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<span class="kw">let </span><font color="Maroon" title="c1">F</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">b1</font>, <font color="Olive" title="b2">c1</font>, <font color="Olive" title="b3">b2</font>, <font color="Olive" title="b4">c2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font> holds  <span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Olive" title="b2">c1</font>,<font color="Olive" title="b1">b1</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Olive" title="b4">c2</font>,<font color="Olive" title="b3">b2</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="field_9.html#K4" title="FIELD_9:func.4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">c1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b4">c2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b4">c2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">b2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">c1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">b2</font></span>)</span></span><a href="field_9.html#K4" title="FIELD_9:func.4">%&gt;</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">b1</font>, <font color="Maroon" title="c3">c1</font>, <font color="Maroon" title="c4">b2</font>, <font color="Maroon" title="c5">c2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c2">b1</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c4">b2</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="field_9.html#K4" title="FIELD_9:func.4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">c1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">c1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">b2</font></span>)</span></span><a href="field_9.html#K4" title="FIELD_9:func.4">%&gt;</a></span></span><br/>



<span class="kw">set </span><font color="Maroon" title="c6">p1</font> = <span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c2">b1</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">p2</font> = <span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c4">b2</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">q</font> = <span class="p1"><a href="field_9.html#K4" title="FIELD_9:func.4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">c1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">c1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">b2</font></span>)</span></span><a href="field_9.html#K4" title="FIELD_9:func.4">%&gt;</a></span>;<br/>
<a NAME="E1:63"/><span class="lab"><font color="Green" title="E26">J</font></span>: 
4 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">XREAL_0:def 2</a></span>;<br/>
<a NAME="E2:63"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">K</font></span>: 
( 2 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; 3 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; 3 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 &amp; 4 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">XREAL_0:def 2</a></span>;<br/>
<span class="lab"><font color="Green" title="E28">A</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c2">b1</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c4">b2</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span></span>)</span> = 
 <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>
<br/>.= 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1"><a href="field_9.html#K4" title="FIELD_9:func.4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">c1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">c1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">b2</font></span>)</span></span><a href="field_9.html#K4" title="FIELD_9:func.4">%&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>
;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">o</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1"><a href="field_9.html#K4" title="FIELD_9:func.4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">c1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">c1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">b2</font></span>)</span></span><a href="field_9.html#K4" title="FIELD_9:func.4">%&gt;</a></span> holds <br/><span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c2">b1</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c4">b2</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="field_9.html#K4" title="FIELD_9:func.4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">c1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">c1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">b2</font></span>)</span></span><a href="field_9.html#K4" title="FIELD_9:func.4">%&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">o</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1"><a href="field_9.html#K4" title="FIELD_9:func.4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">c1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">c1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">b2</font></span>)</span></span><a href="field_9.html#K4" title="FIELD_9:func.4">%&gt;</a></span> implies <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c2">b1</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c4">b2</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="field_9.html#K4" title="FIELD_9:func.4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">c1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">c1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">b2</font></span>)</span></span><a href="field_9.html#K4" title="FIELD_9:func.4">%&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">assume </span><a NAME="E1:63_2"/>
<font color="Maroon" title="c9">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1"><a href="field_9.html#K4" title="FIELD_9:func.4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">c1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">c1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">b2</font></span>)</span></span><a href="field_9.html#K4" title="FIELD_9:func.4">%&gt;</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c2">b1</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c4">b2</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="field_9.html#K4" title="FIELD_9:func.4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">c1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">c1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">b2</font></span>)</span></span><a href="field_9.html#K4" title="FIELD_9:func.4">%&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">i</font> = <font color="Maroon" title="c9">o</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  ;<br/><span class="kw">consider </span><font color="Maroon" title="c11">r</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="c1">F</font><span class="kw"> such that </span><br/><a NAME="E4:63_2"/><span class="lab"><font color="Green" title="E29">B1</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c2">b1</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c4">b2</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c11">r</font> &amp; (  for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">r</font> holds <br/><font color="Maroon" title="c11">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c2">b1</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c4">b2</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#D9" onmouseover="rs('polynom3/D9')" onmouseout="rh()">POLYNOM3:def 9</a></span>;<br/><a NAME="E5:63_2"/>
( <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 implies  not  not <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  ...  &amp;  not <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:63_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:63_2_1_1"/><span class="lab"><font color="Green" title="E30">C</font></span>: 
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c2">b1</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c4">b2</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="field_9.html#K4" title="FIELD_9:func.4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">c1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">c1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">b2</font></span>)</span></span><a href="field_9.html#K4" title="FIELD_9:func.4">%&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:63_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">B2</font></span>: 
<font color="Maroon" title="c11">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:63_2"><span class="lab"><font color="Green" title="E29">B1</font></span></a>, <a class="ref" href="finseq_1.html#T40" onmouseover="rs('finseq_1/T40')" onmouseout="rh()">FINSEQ_1:40</a></span>;<br/><a NAME="E3:63_2_1_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">r</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">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T2" onmouseover="rs('finseq_1/T2')" onmouseout="rh()">FINSEQ_1:2</a>, <a class="ref" href="finseq_1.html#T38" onmouseover="rs('finseq_1/T38')" onmouseout="rh()">FINSEQ_1:38</a></span>;<br/><a NAME="E4:63_2_1_1"/><span class="kw">then </span>
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c11">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c2">b1</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c4">b2</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63_2_1_1"><span class="lab"><font color="Green" title="E30">C</font></span></a>, <a class="txt" href="#E4:63_2"><span class="lab"><font color="Green" title="E29">B1</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c2">b1</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c4">b2</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_2.html#T8" onmouseover="rs('nat_2/T8')" onmouseout="rh()">NAT_2:8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c2">b1</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c4">b2</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_2.html#T8" onmouseover="rs('nat_2/T8')" onmouseout="rh()">NAT_2:8</a></span>
<br/>.= 
<font color="Maroon" title="c3">c1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c4">b2</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T38" onmouseover="rs('polynom5/T38')" onmouseout="rh()">POLYNOM5:38</a></span>
<br/>.= 
<font color="Maroon" title="c3">c1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T38" onmouseover="rs('polynom5/T38')" onmouseout="rh()">POLYNOM5:38</a></span>
;<br/><a NAME="E6:63_2_1_1"/><span class="kw">then </span>
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c11">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63_2_1_1"><span class="lab"><font color="Green" title="E31">B2</font></span></a>, <a class="ref" href="rlvect_1.html#T44" onmouseover="rs('rlvect_1/T44')" onmouseout="rh()">RLVECT_1:44</a></span>;<br/><span class="kw">hence </span><a NAME="E7:63_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c2">b1</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c4">b2</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="field_9.html#K4" title="FIELD_9:func.4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">c1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">c1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">b2</font></span>)</span></span><a href="field_9.html#K4" title="FIELD_9:func.4">%&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:63_2"><span class="lab"><font color="Green" title="E29">B1</font></span></a>, <a class="txt" href="#E1:63_2_1_1"><span class="lab"><font color="Green" title="E30">C</font></span></a>, <a class="txt" href="field_9.html#E22"><span class="lab"><font color="Green" title="E15">qua1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:63_2_1_2"/><span class="lab"><font color="Green" title="E30">C</font></span>: 
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c2">b1</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c4">b2</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="field_9.html#K4" title="FIELD_9:func.4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">c1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">c1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">b2</font></span>)</span></span><a href="field_9.html#K4" title="FIELD_9:func.4">%&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:63_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">B3</font></span>: 
<font color="Maroon" title="c11">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:63_2"><span class="lab"><font color="Green" title="E29">B1</font></span></a>, <a class="ref" href="finseq_1.html#T44" onmouseover="rs('finseq_1/T44')" onmouseout="rh()">FINSEQ_1:44</a></span>;<br/><a NAME="E3:63_2_1_2"/><span class="lab"><font color="Green" title="E32">B4</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">1,2</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:63_2"><span class="lab"><font color="Green" title="E29">B1</font></span></a>, <a class="txt" href="#E1:63_2_1_2"><span class="lab"><font color="Green" title="E30">C</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a>, <a class="ref" href="finseq_1.html#T2" onmouseover="rs('finseq_1/T2')" onmouseout="rh()">FINSEQ_1:2</a></span>;<br/><a NAME="E4:63_2_1_2"/><span class="kw">then </span>
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">B5</font></span>: <font color="Maroon" title="c11">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c2">b1</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c4">b2</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63_2_1_2"><span class="lab"><font color="Green" title="E30">C</font></span></a>, <a class="txt" href="#E4:63_2"><span class="lab"><font color="Green" title="E29">B1</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c2">b1</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c4">b2</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63"><span class="lab"><font color="Green" title="E27">K</font></span></a>, <a class="ref" href="nat_2.html#T8" onmouseover="rs('nat_2/T8')" onmouseout="rh()">NAT_2:8</a></span>
<br/>.= 
<font color="Maroon" title="c3">c1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c4">b2</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T38" onmouseover="rs('polynom5/T38')" onmouseout="rh()">POLYNOM5:38</a></span>
<br/>.= 
<font color="Maroon" title="c3">c1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">b2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T38" onmouseover="rs('polynom5/T38')" onmouseout="rh()">POLYNOM5:38</a></span>
;<br/><a NAME="E6:63_2_1_2"/>
2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63_2_1_2"><span class="lab"><font color="Green" title="E32">B4</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c11">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 = 
<span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c2">b1</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default">2 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c4">b2</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 2</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63_2_1_2"><span class="lab"><font color="Green" title="E30">C</font></span></a>, <a class="txt" href="#E4:63_2"><span class="lab"><font color="Green" title="E29">B1</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c2">b1</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> 1</span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c4">b2</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63"><span class="lab"><font color="Green" title="E27">K</font></span></a>, <a class="ref" href="nat_2.html#T8" onmouseover="rs('nat_2/T8')" onmouseout="rh()">NAT_2:8</a></span>
<br/>.= 
<font color="Maroon" title="c2">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c4">b2</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T38" onmouseover="rs('polynom5/T38')" onmouseout="rh()">POLYNOM5:38</a></span>
<br/>.= 
<font color="Maroon" title="c2">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T38" onmouseover="rs('polynom5/T38')" onmouseout="rh()">POLYNOM5:38</a></span>
;<br/><span class="kw">then </span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c11">r</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">c1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">b2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63_2_1_2"><span class="lab"><font color="Green" title="E31">B3</font></span></a>, <a class="txt" href="#E5:63_2_1_2"><span class="lab"><font color="Green" title="E33">B5</font></span></a>, <a class="ref" href="rlvect_1.html#T45" onmouseover="rs('rlvect_1/T45')" onmouseout="rh()">RLVECT_1:45</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">c1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D12" onmouseover="rs('group_1/D12')" onmouseout="rh()">GROUP_1:def 12</a></span>
;<br/><span class="kw">hence </span><a NAME="E9:63_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c2">b1</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c4">b2</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="field_9.html#K4" title="FIELD_9:func.4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">c1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">c1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">b2</font></span>)</span></span><a href="field_9.html#K4" title="FIELD_9:func.4">%&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:63_2"><span class="lab"><font color="Green" title="E29">B1</font></span></a>, <a class="txt" href="#E1:63_2_1_2"><span class="lab"><font color="Green" title="E30">C</font></span></a>, <a class="txt" href="field_9.html#E22"><span class="lab"><font color="Green" title="E15">qua1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_1_3"><span class="kw">suppose </span></a><a NAME="E1:63_2_1_3"/><span class="lab"><font color="Green" title="E30">C</font></span>: 
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c2">b1</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c4">b2</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="field_9.html#K4" title="FIELD_9:func.4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">c1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">c1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">b2</font></span>)</span></span><a href="field_9.html#K4" title="FIELD_9:func.4">%&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:63_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">B3</font></span>: 
<font color="Maroon" title="c11">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3</span>)</span></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:63_2"><span class="lab"><font color="Green" title="E29">B1</font></span></a>, <a class="ref" href="finseq_1.html#T45" onmouseover="rs('finseq_1/T45')" onmouseout="rh()">FINSEQ_1:45</a></span>;<br/><span class="lab"><font color="Green" title="E32">B4</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">r</font> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 3
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:63_2"><span class="lab"><font color="Green" title="E29">B1</font></span></a>, <a class="txt" href="#E1:63_2_1_3"><span class="lab"><font color="Green" title="E30">C</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2</span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T9" onmouseover="rs('finseq_1/T9')" onmouseout="rh()">FINSEQ_1:9</a></span>
<br/>.= 
<span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default">1,2,3</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T2" onmouseover="rs('finseq_1/T2')" onmouseout="rh()">FINSEQ_1:2</a>, <a class="ref" href="enumset1.html#T3" onmouseover="rs('enumset1/T3')" onmouseout="rh()">ENUMSET1:3</a></span>
;<br/><a NAME="E4:63_2_1_3"/><span class="kw">then </span>
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">B5</font></span>: <font color="Maroon" title="c11">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c2">b1</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c4">b2</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63_2_1_3"><span class="lab"><font color="Green" title="E30">C</font></span></a>, <a class="txt" href="#E4:63_2"><span class="lab"><font color="Green" title="E29">B1</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c2">b1</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c4">b2</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> 2</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63"><span class="lab"><font color="Green" title="E27">K</font></span></a>, <a class="ref" href="nat_2.html#T8" onmouseover="rs('nat_2/T8')" onmouseout="rh()">NAT_2:8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c2">b1</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T38" onmouseover="rs('polynom5/T38')" onmouseout="rh()">POLYNOM5:38</a></span>
;<br/><a NAME="E6:63_2_1_3"/>
2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63_2_1_3"><span class="lab"><font color="Green" title="E32">B4</font></span></a>, <a class="ref" href="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">B6</font></span>: <font color="Maroon" title="c11">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 = 
<span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c2">b1</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> 1</span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c4">b2</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63"><span class="lab"><font color="Green" title="E27">K</font></span></a>, <a class="txt" href="#E1:63_2_1_3"><span class="lab"><font color="Green" title="E30">C</font></span></a>, <a class="txt" href="#E4:63_2"><span class="lab"><font color="Green" title="E29">B1</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c2">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c4">b2</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T38" onmouseover="rs('polynom5/T38')" onmouseout="rh()">POLYNOM5:38</a></span>
<br/>.= 
<font color="Maroon" title="c2">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">b2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T38" onmouseover="rs('polynom5/T38')" onmouseout="rh()">POLYNOM5:38</a></span>
;<br/><a NAME="E8:63_2_1_3"/>
3 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63_2_1_3"><span class="lab"><font color="Green" title="E32">B4</font></span></a>, <a class="ref" href="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c11">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 = 
<span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c2">b1</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> 2</span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c4">b2</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 3</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63"><span class="lab"><font color="Green" title="E27">K</font></span></a>, <a class="txt" href="#E1:63_2_1_3"><span class="lab"><font color="Green" title="E30">C</font></span></a>, <a class="txt" href="#E4:63_2"><span class="lab"><font color="Green" title="E29">B1</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c4">b2</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 3</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T38" onmouseover="rs('polynom5/T38')" onmouseout="rh()">POLYNOM5:38</a></span>
;<br/><a NAME="E10:63_2_1_3"/><span class="kw">then </span>
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c11">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">b2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63_2_1_3"><span class="lab"><font color="Green" title="E31">B3</font></span></a>, <a class="txt" href="#E5:63_2_1_3"><span class="lab"><font color="Green" title="E33">B5</font></span></a>, <a class="txt" href="#E7:63_2_1_3"><span class="lab"><font color="Green" title="E34">B6</font></span></a>, <a class="ref" href="rlvect_1.html#T46" onmouseover="rs('rlvect_1/T46')" onmouseout="rh()">RLVECT_1:46</a></span>;<br/><span class="kw">hence </span><a NAME="E11:63_2_1_3"/>
<span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c2">b1</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c4">b2</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="field_9.html#K4" title="FIELD_9:func.4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">c1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">c1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">b2</font></span>)</span></span><a href="field_9.html#K4" title="FIELD_9:func.4">%&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:63_2"><span class="lab"><font color="Green" title="E29">B1</font></span></a>, <a class="txt" href="#E1:63_2_1_3"><span class="lab"><font color="Green" title="E30">C</font></span></a>, <a class="txt" href="field_9.html#E22"><span class="lab"><font color="Green" title="E15">qua1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_1_4"><span class="kw">suppose </span></a><a NAME="E1:63_2_1_4"/><span class="lab"><font color="Green" title="E30">C</font></span>: 
<font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c2">b1</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c4">b2</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="field_9.html#K4" title="FIELD_9:func.4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">c1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">c1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">b2</font></span>)</span></span><a href="field_9.html#K4" title="FIELD_9:func.4">%&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:63_2_1_4"/><span class="lab"><font color="Green" title="E31">D</font></span>: 
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1"><a href="field_9.html#K4" title="FIELD_9:func.4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">c1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">c1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">b2</font></span>)</span></span><a href="field_9.html#K4" title="FIELD_9:func.4">%&gt;</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="field_9.html#E24"><span class="lab"><font color="Green" title="E16">qua2</font></span></a></span>;<br/><a NAME="E3:63_2_1_4"/><span class="lab"><font color="Green" title="E32">C1</font></span>: 
<font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63_2_1_4"><span class="lab"><font color="Green" title="E30">C</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E4:63_2_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">E</font></span>: 
<span class="p1"><a href="field_9.html#K4" title="FIELD_9:func.4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">c1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">c1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">b2</font></span>)</span></span><a href="field_9.html#K4" title="FIELD_9:func.4">%&gt;</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63_2_1_4"><span class="lab"><font color="Green" title="E31">D</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a>, <a class="ref" href="algseq_1.html#T8" onmouseover="rs('algseq_1/T8')" onmouseout="rh()">ALGSEQ_1:8</a></span>;<br/><a NAME="E5:63_2_1_4"/>
(  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c2">b1</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 &amp;  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c4">b2</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T39" onmouseover="rs('polynom5/T39')" onmouseout="rh()">POLYNOM5:39</a></span>;<br/><a NAME="E6:63_2_1_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c2">b1</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c4">b2</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/><a NAME="E7:63_2_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">F</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p3"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c2">b1</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p3"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c4">b2</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_9.html#T1" target="_self" onmouseover="rs('field_9/T1')" onmouseout="rh()">lemmonus</a></span>;<br/><a NAME="E8:63_2_1_4"/>
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c2">b1</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c4">b2</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p3"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c2">b1</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p3"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c4">b2</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="field_9.html#E12"><span class="lab"><font color="Green" title="E10">leng</font></span></a></span>;<br/><a NAME="E9:63_2_1_4"/><span class="kw">then </span>
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c2">b1</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c4">b2</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:63_2_1_4"><span class="lab"><font color="Green" title="E34">F</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E10:63_2_1_4"/>
<span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c2">b1</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c4">b2</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="field_9.html#K4" title="FIELD_9:func.4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">c1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">c1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">b2</font></span>)</span></span><a href="field_9.html#K4" title="FIELD_9:func.4">%&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63"><span class="lab"><font color="Green" title="E26">J</font></span></a>, <a class="txt" href="#E3:63_2_1_4"><span class="lab"><font color="Green" title="E32">C1</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a>, <a class="txt" href="#E4:63_2_1_4"><span class="lab"><font color="Green" title="E33">E</font></span></a>, <a class="ref" href="algseq_1.html#T8" onmouseover="rs('algseq_1/T8')" onmouseout="rh()">ALGSEQ_1:8</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:63"/>
<span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c2">b1</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c4">b2</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="field_9.html#K4" title="FIELD_9:func.4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">c1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">c2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">b2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">c1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">b2</font></span>)</span></span><a href="field_9.html#K4" title="FIELD_9:func.4">%&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63"><span class="lab"><font color="Green" title="E28">A</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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