroon" title="c9">yb1b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="polynom2.html#K2" title="POLYNOM2:func.2">*</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="polynom2.html#D2" target="_self" title="POLYNOM2:def.2">Def2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">yb2</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font><b> such that </b><br/><a NAME="E9:35"/><i><font color="Green" title="E31">A7</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">yb2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span> &amp;  <a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c4">b2</font>,<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c10">yb2</font> &amp; ( for <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 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">yb2</font> holds <br/><font color="Maroon" title="c10">yb2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="polynom2.html#K2" title="POLYNOM2:func.2">*</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="polynom2.html#D2" target="_self" title="POLYNOM2:def.2">Def2</a></i>;<br/>
<a NAME="E10:35"/><i><font color="Green" title="E32">A8</font></i>: 
(  <a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font> <a href="orders_1.html#R3" title="ORDERS_1:pred.3">linearly_orders</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font> &amp;  <a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font> <a href="orders_1.html#R3" title="ORDERS_1:pred.3">linearly_orders</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="polynom2.html#T15" target="_self" title="POLYNOM2:th.15">Th15</a></i>;<br/>
<a NAME="E11:35"/><i><font color="Green" title="E33">A9</font></i>: 
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">u</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="#E4:35"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="polynom1.html#T42" title="POLYNOM1:th.42">POLYNOM1:42</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_2"/>
( <font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font> or  not <font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_2_1"><b>suppose </b></a><a NAME="E1:35_2_1"/><i><font color="Green" title="E34">A10</font></i>: 
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span>,<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c4">b2</font>,<font color="Maroon" title="c5">x</font></span>)</span></span><br/><div class="add"><a NAME="E2:35_2_1"/><i><font color="Green" title="E35">A11</font></i>: 
for <font color="Olive" title="b1">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span> holds <br/><font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span> implies <font color="Maroon" title="c11">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font> )</span><br/>

<b>assume </b><a NAME="E1:35_2_1_1"/><i><font color="Green" title="E36">A12</font></i>: 
<font color="Maroon" title="c11">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_2_1_1_1_1"/>
( <font color="Maroon" title="c11">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">u</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c11">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font> )
 </span><b>by </b><i><a class="txt" href="#E11:35"><i><font color="Green" title="E33">A9</font></i></a>, <a class="txt" href="#E1:35_2_1_1"><i><font color="Green" title="E36">A12</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_2_1_1_1_1_1"><b>case </b></a><a NAME="E1:35_2_1_1_1_1_1"/>
<font color="Maroon" title="c11">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">u</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span><br/><div class="add"><b>hence </b><a NAME="E2:35_2_1_1_1_1_1"/>
<font color="Maroon" title="c11">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font>
 <b>by </b><i><a class="txt" href="#E1:35_2_1"><i><font color="Green" title="E34">A10</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_2_1_1_1_1_2"><b>case </b></a><a NAME="E1:35_2_1_1_1_1_2"/>
<font color="Maroon" title="c11">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span><br/><div class="add"><b>hence </b><a NAME="E2:35_2_1_1_1_1_2"/>
<font color="Maroon" title="c11">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:35_2_1_1"/>
<font color="Maroon" title="c11">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E3:35_2_1"/><i><font color="Green" title="E36">A13</font></i>: 
for <font color="Olive" title="b1">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font> holds <br/><font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font> implies <font color="Maroon" title="c11">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:35_2_1_2"/><i><font color="Green" title="E37">A14</font></i>: 
<font color="Maroon" title="c11">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_2_1_2_1_1"/>
( <font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">v</font> or <font color="Maroon" title="c6">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">v</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_2_1_2_1_1_1"><b>case </b></a><a NAME="E1:35_2_1_2_1_1_1"/>
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">v</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span></span><br/><div class="add"><a NAME="E2:35_2_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c11">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">u</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:35_2_1_2_1_1_1"/>
<font color="Maroon" title="c11">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E11:35"><i><font color="Green" title="E33">A9</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_2_1_2_1_1_2"><b>case </b></a><a NAME="E1:35_2_1_2_1_1_2"/>
<font color="Maroon" title="c6">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">v</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span></span><br/><div class="add"><a NAME="E2:35_2_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">v</font>
 <b>by </b><i><a class="txt" href="#E5:35"><i><font color="Green" title="E29">A4</font></i></a></i>;<br/><a NAME="E3:35_2_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E1:35_2_1_2"><i><font color="Green" title="E37">A14</font></i></a>, <a class="ref" href="polynom1.html#D7" title="POLYNOM1:def.7">POLYNOM1:def 7</a></i>;<br/><b>hence </b><a NAME="E4:35_2_1_2_1_1_2"/>
<font color="Maroon" title="c11">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#D7" title="POLYNOM1:def.7">POLYNOM1:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:35_2_1_2"/>
<font color="Maroon" title="c11">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E4:35_2_1"/><b>then </b><i><font color="Green" title="E37">A15</font></i>: 
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font>
 <b>by </b><i><a class="txt" href="#E2:35_2_1"><i><font color="Green" title="E35">A11</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/><a NAME="E5:35_2_1"/><i><font color="Green" title="E38">A16</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">yb1b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">yb2</font>
 <b>by </b><i><a class="txt" href="#E7:35"><i><font color="Green" title="E30">A6</font></i></a>, <a class="txt" href="#E9:35"><i><font color="Green" title="E31">A7</font></i></a>, <a class="txt" href="#E2:35_2_1"><i><font color="Green" title="E35">A11</font></i></a>, <a class="txt" href="#E3:35_2_1"><i><font color="Green" title="E36">A13</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/><a NAME="E6:35_2_1"/>
<font color="Maroon" title="c6">u</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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E10:35"><i><font color="Green" title="E32">A8</font></i></a>, <a class="txt" href="#E1:35_2_1"><i><font color="Green" title="E34">A10</font></i></a>, <a class="ref" href="triang_1.html#D2" title="TRIANG_1:def.2">TRIANG_1:def 2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E8:35_2_1"/><i><font color="Green" title="E39">A17</font></i>: 
( <font color="Maroon" title="c11">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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">u</font> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">k</font> = <font color="Maroon" title="c11">k</font> as    <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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>consider </b><font color="Maroon" title="c13">sb2k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E11:35_2_1"/><i><font color="Green" title="E40">A18</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c13">sb2k</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></i>;<br/><b>consider </b><font color="Maroon" title="c14">sb1b2k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E13:35_2_1"/><i><font color="Green" title="E41">A19</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c14">sb1b2k</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c15">sb2k</font> = <font color="Maroon" title="c13">sb2k</font>, <font color="Maroon" title="c16">sb1b2k</font> = <font color="Maroon" title="c14">sb1b2k</font> as    <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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><i><font color="Green" title="E42">A20</font></i>: <font color="Maroon" title="c15">sb2k</font> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E11:35_2_1"><i><font color="Green" title="E40">A18</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E2:35_2_1"><i><font color="Green" title="E35">A11</font></i></a>, <a class="txt" href="#E3:35_2_1"><i><font color="Green" title="E36">A13</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>
<br/>.= 
<font color="Maroon" title="c16">sb1b2k</font>
<b>by </b><i><a class="txt" href="#E13:35_2_1"><i><font color="Green" title="E41">A19</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/><a NAME="E16:35_2_1"/><i><font color="Green" title="E43">A21</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">k</font> &amp; <font color="Maroon" title="c12">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">sb2k</font> )
 <b>by </b><i><a class="txt" href="#E8:35_2_1"><i><font color="Green" title="E39">A17</font></i></a>, <a class="txt" href="#E11:35_2_1"><i><font color="Green" title="E40">A18</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><b>consider </b><font color="Maroon" title="c17">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E18:35_2_1"/><i><font color="Green" title="E44">A22</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">yb2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c17">i</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></i>;<br/><a NAME="E19:35_2_1"/>
<font color="Maroon" title="c17">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E20:35_2_1"/><b>then </b><i><font color="Green" title="E45">A23</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">yb2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i</font>
 <b>by </b><i><a class="txt" href="#E18:35_2_1"><i><font color="Green" title="E44">A22</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E21:35_2_1"/><i><font color="Green" title="E46">A24</font></i>: 
<font color="Maroon" title="c15">sb2k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">yb2</font>
 <b>by </b><i><a class="txt" href="#E9:35"><i><font color="Green" title="E31">A7</font></i></a>, <a class="txt" href="#E11:35_2_1"><i><font color="Green" title="E40">A18</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E22:35_2_1"/><b>then </b><i><font color="Green" title="E47">A25</font></i>: 
<font color="Maroon" title="c12">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> <font color="Maroon" title="c10">yb2</font>
 <b>by </b><i><a class="txt" href="#E8:35_2_1"><i><font color="Green" title="E39">A17</font></i></a>, <a class="txt" href="#E11:35_2_1"><i><font color="Green" title="E40">A18</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E23:35_2_1"/><i><font color="Green" title="E48">A26</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">yb1b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">sb2k</font>
 <b>by </b><i><a class="txt" href="#E7:35"><i><font color="Green" title="E30">A6</font></i></a>, <a class="txt" href="#E13:35_2_1"><i><font color="Green" title="E41">A19</font></i></a>, <a class="txt" href="#E15:35_2_1"><i><font color="Green" title="E42">A20</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E24:35_2_1"/><i><font color="Green" title="E49">A27</font></i>: 
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">b2</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T41" title="POLYNOM1:th.41">POLYNOM1:41</a></i>;<br/><a NAME="E25:35_2_1"/><b>then </b><i><font color="Green" title="E50">A28</font></i>: 
<font color="Maroon" title="c12">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="polynom2.html#K2" title="POLYNOM2:func.2">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:35_2_1"><i><font color="Green" title="E34">A10</font></i></a>, <a class="txt" href="#E8:35_2_1"><i><font color="Green" title="E39">A17</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><a NAME="E26:35_2_1"/><i><font color="Green" title="E51">A29</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E27:35_2_1"/>
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#T41" title="POLYNOM1:th.41">POLYNOM1:41</a></i>;<br/><a NAME="E28:35_2_1"/><b>then </b><i><font color="Green" title="E52">A30</font></i>: 
<font color="Maroon" title="c12">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="polynom2.html#K2" title="POLYNOM2:func.2">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:35_2_1"><i><font color="Green" title="E34">A10</font></i></a>, <a class="txt" href="#E4:35_2_1"><i><font color="Green" title="E37">A15</font></i></a>, <a class="txt" href="#E8:35_2_1"><i><font color="Green" title="E39">A17</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><b>then </b><i><font color="Green" title="E53">A31</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="polynom2.html#K2" title="POLYNOM2:func.2">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="polynom2.html#K2" title="POLYNOM2:func.2">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">k</font>
<b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">u</font>
<b>by </b><i><a class="txt" href="#E8:35_2_1"><i><font color="Green" title="E39">A17</font></i></a>, <a class="txt" href="#E28:35_2_1"><i><font color="Green" title="E52">A30</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
;<br/><i><font color="Green" title="E54">A32</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="polynom2.html#K2" title="POLYNOM2:func.2">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="polynom2.html#K2" title="POLYNOM2:func.2">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">k</font>
<b>by </b><i><a class="txt" href="#E25:35_2_1"><i><font color="Green" title="E50">A28</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c4">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">u</font>
<b>by </b><i><a class="txt" href="#E8:35_2_1"><i><font color="Green" title="E39">A17</font></i></a>, <a class="txt" href="#E25:35_2_1"><i><font color="Green" title="E50">A28</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
;<br/><i><font color="Green" title="E55">A33</font></i>: <font color="Maroon" title="c9">yb1b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">k</font> = 
<span class="p1">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="polynom2.html#K2" title="POLYNOM2:func.2">*</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">k</font></span>)</span>
<b>by </b><i><a class="txt" href="#E7:35"><i><font color="Green" title="E30">A6</font></i></a>, <a class="txt" href="#E4:35_2_1"><i><font color="Green" title="E37">A15</font></i></a>, <a class="txt" href="#E16:35_2_1"><i><font color="Green" title="E43">A21</font></i></a>, <a class="txt" href="#E23:35_2_1"><i><font color="Green" title="E48">A26</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">u</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">u</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E29:35_2_1"><i><font color="Green" title="E53">A31</font></i></a>, <a class="ref" href="polynom1.html#D5" title="POLYNOM1:def.5">POLYNOM1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p5">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">u</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p5">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">k</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="polynom2.html#K2" title="POLYNOM2:func.2">*</a> <span class="p4">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p5">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E30:35_2_1"><i><font color="Green" title="E54">A32</font></i></a>, <a class="ref" href="polynom2.html#T2" target="_self" title="POLYNOM2:th.2">Th2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p5">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">u</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">yb2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">k</font></span>)</span>
<b>by </b><i><a class="txt" href="#E9:35"><i><font color="Green" title="E31">A7</font></i></a>, <a class="txt" href="#E16:35_2_1"><i><font color="Green" title="E43">A21</font></i></a>, <a class="txt" href="#E21:35_2_1"><i><font color="Green" title="E46">A24</font></i></a></i>
;<br/><a NAME="E32:35_2_1"/><i><font color="Green" title="E56">A34</font></i>: 
for <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 <font color="Olive" title="b1">i'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">yb2</font> &amp; <font color="Olive" title="b1">i'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">k</font> holds <br/><font color="Maroon" title="c9">yb1b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">yb2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">i'</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c18">i'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">yb2</font> &amp; <font color="Maroon" title="c18">i'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">k</font> implies <font color="Maroon" title="c9">yb1b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">i'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">yb2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">i'</font> )</span><br/>

<b>assume </b><a NAME="E1:35_2_1_7"/><i><font color="Green" title="E57">A35</font></i>: 
( <font color="Maroon" title="c18">i'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">yb2</font> &amp; <font color="Maroon" title="c18">i'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">k</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">yb1b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">i'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">yb2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">i'</font></span><br/>

<a NAME="E2:35_2_1_7"/><b>then </b><i><font color="Green" title="E58">A36</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">i'</font> &amp; <font color="Maroon" title="c18">i'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">yb2</font> )
 
<b>by </b><i><a class="txt" href="#E18:35_2_1"><i><font color="Green" title="E44">A22</font></i></a>, <a class="txt" href="#E20:35_2_1"><i><font color="Green" title="E45">A23</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:35_2_1_7"/><i><font color="Green" title="E59">A37</font></i>: 
<font color="Maroon" title="c18">i'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c16">sb1b2k</font>
 
<b>by </b><i><a class="txt" href="#E15:35_2_1"><i><font color="Green" title="E42">A20</font></i></a>, <a class="txt" href="#E21:35_2_1"><i><font color="Green" title="E46">A24</font></i></a>, <a class="txt" href="#E1:35_2_1_7"><i><font color="Green" title="E57">A35</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:35_2_1_7"/><i><font color="Green" title="E60">A38</font></i>: 
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">i'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">u</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_1_7_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:35_2_1_7_1"/>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">i'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">u</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:35_2_1_7_1"/><b>then </b><i><font color="Green" title="E61">A39</font></i>: 
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">u</font>
 
<b>by </b><i><a class="txt" href="#E13:35_2_1"><i><font color="Green" title="E41">A19</font></i></a>, <a class="txt" href="#E3:35_2_1_7"><i><font color="Green" title="E59">A37</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E3:35_2_1_7_1"/><i><font color="Green" title="E62">A40</font></i>: 
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">u</font>
 
<b>by </b><i><a class="txt" href="#E2:35_2_1"><i><font color="Green" title="E35">A11</font></i></a>, <a class="txt" href="#E3:35_2_1"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="#E8:35_2_1"><i><font color="Green" title="E39">A17</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<a NAME="E4:35_2_1_7_1"/>
 <a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="polynom2.html#T15" target="_self" title="POLYNOM2:th.15">Th15</a>, <a class="ref" href="triang_1.html#T8" title="TRIANG_1:th.8">TRIANG_1:8</a></i>;<br/>
<b>hence </b><a NAME="E5:35_2_1_7_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E8:35_2_1"><i><font color="Green" title="E39">A17</font></i></a>, <a class="txt" href="#E11:35_2_1"><i><font color="Green" title="E40">A18</font></i></a>, <a class="txt" href="#E13:35_2_1"><i><font color="Green" title="E41">A19</font></i></a>, <a class="txt" href="#E15:35_2_1"><i><font color="Green" title="E42">A20</font></i></a>, <a class="txt" href="#E1:35_2_1_7"><i><font color="Green" title="E57">A35</font></i></a>, <a class="txt" href="#E3:35_2_1_7"><i><font color="Green" title="E59">A37</font></i></a>, <a class="txt" href="#E2:35_2_1_7_1"><i><font color="Green" title="E61">A39</font></i></a>, <a class="txt" href="#E3:35_2_1_7_1"><i><font color="Green" title="E62">A40</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E5:35_2_1_7"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">b2</font>
 
<b>by </b><i><a class="txt" href="#E10:35"><i><font color="Green" title="E32">A8</font></i></a>, <a class="txt" href="#E4:35_2_1"><i><font color="Green" title="E37">A15</font></i></a>, <a class="txt" href="#E24:35_2_1"><i><font color="Green" title="E49">A27</font></i></a>, <a class="ref" href="triang_1.html#D2" title="TRIANG_1:def.2">TRIANG_1:def 2</a></i>;<br/>
<a NAME="E6:35_2_1_7"/><b>then </b><i><font color="Green" title="E61">A41</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E7:35_2_1_7"/>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i'</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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E13:35_2_1"><i><font color="Green" title="E41">A19</font></i></a>, <a class="txt" href="#E3:35_2_1_7"><i><font color="Green" title="E59">A37</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:35_2_1_7"/><b>then </b><i><font color="Green" title="E62">A42</font></i>: 
<font color="Maroon" title="c18">i'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="polynom2.html#K2" title="POLYNOM2:func.2">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E13:35_2_1"><i><font color="Green" title="E41">A19</font></i></a>, <a class="txt" href="#E26:35_2_1"><i><font color="Green" title="E51">A29</font></i></a>, <a class="txt" href="#E3:35_2_1_7"><i><font color="Green" title="E59">A37</font></i></a>, <a class="txt" href="#E6:35_2_1_7"><i><font color="Green" title="E61">A41</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/>
<b>then </b><i><font color="Green" title="E63">A43</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="polynom2.html#K2" title="POLYNOM2:func.2">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">i'</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="polynom2.html#K2" title="POLYNOM2:func.2">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c18">i'</font>
<b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i'</font></span>)</span>
<b>by </b><i><a class="txt" href="#E8:35_2_1_7"><i><font color="Green" title="E62">A42</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">i'</font></span>)</span>
<b>by </b><i><a class="txt" href="#E13:35_2_1"><i><font color="Green" title="E41">A19</font></i></a>, <a class="txt" href="#E3:35_2_1_7"><i><font color="Green" title="E59">A37</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
;<br/>
<a NAME="E10:35_2_1_7"/><i><font color="Green" title="E64">A44</font></i>: 
<font color="Maroon" title="c18">i'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E9:35"><i><font color="Green" title="E31">A7</font></i></a>, <a class="txt" href="#E18:35_2_1"><i><font color="Green" title="E44">A22</font></i></a>, <a class="txt" href="#E20:35_2_1"><i><font color="Green" title="E45">A23</font></i></a>, <a class="txt" href="#E1:35_2_1_7"><i><font color="Green" title="E57">A35</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E11:35_2_1_7"/><i><font color="Green" title="E65">A45</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">b2</font>
 
<b>by </b><i><a class="txt" href="#E10:35"><i><font color="Green" title="E32">A8</font></i></a>, <a class="txt" href="#E24:35_2_1"><i><font color="Green" title="E49">A27</font></i></a>, <a class="ref" href="triang_1.html#D2" title="TRIANG_1:def.2">TRIANG_1:def 2</a></i>;<br/>
<a NAME="E12:35_2_1_7"/>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i'</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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E10:35_2_1_7"><i><font color="Green" title="E64">A44</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E13:35_2_1_7"/><b>then </b><i><font color="Green" title="E66">A46</font></i>: 
<font color="Maroon" title="c18">i'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="polynom2.html#K2" title="POLYNOM2:func.2">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E10:35_2_1_7"><i><font color="Green" title="E64">A44</font></i></a>, <a class="txt" href="#E11:35_2_1_7"><i><font color="Green" title="E65">A45</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/>
<b>then </b><i><font color="Green" title="E67">A47</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="polynom2.html#K2" title="POLYNOM2:func.2">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">i'</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="polynom2.html#K2" title="POLYNOM2:func.2">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c18">i'</font>
<b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c4">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i'</font></span>)</span>
<b>by </b><i><a class="txt" href="#E13:35_2_1_7"><i><font color="Green" title="E66">A46</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon" title="c4">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">i'</font></span>)</span>
<b>by </b><i><a class="txt" href="#E10:35_2_1_7"><i><font color="Green" title="E64">A44</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
;<br/>
<b>thus </b><font color="Maroon" title="c9">yb1b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">i'</font> = 
<span class="p1">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">i'</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">i'</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E7:35"><i><font color="Green" title="E30">A6</font></i></a>, <a class="txt" href="#E5:35_2_1"><i><font color="Green" title="E38">A16</font></i></a>, <a class="txt" href="#E2:35_2_1_7"><i><font color="Green" title="E58">A36</font></i></a>, <a class="txt" href="#E9:35_2_1_7"><i><font color="Green" title="E63">A43</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">i'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">i'</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E5:35"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="#E4:35_2_1"><i><font color="Green" title="E37">A15</font></i></a>, <a class="txt" href="#E4:35_2_1_7"><i><font color="Green" title="E60">A38</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c10">yb2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">i'</font>
<b>by </b><i><a class="txt" href="#E9:35"><i><font color="Green" title="E31">A7</font></i></a>, <a class="txt" href="#E2:35_2_1_7"><i><font color="Green" title="E58">A36</font></i></a>, <a class="txt" href="#E14:35_2_1_7"><i><font color="Green" title="E67">A47</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E33:35_2_1"/><i><font color="Green" title="E57">A48</font></i>: 
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c3">b1</font>
 <b>by </b><i><a class="txt" href="#E4:35"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E34:35_2_1"/>
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c3">b1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">b1</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T41" title="POLYNOM1:th.41">POLYNOM1:41</a></i>;<br/><a NAME="E35:35_2_1"/><b>then </b><i><font color="Green" title="E58">A49</font></i>: 
<font color="Maroon" title="c6">u</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">b1</font>
 <b>by </b><i><a class="txt" href="#E33:35_2_1"><i><font color="Green" title="E57">A48</font></i></a></i>;<br/><a NAME="E36:35_2_1"/><i><font color="Green" title="E59">A50</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E37:35_2_1"/><b>then </b><i><font color="Green" title="E60">A51</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">u</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> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="txt" href="#E2:35"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="#E35:35_2_1"><i><font color="Green" title="E58">A49</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E38:35_2_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">x</font> <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="c2">L</font>
 <b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c18">xu</font> = <font color="Maroon" title="c5">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">u</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font> <b>by </b><i><a class="txt" href="#E37:35_2_1"><i><font color="Green" title="E60">A51</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c19">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font><b> such that </b><br/><a NAME="E41:35_2_1"/><i><font color="Green" title="E61">A52</font></i>: 
<font color="Maroon" title="c19">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c18">xu</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">u</font></span>)</span>
 ;<br/><a NAME="E42:35_2_1"/><i><font color="Green" title="E62">A53</font></i>: 
<font color="Maroon" title="c12">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:35"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="#E8:35_2_1"><i><font color="Green" title="E39">A17</font></i></a>, <a class="txt" href="#E35:35_2_1"><i><font color="Green" title="E58">A49</font></i></a>, <a class="txt" href="#E36:35_2_1"><i><font color="Green" title="E59">A50</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><a NAME="E43:35_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/><a NAME="E44:35_2_1"/><b>then </b>
<font color="Maroon" title="c9">yb1b2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">yb2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E8:35_2_1"><i><font color="Green" title="E39">A17</font></i></a>, <a class="txt" href="#E31:35_2_1"><i><font color="Green" title="E55">A33</font></i></a>, <a class="txt" href="#E41:35_2_1"><i><font color="Green" title="E61">A52</font></i></a>, <a class="txt" href="#E42:35_2_1"><i><font color="Green" title="E62">A53</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><b>hence </b> <a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span>,<font color="Maroon" title="c5">x</font> = 
<font color="Maroon" title="c19">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c10">yb2</font></span>)</span>
<b>by </b><i><a class="txt" href="#E7:35"><i><font color="Green" title="E30">A6</font></i></a>, <a class="txt" href="#E5:35_2_1"><i><font color="Green" title="E38">A16</font></i></a>, <a class="txt" href="#E22:35_2_1"><i><font color="Green" title="E47">A25</font></i></a>, <a class="txt" href="#E32:35_2_1"><i><font color="Green" title="E56">A34</font></i></a>, <a class="ref" href="polynom2.html#T8" target="_self" title="POLYNOM2:th.8">Th8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c4">b2</font>,<font color="Maroon" title="c5">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E4:35"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="#E9:35"><i><font color="Green" title="E31">A7</font></i></a>, <a class="txt" href="#E41:35_2_1"><i><font color="Green" title="E61">A52</font></i></a>, <a class="ref" href="polynom2.html#T17" target="_self" title="POLYNOM2:th.17">Th17</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_2_2"><b>suppose </b></a><a NAME="E1:35_2_2"/><i><font color="Green" title="E34">A54</font></i>: 
not <font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c4">b2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span>,<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c4">b2</font>,<font color="Maroon" title="c5">x</font></span>)</span></span><br/><div class="add"><i><font color="Green" title="E35">A55</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">u</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">u</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">u</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D5" title="POLYNOM1:def.5">POLYNOM1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">u</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="#E1:35_2_2"><i><font color="Green" title="E34">A54</font></i></a>, <a class="ref" href="polynom1.html#D7" title="POLYNOM1:def.7">POLYNOM1:def 7</a></i>
;<br/><a NAME="E3:35_2_2"/><i><font color="Green" title="E36">A56</font></i>: 
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c3">b1</font>
 <b>by </b><i><a class="txt" href="#E4:35"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E4:35_2_2"/>
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c3">b1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">b1</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T41" title="POLYNOM1:th.41">POLYNOM1:41</a></i>;<br/><a NAME="E5:35_2_2"/><b>then </b><i><font color="Green" title="E37">A57</font></i>: 
<font color="Maroon" title="c6">u</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">b1</font>
 <b>by </b><i><a class="txt" href="#E3:35_2_2"><i><font color="Green" title="E36">A56</font></i></a></i>;<br/><a NAME="E6:35_2_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E7:35_2_2"/><b>then </b><i><font color="Green" title="E38">A58</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">u</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> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="txt" href="#E2:35"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="#E5:35_2_2"><i><font color="Green" title="E37">A57</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E8:35_2_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">x</font> <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="c2">L</font>
 <b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c11">xu</font> = <font color="Maroon" title="c5">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">u</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font> <b>by </b><i><a class="txt" href="#E7:35_2_2"><i><font color="Green" title="E38">A58</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c12">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font><b> such that </b><br/><a NAME="E11:35_2_2"/><i><font color="Green" title="E39">A59</font></i>: 
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">xu</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">u</font></span>)</span>
 ;<br/><b>thus </b> <a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">+</a> <font color="Maroon" title="c4">b2</font></span>)</span>,<font color="Maroon" title="c5">x</font> = 
<font color="Maroon" title="c12">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c4">b2</font>,<font color="Maroon" title="c5">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E5:35"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="#E11:35"><i><font color="Green" title="E33">A9</font></i></a>, <a class="txt" href="#E1:35_2_2"><i><font color="Green" title="E34">A54</font></i></a>, <a class="txt" href="#E11:35_2_2"><i><font color="Green" title="E39">A59</font></i></a>, <a class="txt" href="polynom2.html#E27"><i><font color="Green" title="E24">Lm8</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c4">b2</font>,<font color="Maroon" title="c5">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E4:35"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="#E2:35_2_2"><i><font color="Green" title="E35">A55</font></i></a>, <a class="txt" href="#E11:35_2_2"><i><font color="Green" title="E39">A59</font></i></a>, <a class="ref" href="polynom2.html#T17" target="_self" title="POLYNOM2:th.17">Th17</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
